Counting and sampling minimum cuts in genus g graphs

Chambers, Erin Wolf; Fox, Kyle; Nayyeri, Amir

doi:10.1145/2493132.2462366

Cited by 3 publications

(3 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Another means of producing connected 2-partitions is via min cuts, since min cuts are always connected. There are polynomial time algorithms for uniformly sampling min s, t-cuts genus g graphs given in [27]. On general graphs, one can also sample min-cuts in a way that is fixed parameter tractable in the size of the min-cut [20], but the running time of this algorithm is practical only for very small min-cut sizes.…”

Section: See Appendix B3mentioning

confidence: 99%

Complexity and Geometry of Sampling Connected Graph Partitions

Najt¹,

DeFord²,

Solomon³

2019

Preprint

View full text Add to dashboard Cite

In this paper, we prove intractability results about sampling from the set of partitions of a planar graph into connected components. Our proofs are motivated by a technique introduced by Jerrum, Valiant, and Vazirani. Moreover, we use gadgets inspired by their technique to provide families of graphs where the "flip walk" Markov chain used in practice for this sampling task exhibits exponentially slow mixing. Supporting our theoretical results we present some empirical evidence demonstrating the slow mixing of the flip walk on grid graphs and on real data. Inspired by connections to the statistical physics of self-avoiding walks, we investigate the sensitivity of certain popular sampling algorithms to the graph topology. Finally, we discuss a few cases where the sampling problem is tractable. Applications to political redistricting have recently brought increased attention to this problem, and we articulate open questions about this application that are highlighted by our results.

show abstract

Section: See Appendix B3mentioning

confidence: 99%

Complexity and Geometry of Sampling Connected Graph Partitions

Najt¹,

DeFord²,

Solomon³

2019

Preprint

View full text Add to dashboard Cite

show abstract

“…Bezáková et al [5] built an algorithm for both directed and undirected graphs with small treewidth λ; its time complexity is O(2 3λ λn). Moreover, Chambers et al [9] designed an algorithm for directed graphs embedded on orientable surfaces of genus g: its execution time is O(2 g n 2 ). We study the fixed-parameter tractability of counting mincuts, parameterized by the size p of the minimum (S, T )-cuts.…”

Section: Introductionmentioning

confidence: 99%

Fixed-parameter tractability of counting small minimum $(S,T)$-cuts

Bergé¹,

Mouscadet²,

Rimmel³

et al. 2019

Preprint

View full text Add to dashboard Cite

The parameterized complexity of counting minimum cuts stands as a natural question because Ball and Provan showed its #Pcompleteness. For any undirected graph G = (V, E) and two disjoint sets of its vertices S, T , we design a fixed-parameter tractable algorithm which counts minimum edge (S, T )-cuts parameterized by their size p. Our algorithm operates on a transformed graph instance. This transformation, called drainage, reveals a collection of at most n = |V | successive minimum (S, T )-cuts Zi. We prove that any minimum (S, T )-cut X contains edges of at least one cut Zi. This observation, together with Menger's theorem, allows us to build the algorithm counting all minimum (S, T )-cuts with running time 2 O(p 2 ) n O(1) . Initially dedicated to counting minimum cuts, it can be modified to obtain an FPT sampling of minimum edge (S, T )-cuts.

show abstract

“…Given a dart d of edge e, we define the homology signature of d so that [d] = [e] if d = e and [d] = −[e] otherwise. Homology signatures given an implicit representation of a cohomology basis in G. See Erickson and Whittlesey[31] and subsequest papers[2,9,15,30]. The homology signature of a flow f is…”

mentioning

confidence: 99%

Holiest minimum-cost paths and flows in surface graphs

Erickson

Fox²,

Lkhamsuren³

2018

Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing

Self Cite

View full text Add to dashboard Cite

Let G be an edge-weighted directed graph with n vertices embedded on an orientable surface of genus g. We describe a simple deterministic lexicographic perturbation scheme that guarantees uniqueness of minimum-cost flows and shortest paths in G. The perturbations take O(gn) time to compute. We use our perturbation scheme in a black box manner to derive a deterministic O(n log log n) time algorithm for minimum cut in directed edge-weighted planar graphs and a deterministic O(g 2 n log n) time proprocessing scheme for the multiple-source shortest paths problem of computing a shortest path oracle for all vertices lying on a common face of a surface embedded graph. The latter result yields faster deterministic near-linear time algorithms for a variety of problems in constant genus surface embedded graphs.Finally, we open the black box in order to generalize a recent linear-time algorithm for multiplesource shortest paths in unweighted undirected planar graphs to work in arbitrary orientable surfaces. Our algorithm runs in O(g 2 n log g) time in this setting, and it can be used to give improved linear time algorithms for several problems in unweighted undirected surface embedded graphs of constant genus including the computation of minimum cuts, shortest topologically non-trivial cycles, and minimum homology bases.Many recent combinatorial optimization algorithms for directed surface embedded graphs rely on a common assumption: the shortest path between any pair of vertices is unique. The most commonly applied consequence of this assumption is that the shortest paths entering (or leaving) a common vertex do not cross one another. From this consequence, one can prove near-linear running time bounds for a variety of problems, including the computation of maximum flows [4,5,24,26] and global minimum cuts [53] in directed planar (genus 0) graphs as well as the computation of minimum cut oracles in planar and more general embedded graphs [3,6] (see also ).This assumption is also used in algorithms for the multiple-source shortest paths problem introduced for planar graphs by Klein [46]. In the multiple-source shortest paths problem, one is given a surface embedded graph G = (V, E, F ) of genus g with vertices V , edges E, and faces F . The goal is to compute a representation of all shortest paths from vertices on a common face r ∈ F to all other vertices in the graph. Assuming uniqueness of shortest paths, multiple-source shortest paths can be computed in only O(g n log n) time [11,46]. Algorithms for this problem can be used to solve a variety of problems in planar and more general surface embedded graphs of constant genus in near-linear time. Such results include the computation of shortest cycles with non-trivial topology [2,11,27,30,32,34], the computation of maximum flows and minimum cuts [5,14,16,28,30,41,49], the computation of exact and approximate distance oracles [10,44,54], and even the computation of single-source shortest paths [47,55].Enforcing uniqueness. Unfortunately, it is often difficult to actually enfo...

show abstract

Counting and sampling minimum cuts in genus g graphs

Cited by 3 publications

References 25 publications

Complexity and Geometry of Sampling Connected Graph Partitions

Complexity and Geometry of Sampling Connected Graph Partitions

Fixed-parameter tractability of counting small minimum $(S,T)$-cuts

Holiest minimum-cost paths and flows in surface graphs

Contact Info

Product

Resources

About