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The Two-Sided Game of Googol and Sample-Based Prophet Inequalities

2019-12-23
José R. Correa , Andrés Cristi , Boris Epstein +1 more
Previous chapter Next chapter Full AccessProceedings Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms (SODA)The Two-Sided Game of Googol and Sample-Based Prophet InequalitiesJosé R. Correa, Andrés Cristi, Boris Epstein, … Previous chapter Next chapter Full AccessProceedings Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms (SODA)The Two-Sided Game of Googol and Sample-Based Prophet InequalitiesJosé R. Correa, Andrés Cristi, Boris Epstein, and José A. SotoJosé R. Correa, Andrés Cristi, Boris Epstein, and José A. Sotopp.2066 - 2081Chapter DOI:https://doi.org/10.1137/1.9781611975994.127PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract The secretary problem or the game of Googol are classic models for online selection problems that have received significant attention in the last five decades. In this paper we consider a variant of the problem and explore its connections to data-driven online selection. Specifically, we are given n cards with arbitrary nonnegative numbers written on both sides. The cards are randomly placed on n consecutive positions on a table, and for each card, the visible side is also selected at random. The player sees the visible side of all cards and wants to select the card with the maximum hidden value. To this end, the player flips the first card, sees its hidden value and decides whether to pick it or drop it and continue with the next card. We study algorithms for two natural objectives. In the first one, similar to the secretary problem, the player wants to maximize the probability of selecting the maximum hidden value. We show that this can be done with probability at least 0.45292. In the second objective, similar to the prophet inequality, the player wants to maximize the expectation of the selected hidden value. Here we show a guarantee of at least 0.63518 with respect to the expected maximum hidden value. Our algorithms result from combining three basic strategies. One is to stop whenever we see a value larger than the initial n visible numbers. The second one is to stop the first time the last flipped card's value is the largest of the currently n visible numbers in the table. And the third one is similar to the latter but to stop it additionally requires that the last flipped value is larger than the value on the other side of its card. We apply our results to the prophet secretary problem with unknown distributions, but with access to a single sample from each distribution. In particular, our guarantee improves upon 1 – 1/e for this problem, which is the currently best known guarantee and only works for the i.i.d. prophet inequality with samples. Previous chapter Next chapter RelatedDetails Published:2020eISBN:978-1-61197-599-4 https://doi.org/10.1137/1.9781611975994Book Series Name:ProceedingsBook Code:PRDA20Book Pages:xxii + 3011
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Strong Algorithms for the Ordinal Matroid Secretary Problem

2018-01-01
José A. Soto , Abner Turkieltaub , Víctor Verdugo
In the ordinal Matroid Secretary Problem (MSP), elements from a weighted matroid are presented in random order to an algorithm that must incrementally select a large weight independent set. However, … In the ordinal Matroid Secretary Problem (MSP), elements from a weighted matroid are presented in random order to an algorithm that must incrementally select a large weight independent set. However, the algorithm can only compare pairs of revealed elements without using its numerical value. An algorithm is $\alpha$ probability-competitive if every element from the optimum appears with probability $1/\alpha$ in the output. We present a technique to design algorithms with strong probability-competitive ratios, improving the guarantees for almost every matroid class considered in the literature: e.g., we get ratios of 4 for graphic matroids (improving on $2e$ by Korula and P\'al [ICALP 2009]) and of 5.19 for laminar matroids (improving on 9.6 by Ma et al. [THEOR COMPUT SYST 2016]). We also obtain new results for superclasses of $k$ column sparse matroids, for hypergraphic matroids, certain gammoids and graph packing matroids, and a $1+O(\sqrt{\log \rho/\rho})$ probability-competitive algorithm for uniform matroids of rank $\rho$ based on Kleinberg's $1+O(\sqrt{1/\rho})$ utility-competitive algorithm [SODA 2005] for that class. Our second contribution are algorithms for the ordinal MSP on arbitrary matroids of rank $\rho$. We devise an $O(\log \rho)$ probability-competitive algorithm and an $O(\log\log \rho)$ ordinal-competitive algorithm, a weaker notion of competitiveness but stronger than the utility variant. These are based on the $O(\log\log \rho)$ utility-competitive algorithm by Feldman et al.~[SODA 2015].

Online Combinatorial Assignment in Independence Systems

2023-01-01
Javier Marinković , José A. Soto , Víctor Verdugo
We consider an online multi-weighted generalization of several classic online optimization problems, called the online combinatorial assignment problem. We are given an independence system over a ground set of elements … We consider an online multi-weighted generalization of several classic online optimization problems, called the online combinatorial assignment problem. We are given an independence system over a ground set of elements and agents that arrive online one by one. Upon arrival, each agent reveals a weight function over the elements of the ground set. If the independence system is given by the matchings of a hypergraph we recover the combinatorial auction problem, where every node represents an item to be sold, and every edge represents a bundle of items. For combinatorial auctions, Kesselheim et al. showed upper bounds of O(loglog(k)/log(k)) and $O(\log \log(n)/\log(n))$ on the competitiveness of any online algorithm, even in the random order model, where $k$ is the maximum bundle size and $n$ is the number of items. We provide an exponential improvement on these upper bounds to show that the competitiveness of any online algorithm in the prophet IID setting is upper bounded by $O(\log(k)/k)$, and $O(\log(n)/\sqrt{n})$. Furthermore, using linear programming, we provide new and improved guarantees for the $k$-bounded online combinatorial auction problem (i.e., bundles of size at most $k$). We show a $(1-e^{-k})/k$-competitive algorithm in the prophet IID model, a $1/(k+1)$-competitive algorithm in the prophet-secretary model using a single sample per agent, and a $k^{-k/(k-1)}$-competitive algorithm in the secretary model. Our algorithms run in polynomial time and work in more general independence systems where the offline combinatorial assignment problem admits the existence of a polynomial-time randomized algorithm that we call certificate sampler. We show that certificate samplers have a nice interplay with random order models, and we also provide new polynomial-time competitive algorithms for some classes of matroids, matroid intersections, and matchoids.

Approximation Algorithms for Vertex-Connectivity Augmentation on the Cycle

2021-01-01
Waldo Gálvez , Francisco Sanhueza-Matamala , José A. Soto
Given a $k$-vertex-connected graph $G$ and a set $S$ of extra edges (links), the goal of the $k$-vertex-connectivity augmentation problem is to find a set $S' \subseteq S$ of minimum … Given a $k$-vertex-connected graph $G$ and a set $S$ of extra edges (links), the goal of the $k$-vertex-connectivity augmentation problem is to find a set $S' \subseteq S$ of minimum size such that adding $S'$ to $G$ makes it $(k+1)$-vertex-connected. Unlike the edge-connectivity augmentation problem, research for the vertex-connectivity version has been sparse. In this work we present the first polynomial time approximation algorithm that improves the known ratio of 2 for $2$-vertex-connectivity augmentation, for the case in which $G$ is a cycle. This is the first step for attacking the more general problem of augmenting a $2$-connected graph. Our algorithm is based on local search and attains an approximation ratio of $1.8704$. To derive it, we prove novel results on the structure of minimal solutions.
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Approximation Algorithms for Vertex-Connectivity Augmentation on the Cycle

2021-01-01
Waldo Gálvez , Francisco Sanhueza-Matamala , José A. Soto
View PDF Chat

The Two-Sided Game of Googol and Sample-Based Prophet Inequalities

2019-01-01
José R. Correa , Andrés Cristi , Boris Epstein +1 more
The secretary problem or the game of Googol are classic models for online selection problems that have received significant attention in the last five decades. We consider a variant of … The secretary problem or the game of Googol are classic models for online selection problems that have received significant attention in the last five decades. We consider a variant of the problem and explore its connections to data-driven online selection. Specifically, we are given $n$ cards with arbitrary non-negative numbers written on both sides. The cards are randomly placed on $n$ consecutive positions on a table, and for each card, the visible side is also selected at random. The player sees the visible side of all cards and wants to select the card with the maximum hidden value. To this end, the player flips the first card, sees its hidden value and decides whether to pick it or drop it and continue with the next card. We study algorithms for two natural objectives. In the first one, as in the secretary problem, the player wants to maximize the probability of selecting the maximum hidden value. We show that this can be done with probability at least $0.45292$. In the second one, similar to the prophet inequality, the player maximizes the expectation of the selected hidden value. We show a guarantee of at least $0.63518$ with respect to the expected maximum hidden value. Our algorithms result from combining three basic strategies. One is to stop whenever we see a value larger than the initial $n$ visible numbers. The second one is to stop the first time the last flipped card's value is the largest of the currently $n$ visible numbers in the table. And the third one is similar to the latter but it additionally requires that the last flipped value is larger than the value on the other side of its card. We apply our results to the prophet secretary problem with unknown distributions, but with access to a single sample from each distribution. Our guarantee improves upon $1-1/e$ for this problem, which is the currently best known guarantee and only works for the i.i.d. case.

Advances on Matroid Secretary Problems: Free Order Model and Laminar Case

2012-01-01
Patrick Jaillet , José A. Soto , Rico Zenklusen
The most well-known conjecture in the context of matroid secretary problems claims the existence of a constant-factor approximation applicable to any matroid. Whereas this conjecture remains open, modified forms of … The most well-known conjecture in the context of matroid secretary problems claims the existence of a constant-factor approximation applicable to any matroid. Whereas this conjecture remains open, modified forms of it were shown to be true, when assuming that the assignment of weights to the secretaries is not adversarial but uniformly random (Soto [SODA 2011], Oveis Gharan and Vondr\'ak [ESA 2011]). However, so far, there was no variant of the matroid secretary problem with adversarial weight assignment for which a constant-factor approximation was found. We address this point by presenting a 9-approximation for the \emph{free order model}, a model suggested shortly after the introduction of the matroid secretary problem, and for which no constant-factor approximation was known so far. The free order model is a relaxed version of the original matroid secretary problem, with the only difference that one can choose the order in which secretaries are interviewed. Furthermore, we consider the classical matroid secretary problem for the special case of laminar matroids. Only recently, a constant-factor approximation has been found for this case, using a clever but rather involved method and analysis (Im and Wang, [SODA 2011]) that leads to a 16000/3-approximation. This is arguably the most involved special case of the matroid secretary problem for which a constant-factor approximation is known. We present a considerably simpler and stronger $3\sqrt{3}e\approx 14.12$-approximation, based on reducing the problem to a matroid secretary problem on a partition matroid.
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Independent and Hitting Sets of Rectangles Intersecting a Diagonal Line : Algorithms and Complexity

2013-01-01
José R. Correa , Laurent Feuilloley , Pablo Pérez-Lantero +1 more
Finding a maximum independent set (MIS) of a given fam- ily of axis-parallel rectangles is a basic problem in computational geom- etry and combinatorics. This problem has attracted significant atten- … Finding a maximum independent set (MIS) of a given fam- ily of axis-parallel rectangles is a basic problem in computational geom- etry and combinatorics. This problem has attracted significant atten- tion since the sixties, when Wegner conjectured that the corresponding duality gap, i.e., the maximum possible ratio between the maximum independent set and the minimum hitting set (MHS), is bounded by a universal constant. An interesting special case, that may prove use- ful to tackling the general problem, is the diagonal-intersecting case, in which the given family of rectangles is intersected by a diagonal. Indeed, Chepoi and Felsner recently gave a factor 6 approximation algorithm for MHS in this setting, and showed that the duality gap is between 3/2 and 6. In this paper we improve upon these results. First we show that MIS in diagonal-intersecting families is NP-complete, providing one smallest subclass for which MIS is provably hard. Then, we derive an $O(n^2)$-time algorithm for the maximum weight independent set when, in addition the rectangles intersect below the diagonal. This improves and extends a classic result of Lubiw, and amounts to obtain a 2-approximation algo- rithm for the maximum weight independent set of rectangles intersecting a diagonal. Finally, we prove that for diagonal-intersecting families the duality gap is between 2 and 4. The upper bound, which implies an approximation algorithm of the same factor, follows from a simple com- binatorial argument, while the lower bound represents the best known lower bound on the duality gap, even in the general case.
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TSP Tours in Cubic Graphs: Beyond 4/3

2013-01-01
José R. Correa , Omar Larré , José A. Soto
After a sequence of improvements Boyd, Sitters, van der Ster, and Stougie proved that any 2-connected graph whose n vertices have degree 3, i.e., a cubic 2-connected graph, has a … After a sequence of improvements Boyd, Sitters, van der Ster, and Stougie proved that any 2-connected graph whose n vertices have degree 3, i.e., a cubic 2-connected graph, has a Hamiltonian tour of length at most (4/3)n, establishing in particular that the integrality gap of the subtour LP is at most 4/3 for cubic 2-connected graphs and matching the conjectured value of the famous 4/3 conjecture. In this paper we improve upon this result by designing an algorithm that finds a tour of length (4/3 - 1/61236)n, implying that cubic 2-connected graphs are among the few interesting classes of graphs for which the integrality gap of the subtour LP is strictly less than 4/3. With the previous result, and by considering an even smaller epsilon, we show that the integrality gap of the TSP relaxation is at most 4/3 - epsilon, even if the graph is not 2-connected (i.e. for cubic connected graphs), implying that the approximability threshold of the TSP in cubic graphs is strictly below 4/3. Finally, using similar techniques we show, as an additional result, that every Barnette graph admits a tour of length at most (4/3 - 1/18)n.
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A simple PTAS for Weighted Matroid Matching on Strongly Base Orderable Matroids

2011-01-01
José A. Soto
We give a simple polynomial time approximation scheme for the weighted matroid matching problem on strongly base orderable matroids. We also show that even the unweighted version of this problem … We give a simple polynomial time approximation scheme for the weighted matroid matching problem on strongly base orderable matroids. We also show that even the unweighted version of this problem is NP-complete and not in oracle-coNP.
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Symmetry exploitation for Online Machine Covering with Bounded Migration

2016-01-01
Waldo Gálvez , José A. Soto , José Verschae
Online models that allow recourse are highly effective in situations where classical models are too pessimistic. One such problem is the online machine covering problem on identical machines. In this … Online models that allow recourse are highly effective in situations where classical models are too pessimistic. One such problem is the online machine covering problem on identical machines. In this setting, jobs arrive one by one and must be assigned to machines with the objective of maximizing the minimum machine load. When a job arrives, we are allowed to reassign some jobs as long as their total size is (at most) proportional to the processing time of the arriving job. The proportionality constant is called the migration factor of the algorithm. Using a rounding procedure with useful structural properties for online packing and covering problems, we design first a simple $(1.7 + \varepsilon)$-competitive algorithm using a migration factor of $O(1/\varepsilon)$ which maintains at every arrival a locally optimal solution with respect to the Jump neighborhood. After that, we present as our main contribution a more involved $(4/3+\varepsilon)$-competitive algorithm using a migration factor of $\tilde{O}(1/\varepsilon^3)$. At every arrival, we run an adaptation of the Largest Processing Time first (LPT) algorithm. Since the new job can cause a complete change of the assignment of smaller jobs in both cases, a low migration factor is achieved by carefully exploiting the highly symmetric structure obtained by the rounding procedure.
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Matroid Secretary Problem in the Random Assignment Model

2010-01-01
José A. Soto
In the Matroid Secretary Problem, introduced by Babaioff et al. [SODA 2007], the elements of a given matroid are presented to an online algorithm in random order. When an element … In the Matroid Secretary Problem, introduced by Babaioff et al. [SODA 2007], the elements of a given matroid are presented to an online algorithm in random order. When an element is revealed, the algorithm learns its weight and decides whether or not to select it under the restriction that the selected elements form an independent set in the matroid. The objective is to maximize the total weight of the chosen elements. In the most studied version of this problem, the algorithm has no information about the weights beforehand. We refer to this as the zero information model. In this paper we study a different model, also proposed by Babaioff et al., in which the relative order of the weights is random in the matroid. To be precise, in the random assignment model, an adversary selects a collection of weights that are randomly assigned to the elements of the matroid. Later, the elements are revealed to the algorithm in a random order independent of the assignment. Our main result is the first constant competitive algorithm for the matroid secretary problem in the random assignment model. This solves an open question of Babaioff et al. Our algorithm achieves a competitive ratio of $2e^2/(e-1)$. It exploits the notion of principal partition of a matroid, its decomposition into uniformly dense minors, and a $2e$-competitive algorithm for uniformly dense matroids we also develop. As additional results, we present simple constant competitive algorithms in the zero information model for various classes of matroids including cographic, low density and the case when every element is in a small cocircuit. In the same model, we also give a $ke$-competitive algorithm for $k$-column sparse linear matroids, and a new $O(\log r)$-competitive algorithm for general matroids of rank $r$ which only uses the relative order of the weights seen and not their numerical value, as previously needed.
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Robust randomized matchings

2017-01-01
Jannik Matuschke , Martin Skutella , José A. Soto
The following game is played on a weighted graph: Alice selects a matching $M$ and Bob selects a number $k$. Alice's payoff is the ratio of the weight of the … The following game is played on a weighted graph: Alice selects a matching $M$ and Bob selects a number $k$. Alice's payoff is the ratio of the weight of the $k$ heaviest edges of $M$ to the maximum weight of a matching of size at most $k$. If $M$ guarantees a payoff of at least $\alpha$ then it is called $\alpha$-robust. In 2002, Hassin and Rubinstein gave an algorithm that returns a $1/\sqrt{2}$-robust matching, which is best possible. We show that Alice can improve her payoff to $1/\ln(4)$ by playing a randomized strategy. This result extends to a very general class of independence systems that includes matroid intersection, b-matchings, and strong 2-exchange systems. It also implies an improved approximation factor for a stochastic optimization variant known as the maximum priority matching problem and translates to an asymptotic robustness guarantee for deterministic matchings, in which Bob can only select numbers larger than a given constant. Moreover, we give a new LP-based proof of Hassin and Rubinstein's bound.

Online Combinatorial Assignment in Independence Systems

2023-01-01
Javier Marinković , José A. Soto , Víctor Verdugo
We consider an online multi-weighted generalization of several classic online optimization problems, called the online combinatorial assignment problem. We are given an independence system over a ground set of elements … We consider an online multi-weighted generalization of several classic online optimization problems, called the online combinatorial assignment problem. We are given an independence system over a ground set of elements and agents that arrive online one by one. Upon arrival, each agent reveals a weight function over the elements of the ground set. If the independence system is given by the matchings of a hypergraph we recover the combinatorial auction problem, where every node represents an item to be sold, and every edge represents a bundle of items. For combinatorial auctions, Kesselheim et al. showed upper bounds of O(loglog(k)/log(k)) and $O(\log \log(n)/\log(n))$ on the competitiveness of any online algorithm, even in the random order model, where $k$ is the maximum bundle size and $n$ is the number of items. We provide an exponential improvement on these upper bounds to show that the competitiveness of any online algorithm in the prophet IID setting is upper bounded by $O(\log(k)/k)$, and $O(\log(n)/\sqrt{n})$. Furthermore, using linear programming, we provide new and improved guarantees for the $k$-bounded online combinatorial auction problem (i.e., bundles of size at most $k$). We show a $(1-e^{-k})/k$-competitive algorithm in the prophet IID model, a $1/(k+1)$-competitive algorithm in the prophet-secretary model using a single sample per agent, and a $k^{-k/(k-1)}$-competitive algorithm in the secretary model. Our algorithms run in polynomial time and work in more general independence systems where the offline combinatorial assignment problem admits the existence of a polynomial-time randomized algorithm that we call certificate sampler. We show that certificate samplers have a nice interplay with random order models, and we also provide new polynomial-time competitive algorithms for some classes of matroids, matroid intersections, and matchoids.

Approximation Algorithms for Vertex-Connectivity Augmentation on the Cycle

2021-01-01
Waldo Gálvez , Francisco Sanhueza-Matamala , José A. Soto
Given a $k$-vertex-connected graph $G$ and a set $S$ of extra edges (links), the goal of the $k$-vertex-connectivity augmentation problem is to find a set $S' \subseteq S$ of minimum … Given a $k$-vertex-connected graph $G$ and a set $S$ of extra edges (links), the goal of the $k$-vertex-connectivity augmentation problem is to find a set $S' \subseteq S$ of minimum size such that adding $S'$ to $G$ makes it $(k+1)$-vertex-connected. Unlike the edge-connectivity augmentation problem, research for the vertex-connectivity version has been sparse. In this work we present the first polynomial time approximation algorithm that improves the known ratio of 2 for $2$-vertex-connectivity augmentation, for the case in which $G$ is a cycle. This is the first step for attacking the more general problem of augmenting a $2$-connected graph. Our algorithm is based on local search and attains an approximation ratio of $1.8704$. To derive it, we prove novel results on the structure of minimal solutions.
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Approximation Algorithms for Vertex-Connectivity Augmentation on the Cycle

2021-01-01
Waldo Gálvez , Francisco Sanhueza-Matamala , José A. Soto
View PDF Chat

The Two-Sided Game of Googol and Sample-Based Prophet Inequalities

2019-12-23
José R. Correa , Andrés Cristi , Boris Epstein +1 more
Previous chapter Next chapter Full AccessProceedings Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms (SODA)The Two-Sided Game of Googol and Sample-Based Prophet InequalitiesJosé R. Correa, Andrés Cristi, Boris Epstein, … Previous chapter Next chapter Full AccessProceedings Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms (SODA)The Two-Sided Game of Googol and Sample-Based Prophet InequalitiesJosé R. Correa, Andrés Cristi, Boris Epstein, and José A. SotoJosé R. Correa, Andrés Cristi, Boris Epstein, and José A. Sotopp.2066 - 2081Chapter DOI:https://doi.org/10.1137/1.9781611975994.127PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract The secretary problem or the game of Googol are classic models for online selection problems that have received significant attention in the last five decades. In this paper we consider a variant of the problem and explore its connections to data-driven online selection. Specifically, we are given n cards with arbitrary nonnegative numbers written on both sides. The cards are randomly placed on n consecutive positions on a table, and for each card, the visible side is also selected at random. The player sees the visible side of all cards and wants to select the card with the maximum hidden value. To this end, the player flips the first card, sees its hidden value and decides whether to pick it or drop it and continue with the next card. We study algorithms for two natural objectives. In the first one, similar to the secretary problem, the player wants to maximize the probability of selecting the maximum hidden value. We show that this can be done with probability at least 0.45292. In the second objective, similar to the prophet inequality, the player wants to maximize the expectation of the selected hidden value. Here we show a guarantee of at least 0.63518 with respect to the expected maximum hidden value. Our algorithms result from combining three basic strategies. One is to stop whenever we see a value larger than the initial n visible numbers. The second one is to stop the first time the last flipped card's value is the largest of the currently n visible numbers in the table. And the third one is similar to the latter but to stop it additionally requires that the last flipped value is larger than the value on the other side of its card. We apply our results to the prophet secretary problem with unknown distributions, but with access to a single sample from each distribution. In particular, our guarantee improves upon 1 – 1/e for this problem, which is the currently best known guarantee and only works for the i.i.d. prophet inequality with samples. Previous chapter Next chapter RelatedDetails Published:2020eISBN:978-1-61197-599-4 https://doi.org/10.1137/1.9781611975994Book Series Name:ProceedingsBook Code:PRDA20Book Pages:xxii + 3011
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The Two-Sided Game of Googol and Sample-Based Prophet Inequalities

2019-01-01
José R. Correa , Andrés Cristi , Boris Epstein +1 more
The secretary problem or the game of Googol are classic models for online selection problems that have received significant attention in the last five decades. We consider a variant of … The secretary problem or the game of Googol are classic models for online selection problems that have received significant attention in the last five decades. We consider a variant of the problem and explore its connections to data-driven online selection. Specifically, we are given $n$ cards with arbitrary non-negative numbers written on both sides. The cards are randomly placed on $n$ consecutive positions on a table, and for each card, the visible side is also selected at random. The player sees the visible side of all cards and wants to select the card with the maximum hidden value. To this end, the player flips the first card, sees its hidden value and decides whether to pick it or drop it and continue with the next card. We study algorithms for two natural objectives. In the first one, as in the secretary problem, the player wants to maximize the probability of selecting the maximum hidden value. We show that this can be done with probability at least $0.45292$. In the second one, similar to the prophet inequality, the player maximizes the expectation of the selected hidden value. We show a guarantee of at least $0.63518$ with respect to the expected maximum hidden value. Our algorithms result from combining three basic strategies. One is to stop whenever we see a value larger than the initial $n$ visible numbers. The second one is to stop the first time the last flipped card's value is the largest of the currently $n$ visible numbers in the table. And the third one is similar to the latter but it additionally requires that the last flipped value is larger than the value on the other side of its card. We apply our results to the prophet secretary problem with unknown distributions, but with access to a single sample from each distribution. Our guarantee improves upon $1-1/e$ for this problem, which is the currently best known guarantee and only works for the i.i.d. case.

Strong Algorithms for the Ordinal Matroid Secretary Problem

2018-01-01
José A. Soto , Abner Turkieltaub , Víctor Verdugo
In the ordinal Matroid Secretary Problem (MSP), elements from a weighted matroid are presented in random order to an algorithm that must incrementally select a large weight independent set. However, … In the ordinal Matroid Secretary Problem (MSP), elements from a weighted matroid are presented in random order to an algorithm that must incrementally select a large weight independent set. However, the algorithm can only compare pairs of revealed elements without using its numerical value. An algorithm is $\alpha$ probability-competitive if every element from the optimum appears with probability $1/\alpha$ in the output. We present a technique to design algorithms with strong probability-competitive ratios, improving the guarantees for almost every matroid class considered in the literature: e.g., we get ratios of 4 for graphic matroids (improving on $2e$ by Korula and P\'al [ICALP 2009]) and of 5.19 for laminar matroids (improving on 9.6 by Ma et al. [THEOR COMPUT SYST 2016]). We also obtain new results for superclasses of $k$ column sparse matroids, for hypergraphic matroids, certain gammoids and graph packing matroids, and a $1+O(\sqrt{\log \rho/\rho})$ probability-competitive algorithm for uniform matroids of rank $\rho$ based on Kleinberg's $1+O(\sqrt{1/\rho})$ utility-competitive algorithm [SODA 2005] for that class. Our second contribution are algorithms for the ordinal MSP on arbitrary matroids of rank $\rho$. We devise an $O(\log \rho)$ probability-competitive algorithm and an $O(\log\log \rho)$ ordinal-competitive algorithm, a weaker notion of competitiveness but stronger than the utility variant. These are based on the $O(\log\log \rho)$ utility-competitive algorithm by Feldman et al.~[SODA 2015].

Robust randomized matchings

2017-01-01
Jannik Matuschke , Martin Skutella , José A. Soto
The following game is played on a weighted graph: Alice selects a matching $M$ and Bob selects a number $k$. Alice's payoff is the ratio of the weight of the … The following game is played on a weighted graph: Alice selects a matching $M$ and Bob selects a number $k$. Alice's payoff is the ratio of the weight of the $k$ heaviest edges of $M$ to the maximum weight of a matching of size at most $k$. If $M$ guarantees a payoff of at least $\alpha$ then it is called $\alpha$-robust. In 2002, Hassin and Rubinstein gave an algorithm that returns a $1/\sqrt{2}$-robust matching, which is best possible. We show that Alice can improve her payoff to $1/\ln(4)$ by playing a randomized strategy. This result extends to a very general class of independence systems that includes matroid intersection, b-matchings, and strong 2-exchange systems. It also implies an improved approximation factor for a stochastic optimization variant known as the maximum priority matching problem and translates to an asymptotic robustness guarantee for deterministic matchings, in which Bob can only select numbers larger than a given constant. Moreover, we give a new LP-based proof of Hassin and Rubinstein's bound.

Symmetry exploitation for Online Machine Covering with Bounded Migration

2016-01-01
Waldo Gálvez , José A. Soto , José Verschae
Online models that allow recourse are highly effective in situations where classical models are too pessimistic. One such problem is the online machine covering problem on identical machines. In this … Online models that allow recourse are highly effective in situations where classical models are too pessimistic. One such problem is the online machine covering problem on identical machines. In this setting, jobs arrive one by one and must be assigned to machines with the objective of maximizing the minimum machine load. When a job arrives, we are allowed to reassign some jobs as long as their total size is (at most) proportional to the processing time of the arriving job. The proportionality constant is called the migration factor of the algorithm. Using a rounding procedure with useful structural properties for online packing and covering problems, we design first a simple $(1.7 + \varepsilon)$-competitive algorithm using a migration factor of $O(1/\varepsilon)$ which maintains at every arrival a locally optimal solution with respect to the Jump neighborhood. After that, we present as our main contribution a more involved $(4/3+\varepsilon)$-competitive algorithm using a migration factor of $\tilde{O}(1/\varepsilon^3)$. At every arrival, we run an adaptation of the Largest Processing Time first (LPT) algorithm. Since the new job can cause a complete change of the assignment of smaller jobs in both cases, a low migration factor is achieved by carefully exploiting the highly symmetric structure obtained by the rounding procedure.
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Independent and Hitting Sets of Rectangles Intersecting a Diagonal Line : Algorithms and Complexity

2013-01-01
José R. Correa , Laurent Feuilloley , Pablo Pérez-Lantero +1 more
Finding a maximum independent set (MIS) of a given fam- ily of axis-parallel rectangles is a basic problem in computational geom- etry and combinatorics. This problem has attracted significant atten- … Finding a maximum independent set (MIS) of a given fam- ily of axis-parallel rectangles is a basic problem in computational geom- etry and combinatorics. This problem has attracted significant atten- tion since the sixties, when Wegner conjectured that the corresponding duality gap, i.e., the maximum possible ratio between the maximum independent set and the minimum hitting set (MHS), is bounded by a universal constant. An interesting special case, that may prove use- ful to tackling the general problem, is the diagonal-intersecting case, in which the given family of rectangles is intersected by a diagonal. Indeed, Chepoi and Felsner recently gave a factor 6 approximation algorithm for MHS in this setting, and showed that the duality gap is between 3/2 and 6. In this paper we improve upon these results. First we show that MIS in diagonal-intersecting families is NP-complete, providing one smallest subclass for which MIS is provably hard. Then, we derive an $O(n^2)$-time algorithm for the maximum weight independent set when, in addition the rectangles intersect below the diagonal. This improves and extends a classic result of Lubiw, and amounts to obtain a 2-approximation algo- rithm for the maximum weight independent set of rectangles intersecting a diagonal. Finally, we prove that for diagonal-intersecting families the duality gap is between 2 and 4. The upper bound, which implies an approximation algorithm of the same factor, follows from a simple com- binatorial argument, while the lower bound represents the best known lower bound on the duality gap, even in the general case.
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TSP Tours in Cubic Graphs: Beyond 4/3

2013-01-01
José R. Correa , Omar Larré , José A. Soto
After a sequence of improvements Boyd, Sitters, van der Ster, and Stougie proved that any 2-connected graph whose n vertices have degree 3, i.e., a cubic 2-connected graph, has a … After a sequence of improvements Boyd, Sitters, van der Ster, and Stougie proved that any 2-connected graph whose n vertices have degree 3, i.e., a cubic 2-connected graph, has a Hamiltonian tour of length at most (4/3)n, establishing in particular that the integrality gap of the subtour LP is at most 4/3 for cubic 2-connected graphs and matching the conjectured value of the famous 4/3 conjecture. In this paper we improve upon this result by designing an algorithm that finds a tour of length (4/3 - 1/61236)n, implying that cubic 2-connected graphs are among the few interesting classes of graphs for which the integrality gap of the subtour LP is strictly less than 4/3. With the previous result, and by considering an even smaller epsilon, we show that the integrality gap of the TSP relaxation is at most 4/3 - epsilon, even if the graph is not 2-connected (i.e. for cubic connected graphs), implying that the approximability threshold of the TSP in cubic graphs is strictly below 4/3. Finally, using similar techniques we show, as an additional result, that every Barnette graph admits a tour of length at most (4/3 - 1/18)n.
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Advances on Matroid Secretary Problems: Free Order Model and Laminar Case

2012-01-01
Patrick Jaillet , José A. Soto , Rico Zenklusen
The most well-known conjecture in the context of matroid secretary problems claims the existence of a constant-factor approximation applicable to any matroid. Whereas this conjecture remains open, modified forms of … The most well-known conjecture in the context of matroid secretary problems claims the existence of a constant-factor approximation applicable to any matroid. Whereas this conjecture remains open, modified forms of it were shown to be true, when assuming that the assignment of weights to the secretaries is not adversarial but uniformly random (Soto [SODA 2011], Oveis Gharan and Vondr\'ak [ESA 2011]). However, so far, there was no variant of the matroid secretary problem with adversarial weight assignment for which a constant-factor approximation was found. We address this point by presenting a 9-approximation for the \emph{free order model}, a model suggested shortly after the introduction of the matroid secretary problem, and for which no constant-factor approximation was known so far. The free order model is a relaxed version of the original matroid secretary problem, with the only difference that one can choose the order in which secretaries are interviewed. Furthermore, we consider the classical matroid secretary problem for the special case of laminar matroids. Only recently, a constant-factor approximation has been found for this case, using a clever but rather involved method and analysis (Im and Wang, [SODA 2011]) that leads to a 16000/3-approximation. This is arguably the most involved special case of the matroid secretary problem for which a constant-factor approximation is known. We present a considerably simpler and stronger $3\sqrt{3}e\approx 14.12$-approximation, based on reducing the problem to a matroid secretary problem on a partition matroid.
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A simple PTAS for Weighted Matroid Matching on Strongly Base Orderable Matroids

2011-01-01
José A. Soto
We give a simple polynomial time approximation scheme for the weighted matroid matching problem on strongly base orderable matroids. We also show that even the unweighted version of this problem … We give a simple polynomial time approximation scheme for the weighted matroid matching problem on strongly base orderable matroids. We also show that even the unweighted version of this problem is NP-complete and not in oracle-coNP.
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Matroid Secretary Problem in the Random Assignment Model

2010-01-01
José A. Soto
In the Matroid Secretary Problem, introduced by Babaioff et al. [SODA 2007], the elements of a given matroid are presented to an online algorithm in random order. When an element … In the Matroid Secretary Problem, introduced by Babaioff et al. [SODA 2007], the elements of a given matroid are presented to an online algorithm in random order. When an element is revealed, the algorithm learns its weight and decides whether or not to select it under the restriction that the selected elements form an independent set in the matroid. The objective is to maximize the total weight of the chosen elements. In the most studied version of this problem, the algorithm has no information about the weights beforehand. We refer to this as the zero information model. In this paper we study a different model, also proposed by Babaioff et al., in which the relative order of the weights is random in the matroid. To be precise, in the random assignment model, an adversary selects a collection of weights that are randomly assigned to the elements of the matroid. Later, the elements are revealed to the algorithm in a random order independent of the assignment. Our main result is the first constant competitive algorithm for the matroid secretary problem in the random assignment model. This solves an open question of Babaioff et al. Our algorithm achieves a competitive ratio of $2e^2/(e-1)$. It exploits the notion of principal partition of a matroid, its decomposition into uniformly dense minors, and a $2e$-competitive algorithm for uniformly dense matroids we also develop. As additional results, we present simple constant competitive algorithms in the zero information model for various classes of matroids including cographic, low density and the case when every element is in a small cocircuit. In the same model, we also give a $ke$-competitive algorithm for $k$-column sparse linear matroids, and a new $O(\log r)$-competitive algorithm for general matroids of rank $r$ which only uses the relative order of the weights seen and not their numerical value, as previously needed.
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Coauthor Papers Together
José R. Correa 4
Waldo Gálvez 3
Víctor Verdugo 2
Andrés Cristi 2
Boris Epstein 2
Francisco Sanhueza-Matamala 2
Martin Skutella 1
José Verschae 1
Laurent Feuilloley 1
Pablo Pérez-Lantero 1
Patrick Jaillet 1
Javier Marinković 1
Omar Larré 1
Rico Zenklusen 1
Abner Turkieltaub 1
Jannik Matuschke 1

Comparisons of Stop Rule and Supremum Expectations of I.I.D. Random Variables

1982-05-01
Theodore P. Hill , Robert P. Kertz
Implicitly defined (and easily approximated) universal constants $1.1 < a_n < 1.6, n = 2,3, \cdots$, are found so that if $X_1, X_2, \cdots$ are i.i.d. non-negative random variables and … Implicitly defined (and easily approximated) universal constants $1.1 < a_n < 1.6, n = 2,3, \cdots$, are found so that if $X_1, X_2, \cdots$ are i.i.d. non-negative random variables and if $T_n$ is the set of stop rules for $X_1, \cdots, X_n$, then $E(\max\{X_1, \cdots, X_n\}) \leq a_n \sup\{EX_t: t \in T_n\}$, and the bound $a_n$ is best possible. Similar universal constants $0 < b_n < \frac{1}{4}$ are found so that if the $\{X_i\}$ are i.i.d. random variables taking values only in $\lbrack a, b\rbrack$, then $E(\max\{X_1, \cdots, X_n\}) \leq \sup\{EX_t: t \in T_n\} + b_n(b - a)$, where again the bound $b_n$ is best possible. In both situations, extremal distributions for which equality is attained (or nearly attained) are given in implicit form.
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Matroid prophet inequalities

2012-05-19
Robert Kleinberg , S. Matthew Weinberg
Consider a gambler who observes a sequence of independent, non-negative random numbers and is allowed to stop the sequence at any time, claiming a reward equal to the most recent … Consider a gambler who observes a sequence of independent, non-negative random numbers and is allowed to stop the sequence at any time, claiming a reward equal to the most recent observation. The famous prophet inequality of Krengel, Sucheston, and Garling asserts that a gambler who knows the distribution of each random variable can achieve at least half as much reward, in expectation, as a "prophet" who knows the sampled values of each random variable and can choose the largest one. We generalize this result to the setting in which the gambler and the prophet are allowed to make more than one selection, subject to a matroid constraint. We show that the gambler can still achieve at least half as much reward as the prophet; this result is the best possible, since it is known that the ratio cannot be improved even in the original prophet inequality, which corresponds to the special case of rank-one matroids. Generalizing the result still further, we show that under an intersection of $p$ matroid constraints, the prophet's reward exceeds the gambler's by a factor of at most $O(p)$, and this factor is also tight.
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Stop rule and supremum expectations of i.i.d. random variables: A complete comparison by conjugate duality

1986-06-01
Robert P. Kertz

A Simplified 1.5-Approximation Algorithm for Augmenting Edge-Connectivity of a Graph from 1 to 2

2015-11-17
Guy Kortsarz , Zeev Nutov
The Tree Augmentation Problem (TAP) is as follows: given a connected graph G =( V , ε ) and an edge set E on V , find a minimum size … The Tree Augmentation Problem (TAP) is as follows: given a connected graph G =( V , ε ) and an edge set E on V , find a minimum size subset of edges F ⊆ E such that ( V , ε ∪ F ) is 2-edge-connected. In the conference version [Even et al. 2001] was sketched a 1.5-approximation algorithm for the problem. Since a full proof was very complex and long, the journal version was cut into two parts. The first part [Even et al. 2009] only proved ratio 1.8. An attempt to simplify the second part produced an error in Even et al. [2011]. Here we give a correct, different, and self-contained proof of the ratio 1.5 that is also substantially simpler and shorter than the previous proofs.
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Beating 1-1/e for ordered prophets

2017-06-15
Melika Abolhassani , Soheil Ehsani , Hossein Esfandiari +3 more
Hill and Kertz studied the prophet inequality on iid distributions [The Annals of Probability 1982]. They proved a theoretical bound of 1 - 1/e on the approximation factor of their … Hill and Kertz studied the prophet inequality on iid distributions [The Annals of Probability 1982]. They proved a theoretical bound of 1 - 1/e on the approximation factor of their algorithm. They conjectured that the best approximation factor for arbitrarily large n is 1/1+1/e ≃ 0.731. This conjecture remained open prior to this paper for over 30 years. In this paper we present a threshold-based algorithm for the prophet inequality with n iid distributions. Using a nontrivial and novel approach we show that our algorithm is a 0.738-approximation algorithm. By beating the bound of 1/1+1/e, this refutes the conjecture of Hill and Kertz. Moreover, we generalize our results to non-uniform distributions and discuss its applications in mechanism design.
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On the Tree Augmentation Problem

2020-09-15
Zeev Nutov
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The Prophet Inequality Can Be Solved Optimally with a Single Set of Samples

2018-01-01
Jack Wang
The setting of the classic prophet inequality is as follows: a gambler is shown the probability distributions of $n$ independent, non-negative random variables with finite expectations. In their indexed order, … The setting of the classic prophet inequality is as follows: a gambler is shown the probability distributions of $n$ independent, non-negative random variables with finite expectations. In their indexed order, a value is drawn from each distribution, and after every draw the gambler may choose to accept the value and end the game, or discard the value permanently and continue the game. What is the best performance that the gambler can achieve in comparison to a prophet who can always choose the highest value? Krengel, Sucheston, and Garling solved this problem in 1978, showing that there exists a strategy for which the gambler can achieve half as much reward as the prophet in expectation. Furthermore, this result is tight. In this work, we consider a setting in which the gambler is allowed much less information. Suppose that the gambler can only take one sample from each of the distributions before playing the game, instead of knowing the full distributions. We provide a simple and intuitive algorithm that recovers the original approximation of $\frac{1}{2}$. Our algorithm works against even an almighty adversary who always chooses a worst-case ordering, rather than the standard offline adversary. The result also has implications for mechanism design -- there is much interest in designing competitive auctions with a finite number of samples from value distributions rather than full distributional knowledge.

Prophet Inequalities for I.I.D. Random Variables from an Unknown Distribution

2019-06-17
José R. Correa , Paul Dütting , Felix Fischer +1 more
A central object in optimal stopping theory is the single-choice prophet inequality for independent, identically distributed random variables: given a sequence of random variables X1, ..., Xn drawn independently from … A central object in optimal stopping theory is the single-choice prophet inequality for independent, identically distributed random variables: given a sequence of random variables X1, ..., Xn drawn independently from a distribution F, the goal is to choose a stopping time τ so as to maximize α such that for all distributions F we have E[Xτ]≥α•E[maxt Xt]. What makes this problem challenging is that the decision whether τ=t may only depend on the values of the random variables X1, ..., Xt and on the distribution F. For a long time the best known bound for the problem had been α≥1-1/e≅0.632, but quite recently a tight bound of α≅0.745 was obtained. The case where F is unknown, such that the decision whether τ=t may depend only on the values of the random variables X1, ..., Xt, is equally well motivated but has received much less attention. A straightforward guarantee for this case of α≥1-1/e≅0.368 can be derived from the solution to the secretary problem, where an arbitrary set of values arrive in random order and the goal is to maximize the probability of selecting the largest value. We show that this bound is in fact tight. We then investigate the case where the stopping time may additionally depend on a limited number of samples from~F, and show that even with o(n) samples α≥1/e. On the other hand, n samples allow for a significant improvement, while O(n2) samples are equivalent to knowledge of the distribution: specifically, with n samples α≥1-1/e≅0.632 and α≥ln(2)≅0.693, and with O(n2) samples α≥0.745-ε for any ε>0.
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Improved approximation for tree augmentation: saving by rewiring

2018-06-20
Fabrizio Grandoni , Christos Kalaitzis , Rico Zenklusen
The Tree Augmentation Problem (TAP) is a fundamental network design problem in which we are given a tree and a set of additional edges, also called links. The task is … The Tree Augmentation Problem (TAP) is a fundamental network design problem in which we are given a tree and a set of additional edges, also called links. The task is to find a set of links, of minimum size, whose addition to the tree leads to a 2-edge-connected graph. A long line of results on TAP culminated in the previously best known approximation guarantee of 1.5 achieved by a combinatorial approach due to Kortsarz and Nutov [ACM Transactions on Algorithms 2016], and also by an SDP-based approach by Cheriyan and Gao [Algorithmica 2017]. Moreover, an elegant LP-based (1.5+є)-approximation has also been found very recently by Fiorini, Groß, K'onemann, and Sanitá [SODA 2018]. In this paper, we show that an approximation factor below 1.5 can be achieved, by presenting a 1.458-approximation that is based on several new techniques.
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Approximating (Unweighted) Tree Augmentation via Lift-and-Project, Part I: Stemless TAP

2017-01-06
Joseph Cheriyan , Zhihan Gao
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Prophet Inequalities Made Easy: Stochastic Optimization by Pricing Non-Stochastic Inputs

2017-10-01
Paul Dütting , Michal Feldman , Thomas Keßelheim +1 more
We present a general framework for stochastic online maximization problems with combinatorial feasibility constraints. The framework establishes prophet inequalities by constructing price-based online approximation algorithms, a natural extension of threshold … We present a general framework for stochastic online maximization problems with combinatorial feasibility constraints. The framework establishes prophet inequalities by constructing price-based online approximation algorithms, a natural extension of threshold algorithms for settings beyond binary selection. Our analysis takes the form of an extension theorem: we derive sufficient conditions on prices when all weights are known in advance, then prove that the resulting approximation guarantees extend directly to stochastic settings. Our framework unifies and simplifies much of the existing literature on prophet inequalities and posted price mechanisms, and is used to derive new and improved results for combinatorial markets (with and without complements), multi-dimensional matroids, and sparse packing problems. Finally, we highlight a surprising connection between the smoothness framework for bounding the price of anarchy of mechanisms and our framework, and show that many smooth mechanisms can be recast as posted price mechanisms with comparable performance guarantees.
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Prophet Secretary for Combinatorial Auctions and Matroids

2018-01-01
Soheil Ehsani , MohammadTaghi Hajiaghayi , Thomas Keßelheim +1 more
The secretary and the prophet inequality problems are central to the field of Stopping Theory. Recently, there has been a lot of work in generalizing these models to multiple items … The secretary and the prophet inequality problems are central to the field of Stopping Theory. Recently, there has been a lot of work in generalizing these models to multiple items because of their applications in mechanism design. The most important of these generalizations are to matroids and to combinatorial auctions (extends bipartite matching). Kleinberg-Weinberg [33] and Feldman et al. [17] show that for adversarial arrival order of random variables the optimal prophet inequalities give a 1/2-approximation. For many settings, however, it's conceivable that the arrival order is chosen uniformly at random, akin to the secretary problem. For such a random arrival model, we improve upon the 1/2-approximation and obtain (1 – 1/e)-approximation prophet inequalities for both matroids and combinatorial auctions. This also gives improvements to the results of Yan [45] and Esfandiari et al. [15] who worked in the special cases where we can fully control the arrival order or when there is only a single item.Our techniques are threshold based. We convert our discrete problem into a continuous setting and then give a generic template on how to dynamically adjust these thresholds to lower bound the expected total welfare.
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Breaching the 2-approximation barrier for connectivity augmentation: a reduction to Steiner tree

2020-06-07
Jarosław Byrka , Fabrizio Grandoni , Afrouz Jabal Ameli
The basic goal of survivable network design is to build a cheap network that maintains the connectivity between given sets of nodes despite the failure of a few edges/nodes. The … The basic goal of survivable network design is to build a cheap network that maintains the connectivity between given sets of nodes despite the failure of a few edges/nodes. The Connectivity Augmentation Problem (CAP) is arguably one of the most basic problems in this area: given a k(-edge)-connected graph G and a set of extra edges (links), select a minimum cardinality subset A of links such that adding A to G increases its edge connectivity to k+1. Intuitively, one wants to make an existing network more reliable by augmenting it with extra edges. The best known approximation factor for this NP-hard problem is 2, and this can be achieved with multiple approaches (the first such result is in [Frederickson and Jájá'81]).
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Prophet secretary through blind strategies

2020-08-03
José R. Correa , Raimundo Saona , Bruno Ziliotto
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Bridging the gap between tree and connectivity augmentation: unified and stronger approaches

2021-06-15
Federica Cecchetto , Vera Traub , Rico Zenklusen
We consider the Connectivity Augmentation Problem (CAP), a classical problem in the area of Survivable Network Design. It is about increasing the edge-connectivity of a graph by one unit in … We consider the Connectivity Augmentation Problem (CAP), a classical problem in the area of Survivable Network Design. It is about increasing the edge-connectivity of a graph by one unit in the cheapest possible way. More precisely, given a k-edge-connected graph G=(V,E) and a set of extra edges, the task is to find a minimum cardinality subset of extra edges whose addition to G makes the graph (k+1)-edge-connected. If k is odd, the problem is known to reduce to the Tree Augmentation Problem (TAP)—i.e., G is a spanning tree—for which significant progress has been achieved recently, leading to approximation factors below 1.5 (the currently best factor is 1.458). However, advances on TAP did not carry over to CAP so far. Indeed, only very recently, Byrka, Grandoni, and Ameli (STOC 2020) managed to obtain the first approximation factor below 2 for CAP by presenting a 1.91-approximation algorithm based on a method that is disjoint from recent advances for TAP.
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Prophet Secretary: Surpassing the $1-1/e$ Barrier

2017-01-01
Yossi Azar , Ashish Chiplunkar , Haim Kaplan
In the Prophet Secretary problem, samples from a known set of probability distributions arrive one by one in a uniformly random order, and an algorithm must irrevocably pick one of … In the Prophet Secretary problem, samples from a known set of probability distributions arrive one by one in a uniformly random order, and an algorithm must irrevocably pick one of the samples as soon as it arrives. The goal is to maximize the expected value of the sample picked relative to the expected maximum of the distributions. This is one of the most simple and fundamental problems in online decision making that models the process selling one item to a sequence of costumers. For a closely related problem called the Prophet Inequality where the order of the random variables is adversarial, it is known that one can achieve in expectation $1/2$ of the expected maximum, and no better ratio is possible. For the Prophet Secretary problem, that is, when the variables arrive in a random order, Esfandiari et al.\ (ESA 2015) showed that one can actually get $1-1/e$ of the maximum. The $1-1/e$ bound was recently extended to more general settings (Ehsani et al., 2017). Given these results, one might be tempted to believe that $1-1/e$ is the correct bound. We show that this is not the case by providing an algorithm for the Prophet Secretary problem that beats the $1-1/e$ bound and achieves $1-1/e+1/400$ of the optimum value. We also prove a hardness result on the performance of algorithms under a natural restriction which we call deterministic distribution-insensitivity.

Fixed-Parameter Algorithms for Minimum-Cost Edge-Connectivity Augmentation

2015-04-13
Dániel Marx , László A. Végh
We consider connectivity-augmentation problems in a setting where each potential new edge has a non-negative cost associated with it, and the task is to achieve a certain connectivity target with … We consider connectivity-augmentation problems in a setting where each potential new edge has a non-negative cost associated with it, and the task is to achieve a certain connectivity target with at most p new edges of minimum total cost. The main result is that the minimum cost augmentation of edge-connectivity from k − 1 to k with at most p new edges is fixed-parameter tractable parameterized by p and admits a polynomial kernel. We also prove the fixed-parameter tractability of increasing edge connectivity from 0 to 2 and increasing node connectivity from 1 to 2.
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