An s-branching flow f in a network N = (D, u), such that u is the capacity function, is a flow that reaches every vertex in V (D) {s} from …
An s-branching flow f in a network N = (D, u), such that u is the capacity function, is a flow that reaches every vertex in V (D) {s} from s while loosing exactly one unit of flow in each vertex other than s. It is known that the hardness of the problem of finding k arc-disjoint s-branching flows in a network N is linked to the capacity u of the arcs in N : for fixed c, the problem is solvable in polynomial time if every arc has capacity n − c and, unless the Exponential Time Hypothesis (ETH) fails, there is no polynomial time algorithm for it for most other choices of the capacity function when every arc has the same capacity. The hardness of a few cases remains open. We further investigate a conjecture that aims to characterize networks admitting k arc-disjoint s-branching flows, generalizing a result that provides such characterization when all arcs have capacity n−1, based on Edmonds' branching theorem. We show that, in general, the conjecture is false. However, it holds for some special classes of digraphs, as branchings and spindles with parallel arcs.
An s-branching flow f in a network N = (D,c) (where c is the capacity function) is a flow that reaches every vertex in V(D) \ {s} from s while …
An s-branching flow f in a network N = (D,c) (where c is the capacity function) is a flow that reaches every vertex in V(D) \ {s} from s while loosing exactly one unit of flow in each vertex other than s. In other words, the difference between the flow entering a vertex v and a flow leaving a vertex v is one whenever v is different from s. It is known that the hardness of the problem of finding k arc-disjoint s-branching flows in network N is linked to the capacity c of the arcs in N: the problem is solvable in polynomial time if every arc has capacity n - l, for fixed l, and NP-complete in most other cases, with very few cases open. We further investigate a conjecture by Costa et al. from 2019 that aims to characterize networks admitting k arc-disjoint s-branching flows, generalizing a classical result by Edmonds that provides such characterization when all arcs have capacity n-1. We show that, in general, the conjecture is false. However, on the positive side, it holds for digraphs formed by out-branchings together with parallel arcs.
We present new min-max relations in digraphs between the number of paths satisfying certain conditions and the order of the corresponding cuts. We define these objects in order to capture, …
We present new min-max relations in digraphs between the number of paths satisfying certain conditions and the order of the corresponding cuts. We define these objects in order to capture, in the context of solving the half-integral linkage problem, the essential properties needed for reaching a large bramble of congestion two (or any other constant) from the terminal set. This strategy has been used ad-hoc in several articles, usually with lengthy technical proofs, and our objective is to abstract it to make it applicable in a simpler and unified way. We provide two proofs of the min-max relations, one consisting in applying Menger's Theorem on appropriately defined auxiliary digraphs, and an alternative simpler one using matroids, however with worse polynomial running time. As an application, we manage to simplify and improve several results of Edwards et al. [ESA 2017] and of Giannopoulou et al. [SODA 2022] about finding half-integral linkages in digraphs. Concerning the former, besides being simpler, our proof provides an almost optimal bound on the strong connectivity of a digraph for it to be half-integrally feasible under the presence of a large bramble of congestion two (or equivalently, if the directed tree-width is large, which is the hard case). Concerning the latter, our proof uses brambles as rerouting objects instead of cylindrical grids, hence yielding much better bounds and being somehow independent of a particular topology. We hope that our min-max relations will find further applications as, in our opinion, they are simple, robust, and versatile to be easily applicable to different types of routing problems in digraphs.
This paper presents a comprehensive overview of monotone priority queues, focusing on their evolution and application in shortest path algorithms. Monotone priority queues are characterized by the property that their …
This paper presents a comprehensive overview of monotone priority queues, focusing on their evolution and application in shortest path algorithms. Monotone priority queues are characterized by the property that their minimum key does not decrease over time, making them particularly effective for label-setting algorithms like Dijkstra's. Some key data structures within this category are explored, emphasizing those derived directly from Dial's algorithm, including variations of multi-level bucket structures and radix heaps. Theoretical complexities and practical considerations of these structures are discussed, with insights into their development and refinement provided through a historical timeline.
We present new min-max relations in digraphs between the number of paths satisfying certain conditions and the order of the corresponding cuts. We define these objects in order to capture, …
We present new min-max relations in digraphs between the number of paths satisfying certain conditions and the order of the corresponding cuts. We define these objects in order to capture, in the context of solving the half-integral linkage problem, the essential properties needed for reaching a large bramble of constant congestion from the terminal set. This strategy has been used ad-hoc in several articles, usually with lengthy technical proofs, and our objective is to abstract it to make it applicable in a simpler and unified way. We provide two proofs of the min-max relations, one consisting in applying Menger’s Theorem on appropriately defined digraphs, and an alternative simpler one using matroids, however with worse polynomial running time. As an application, we manage to simplify and improve several results of Edwards et al. [ESA 2017] and of Giannopoulou et al. [SODA 2022] about finding half-integral linkages in digraphs. Concerning the former, besides being simpler, our proof provides an almost optimal bound on the strong connectivity of a digraph for it to be half-integrally feasible under the presence of a large bramble of congestion two (or equivalently, if the directed tree-width is large). Concerning the latter, our proof uses brambles as rerouting objects instead of cylindrical grids, hence yielding much better bounds and being somehow independent of a particular topology. We hope that our min-max relations will find further applications as, in our opinion, they are simple, robust, and versatile to be easily applicable to different types of routing problems in digraphs.
We present new min-max relations in digraphs between the number of paths satisfying certain conditions and the order of the corresponding cuts. We define these objects in order to capture, …
We present new min-max relations in digraphs between the number of paths satisfying certain conditions and the order of the corresponding cuts. We define these objects in order to capture, in the context of solving the half-integral linkage problem, the essential properties needed for reaching a large bramble of constant congestion from the terminal set. This strategy has been used ad-hoc in several articles, usually with lengthy technical proofs, and our objective is to abstract it to make it applicable in a simpler and unified way. We provide two proofs of the min-max relations, one consisting in applying Menger’s Theorem on appropriately defined digraphs, and an alternative simpler one using matroids, however with worse polynomial running time. As an application, we manage to simplify and improve several results of Edwards et al. [ESA 2017] and of Giannopoulou et al. [SODA 2022] about finding half-integral linkages in digraphs. Concerning the former, besides being simpler, our proof provides an almost optimal bound on the strong connectivity of a digraph for it to be half-integrally feasible under the presence of a large bramble of congestion two (or equivalently, if the directed tree-width is large). Concerning the latter, our proof uses brambles as rerouting objects instead of cylindrical grids, hence yielding much better bounds and being somehow independent of a particular topology. We hope that our min-max relations will find further applications as, in our opinion, they are simple, robust, and versatile to be easily applicable to different types of routing problems in digraphs.
This paper presents a comprehensive overview of monotone priority queues, focusing on their evolution and application in shortest path algorithms. Monotone priority queues are characterized by the property that their …
This paper presents a comprehensive overview of monotone priority queues, focusing on their evolution and application in shortest path algorithms. Monotone priority queues are characterized by the property that their minimum key does not decrease over time, making them particularly effective for label-setting algorithms like Dijkstra's. Some key data structures within this category are explored, emphasizing those derived directly from Dial's algorithm, including variations of multi-level bucket structures and radix heaps. Theoretical complexities and practical considerations of these structures are discussed, with insights into their development and refinement provided through a historical timeline.
We present new min-max relations in digraphs between the number of paths satisfying certain conditions and the order of the corresponding cuts. We define these objects in order to capture, …
We present new min-max relations in digraphs between the number of paths satisfying certain conditions and the order of the corresponding cuts. We define these objects in order to capture, in the context of solving the half-integral linkage problem, the essential properties needed for reaching a large bramble of congestion two (or any other constant) from the terminal set. This strategy has been used ad-hoc in several articles, usually with lengthy technical proofs, and our objective is to abstract it to make it applicable in a simpler and unified way. We provide two proofs of the min-max relations, one consisting in applying Menger's Theorem on appropriately defined auxiliary digraphs, and an alternative simpler one using matroids, however with worse polynomial running time. As an application, we manage to simplify and improve several results of Edwards et al. [ESA 2017] and of Giannopoulou et al. [SODA 2022] about finding half-integral linkages in digraphs. Concerning the former, besides being simpler, our proof provides an almost optimal bound on the strong connectivity of a digraph for it to be half-integrally feasible under the presence of a large bramble of congestion two (or equivalently, if the directed tree-width is large, which is the hard case). Concerning the latter, our proof uses brambles as rerouting objects instead of cylindrical grids, hence yielding much better bounds and being somehow independent of a particular topology. We hope that our min-max relations will find further applications as, in our opinion, they are simple, robust, and versatile to be easily applicable to different types of routing problems in digraphs.
An s-branching flow f in a network N = (D, u), such that u is the capacity function, is a flow that reaches every vertex in V (D) {s} from …
An s-branching flow f in a network N = (D, u), such that u is the capacity function, is a flow that reaches every vertex in V (D) {s} from s while loosing exactly one unit of flow in each vertex other than s. It is known that the hardness of the problem of finding k arc-disjoint s-branching flows in a network N is linked to the capacity u of the arcs in N : for fixed c, the problem is solvable in polynomial time if every arc has capacity n − c and, unless the Exponential Time Hypothesis (ETH) fails, there is no polynomial time algorithm for it for most other choices of the capacity function when every arc has the same capacity. The hardness of a few cases remains open. We further investigate a conjecture that aims to characterize networks admitting k arc-disjoint s-branching flows, generalizing a result that provides such characterization when all arcs have capacity n−1, based on Edmonds' branching theorem. We show that, in general, the conjecture is false. However, it holds for some special classes of digraphs, as branchings and spindles with parallel arcs.
An s-branching flow f in a network N = (D,c) (where c is the capacity function) is a flow that reaches every vertex in V(D) \ {s} from s while …
An s-branching flow f in a network N = (D,c) (where c is the capacity function) is a flow that reaches every vertex in V(D) \ {s} from s while loosing exactly one unit of flow in each vertex other than s. In other words, the difference between the flow entering a vertex v and a flow leaving a vertex v is one whenever v is different from s. It is known that the hardness of the problem of finding k arc-disjoint s-branching flows in network N is linked to the capacity c of the arcs in N: the problem is solvable in polynomial time if every arc has capacity n - l, for fixed l, and NP-complete in most other cases, with very few cases open. We further investigate a conjecture by Costa et al. from 2019 that aims to characterize networks admitting k arc-disjoint s-branching flows, generalizing a classical result by Edmonds that provides such characterization when all arcs have capacity n-1. We show that, in general, the conjecture is false. However, on the positive side, it holds for digraphs formed by out-branchings together with parallel arcs.