Path problems in skew-symmetric graphs

We discuss an adaptation of the famous primal-dual 1-matching algorithm to balanced network flows which can be viewed as a network flow description of capacitated matching problems. This method is endowed with a sophisticated start-up procedure which eventually makes the algorithm strongly polynomial. We apply the primal-dual algorithm to the shortest valid path problem with arbitrary arc lengths, and so end up with a new complexity bound for this problem.

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“…In accordance with [7], and as an extension of the reduced-length labels for ordinary min-cost flow problems, we call…”

Section: Complementaritymentioning

confidence: 95%

“…As in [7], a fragment of a balanced network N is a pair (U, a), where U is a self-complementary node set, called the interior, and a is an arc in N[Ū, U], called the prop. Blossoms and nuclei together with their props are reasonable examples of fragments.…”

Section: Shrinking Familiesmentioning

confidence: 99%

Balanced network flows. VII. Primal‐dual algorithms

Fremuth-Paeger

Jungnickel

2001

Networks

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“…In accordance with [10], we call the modified cost of the arc a. Note that the modified cost labels are the reduced cost labels known from linear programming.…”

Section: Dualitymentioning

confidence: 99%

Balanced network flows. VI. Polyhedral descriptions

Fremuth-Paeger

Jungnickel

2001

Networks

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This paper discusses the balanced circulation polytope, that is, the convex hull of balanced circulations of a given balanced flow network. The LP description of this polytope is the LP description of ordinary circulations plus some odd-set constraints. The paper starts with an exposition of several classes of odd-set inequalities. These inequalities are described in terms of balanced network flows as well as matchings and put into relation to each other.Step by step, the problem of finding a cost minimum balanced circulation can be reduced to the b-matching problem. We present an LP characterization of the b-matching polytope by blossom inequalities. With a moderate effort, these odd sets are lifted to the setting of balanced-network flows. We finish with the dualization of the derived LP formulation, an introduction of reduced-cost labels, and a corresponding optimality condition.

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“…Skew-symmetric graphs were introduced under the name of antisymmetrical digraphs by Tutte [36] along with a notion of self-conjugate flows as a generalization of maximum flows in networks and matchings in graphs and subsequently by Zelinka [40] and Zaslavsky [39]. Goldberg and Karzanov [11,12] revisited the work of Tutte and gave unified proofs for the analogues of the flow-decomposition and max-flow min-cut theorems on these graphs.…”

Section: Introductionmentioning

confidence: 99%

“…Flows on skew-symmetric graphs have been used to generalize maximum flow and maximum matching problems on graphs, initially by Tutte [1967], and later by Goldberg and Karzanov [1994, 1995]. In this paper, we introduce a separation problem, d-SkewSymmetric Multicut, where we are given a skewsymmetric graph D, a family of T of d-sized subsets of vertices and an integer k. The objective is to decide if there is a set X ⊆ A of k arcs such that every set J in the family has a vertex v such that v and σ(v) are in different strongly connected components of D = (V, A \ (X ∪ σ(X)).…”

mentioning

confidence: 99%

Linear Time Parameterized Algorithms via Skew-Symmetric Multicuts

Ramanujan¹,

Saurabh²

2013

Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms

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A skew-symmetric graph (D = (V, A), σ) is a directed graph D with an involution σ on the set of vertices and arcs. Flows on skew-symmetric graphs have been used to generalize maximum flow and maximum matching problems on graphs, initially by Tutte [1967], and later by Goldberg and Karzanov [1994, 1995]. In this paper, we introduce a separation problem, d-SkewSymmetric Multicut, where we are given a skewsymmetric graph D, a family of T of d-sized subsets of vertices and an integer k. The objective is to decide if there is a set X ⊆ A of k arcs such that every set J in the family has a vertex v such that v and σ(v) are in different strongly connected components of D = (V, A \ (X ∪ σ(X)). In this paper, we give an algorithm for d-Skew-Symmetric Multicut which runs in time O((4d) k (m+n+ )), where m is the number of arcs in the graph, n the number of vertices and the length of the family given in the input.This problem, apart from being independently interesting, also abstracts out and captures the main combinatorial obstacles towards solving numerous classical problems. Our algorithm for d-Skew-Symmetric Multicut paves the way for the first linear time parameterized algorithms for several problems. We demonstrate its utility by obtaining the following linear time parameterized algorithms.• We show that Almost 2-SAT is a special case of 1-Skew-Symmetric Multicut, resulting in an algorithm for Almost 2-SAT which runs in time O(4 k k 4 ) where k is the size of the solution and is the length of the input formula. Then, using linear time parameter preserving reductions to Almost 2-SAT, we obtain algorithms for Odd Cycle Transversal and Edge Bipartization which run in time O(4 k k 4 (m+n)) and O(4 k k 5 (m+ n)) respectively where k is size of the solution, * Supported by Parameterized Approximation, ERC Starting Grant 306992.

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Path problems in skew-symmetric graphs

Cited by 30 publications

References 19 publications

Balanced network flows. VII. Primal‐dual algorithms

Balanced network flows. VII. Primal‐dual algorithms

Balanced network flows. VI. Polyhedral descriptions

Linear Time Parameterized Algorithms via Skew-Symmetric Multicuts

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