A simple algorithm for finding a maximum triangle-free <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" display="inline" overflow="scroll"><mml:mn>2</mml:mn></mml:math>-matching in subcubic graphs

Kobayashi, Yusuke

doi:10.1016/j.disopt.2010.04.001

Cited by 16 publications

(7 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is still open whether the weighted C 3 -free 2-matching problem can be solved in polynomial time. For the weighted C 3 -free 2-matching problem in subcubic graphs, Hartvigsen and Li [17] gave a polyhedral description and a polynomial-time algorithm, and faster polynomial-time algorithms were presented by Kobayashi [21] and by Paluch and Wasylkiewicz [36]. Recently, Kobayashi [23] designed a polynomial-time algorithm for the weighted C 3 -free 2-matching problem in which the cycles of length three are edge-disjoint.…”

Section: Further Related Workmentioning

confidence: 99%

See 1 more Smart Citation

Optimal matroid bases with intersection constraints: valuated matroids, M-convex functions, and their applications

Iwamasa

Takazawa

2021

Math. Program.

View full text Add to dashboard Cite

The problem of finding a maximum 2-matching without short cycles has received significant attention due to its relevance to the Hamilton cycle problem. This problem is generalized to finding a maximum t-matching which excludes specified complete t-partite subgraphs, where t is a fixed positive integer. The polynomial solvability of this generalized problem remains an open question. In this paper, we present polynomial-time algorithms for the following two cases of this problem: in the first case the forbidden complete t-partite subgraphs are edge-disjoint; and in the second case the maximum degree of the input graph is at most 2t − 1. Our result for the first case extends the previous work of Nam (1994) showing the polynomial solvability of the problem of finding a maximum 2-matching without cycles of length four, where the cycles of length four are vertex-disjoint. The second result expands upon the works of Bérczi and Végh (2010) and Kobayashi and Yin (2012), which focused on graphs with maximum degree at most t + 1. Our algorithms are obtained from exploiting the discrete structure of restricted t-matchings and employing an algorithm for the Boolean edge-CSP.

show abstract

Section: Further Related Workmentioning

confidence: 99%

“…The relationship between K-free t-matchings and jump systems has been studied in [3,9,27], some of which will be used in this paper. More generally, the relationship between weighted K-free t-matchings and discrete convexity has been studied in [3,21,22,27].…”

Section: Further Related Workmentioning

confidence: 99%

Optimal matroid bases with intersection constraints: valuated matroids, M-convex functions, and their applications

Iwamasa

Takazawa

2021

Math. Program.

View full text Add to dashboard Cite

show abstract

“…The case of k = 3, which we call the weighted triangle-free 2-matching problem, is a long-standing open problem. For the weighted triangle-free 2-matching problem in subcubic graphs, Hartvigsen and Li [15] gave a polyhedral description and a polynomial-time algorithm, followed by a slight generalized polyhedral description by Bérczi [2] and another polynomial-time algorithm by Kobayashi [19]. Relationship between C ≤k -free 2-matchings and discrete convexity is studied in [19,20,21].…”

Section: -Matchings Without Short Cyclesmentioning

confidence: 99%

Weighted Triangle-Free 2-Matching Problem with Edge-Disjoint Forbidden Triangles

Kobayashi

2020

Integer Programming and Combinatorial Optimization

Self Cite

View full text Add to dashboard Cite

The weighted T -free 2-matching problem is the following problem: given an undirected graph G, a weight function on its edge set, and a set T of triangles in G, find a maximum weight 2matching containing no triangle in T . When T is the set of all triangles in G, this problem is known as the weighted triangle-free 2-matching problem, which is a long-standing open problem. A main contribution of this paper is to give a first polynomial-time algorithm for the weighted T -free 2matching problem under the assumption that T is a set of edge-disjoint triangles. In our algorithm, a key ingredient is to give an extended formulation representing the solution set, that is, we introduce new variables and represent the convex hull of the feasible solutions as a projection of another polytope in a higher dimensional space. Although our extended formulation has exponentially many inequalities, we show that the separation problem can be solved in polynomial time, which leads to a polynomial-time algorithm for the weighted T -free 2-matching problem. 1 Although such an edge set is often called a simple 2-matching in the literature, we call it a 2-matching to simplify the description.

show abstract

“…(The algorithm and polytope for weighted‐

{P}_{3}

are considerably more complex than the algorithm and polytope for weighted‐

{Q}_{3}

.) Subsequently, for weighted‐

{P}_{3}

on subcubic graphs, a slightly more general algorithm and polytope were presented by Bérczi [4] and a different, simpler algorithm was discovered by Kobayashi [32].…”

Section: Introductionmentioning

confidence: 99%

Finding triangle‐free 2‐factors in general graphs

Hartvigsen

2024

Journal of Graph Theory

View full text Add to dashboard Cite

A 2‐factor in a graph is a subset of edges such that every node of is incident with exactly two edges of . Many results are known concerning 2‐factors including a polynomial‐time algorithm for finding 2‐factors and a characterization of those graphs that have a 2‐factor. The problem of finding a 2‐factor in a graph is a relaxation of the NP‐hard problem of finding a Hamilton cycle. A stronger relaxation is the problem of finding a triangle‐free 2‐factor, that is, a 2‐factor whose edges induce no cycle of length 3. In this paper, we present a polynomial‐time algorithm for the problem of finding a triangle‐free 2‐factor as well as a characterization of the graphs that have such a 2‐factor and related min–max and augmenting path theorems.

show abstract

A simple algorithm for finding a maximum triangle-free 2-matching in subcubic graphs

Cited by 16 publications

References 27 publications

Optimal matroid bases with intersection constraints: valuated matroids, M-convex functions, and their applications

Optimal matroid bases with intersection constraints: valuated matroids, M-convex functions, and their applications

Weighted Triangle-Free 2-Matching Problem with Edge-Disjoint Forbidden Triangles

Finding triangle‐free 2‐factors in general graphs

Contact Info

Product

Resources

About