Packing Non-Zero A-Paths In Group-Labelled Graphs

Chudnovsky, Maria; Geelen, Jim; Gerards, Bert; Goddyn, Luis; Lohman, Michael; Seymour, Paul

doi:10.1007/s00493-006-0030-1

Cited by 58 publications

(63 citation statements)

References 3 publications

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“…Here S is a partition of A and an S-path is an A-path between terminals in distinct subsets in S. Hence, for any disjoint S, T ⊆ V , the concept of S-path includes that of S-T path as a special case with S = {S, T }. As a generalization of Mader's problem, Chudnovsky, et al [3] introduced a framework of packing A-paths in group-labelled graphs, called the non-zero model in this paper, and proved a min-max theorem which generalizes Mader's theorem. Later Pap [11] introduced a slightly more generalized model, called the non-returning model in this paper, and gave a simpler proof of a further generalized min-max theorem for his model.…”

Section: Main Theoremmentioning

confidence: 97%

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Packing A-paths in Group-Labelled Graphs via Linear Matroid Parity

Yamaguchi

2013

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

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Mader's disjoint S-paths problem is a common generalization of non-bipartite matching and Menger's disjoint paths problems. Lovász (1980) suggested a polynomial-time algorithm for this problem through a reduction to matroid matching. A more direct reduction to the linear matroid parity problem was given later by Schrijver (2003), which leads to faster algorithms.As a generalization of Mader's problem, Chudnovsky, Geelen, Gerards, Goddyn, Lohman, and Seymour (2006) introduced a framework of packing non-zero A-paths in group-labelled graphs, and proved a min-max theorem. Chudnovsky, Cunningham, and Geelen (2008) provided an efficient combinatorial algorithm for this generalized problem. On the other hand, Pap (2007) introduced a framework of packing non-returning A-paths as a further genaralization.In this paper, we discuss a possible extension of Schrijver's reduction technique to another framework introduced by Pap (2006), under the name of the subgroup model, which apparently generalizes but in fact is equivalent to packing non-returning A-paths. We provide a necessary and sufficient condition for the groups in question to admit a reduction to the linear matroid parity problem. As a consequence, we give faster algorithms for important special cases of packing non-zero A-paths such as odd-length A-paths. In addition, it turns out that packing non-returning A-paths admits a reduction to the linear matroid parity problem, which leads to the quite efficient solvability, if and only if the size of the input label set is at most four.

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Section: Main Theoremmentioning

confidence: 97%

“…The non-zero model dealt with in [3,2] is a simple special case of the subgroup model with Γ ′ trivial. The previous two examples are included in this model.…”

Section: Non-zero Modelmentioning

confidence: 99%

Packing A-paths in Group-Labelled Graphs via Linear Matroid Parity

Yamaguchi

2013

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

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“…Group-labelled graphs (as in Group Feedback Vertex Set) and bijection-labelled graphs (as in Unique Label Cover) have been explored from a graph-theory perspective, in particular with respect to path-packing; see [13,14,27] and [44,45].…”

Section: Related Workmentioning

confidence: 99%

Half-integrality, LP-branching and FPT Algorithms

Wahlström

2013

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

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A recent trend in parameterized algorithms is the application of polytope tools (specifically, LPbranching) to FPT algorithms (e.g., Cygan et al., 2011;. Though the list of work in this direction is short, the results are already interesting, yielding significant speedups for a range of important problems. However, the existing approaches require the underlying polytope to have very restrictive properties, including half-integrality and Nemhauser-Trotter-style persistence properties. To date, these properties are essentially known to hold only for two classes of polytopes, covering the cases of Vertex Cover (Nemhauser and Trotter, 1975) and Node Multiway Cut (Garg et al., 1994).Taking a slightly different approach, we view half-integrality as a discrete relaxation of a problem, e.g., a relaxation of the search space from {0, 1}V to {0, 1 /2, 1} V such that the new problem admits a polynomial-time exact solution. Using tools from CSP (in particular Thapper andŽivný, 2012) to study the existence of such relaxations, we are able to provide a much broader class of half-integral polytopes with the required properties.Our results unify and significantly extend the previously known cases. In addition to the new insight into problems with half-integral relaxations, our results yield a range of new and improved FPT algorithms, including an O * (|Σ| 2k )-time algorithm for node-deletion Unique Label Cover with label set

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“…structures under parity constraints; for example, having vertex-or edge-disjoint odd cycles [29,28], parity linkages [48,30,25], odd minors [19,31,33], odd subdivisions [27], vertex-disjoint packings of odd paths with ends in a specified vertex-set [4,3], and packing non-zero cycles in group-labelled graphs [22,32]. A key question for each type of structure is whether the Erdős-Pósa property holds: is there a function f such that there are either k vertex-disjoint (or edge-disjoint) instances of the structure or a set of f (k) vertices (or edges) intersecting all such instances?…”

mentioning

confidence: 99%

“…Churchley, Mohar, and Wu [6] first proved that the maximum number of edge-disjoint odd (u, v)-trails in a graph is, up to a constant multiplicative factor, approximately equal to the minimum number of edges needed to cover all such trails. Using ideas from [6] and [4], it is possible to express an approximate packing-covering duality in terms of a quantity that measures "how far from a bipartite graph" the part of the graph around u and v is. This paper formalizes this measure, which we call the perimeter of a vertex-set.…”

mentioning

confidence: 99%

A submodular measure and approximate Gomory-Hu theorem for packing odd trails

Churchley

Mohar

2018

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

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Motivated by a problem about totally odd immersions of graphs, we define the odd edge-connectivity λ o (u, v) as the maximum number of edge-disjoint trails of odd length from u to v. It was recently discovered that λ o (u, v) can be approximated up to a constant multiplicative factor using the usual edge-connectivity between u and v and the minimum value of another parameter that measures "how far from a bipartite graph" the part of the graph around u and v is.In this paper, we formalize this second ingredient and call it the perimeter. We prove that perimeter is a submodular function on the vertex-sets of a graph. Using this fact, we obtain a version of the Gomory-Hu Theorem in which minimum edge-cuts are replaced by sets of minimum perimeter. We construct (in polynomial time) a rooted forest structure, analogous to the Gomory-Hu tree of a graph, which encodes a collection of minimum-perimeter vertex-sets. Although the classical Gomory-Hu Theorem extends to arbitrary symmetric submodular functions, our result is novel and indicates a possibility for further generalizations.These results have significant implications for the study of path and trail systems with parity constraints. We present two such applications: an efficient data structure for storing approximate odd edge-connectivities for all pairs of vertices in a graph, and a rough structure theorem for graphs with no "totally odd" immersion of a large complete graph.

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Packing Non-Zero A-Paths In Group-Labelled Graphs

Cited by 58 publications

References 3 publications

Packing A-paths in Group-Labelled Graphs via Linear Matroid Parity

Packing A-paths in Group-Labelled Graphs via Linear Matroid Parity

Half-integrality, LP-branching and FPT Algorithms

A submodular measure and approximate Gomory-Hu theorem for packing odd trails

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