Weighted Triangle-free 2-matching Problem with Edge-disjoint Forbidden Triangles
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 2-matching containing no triangle in T . When T is the set of all triangles in G , this problem is known as the wei...
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Veröffentlicht in: | Mathematical programming 2022-03, Vol.192 (1-2), p.675-702 |
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Format: | Artikel |
Sprache: | eng |
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Zusammenfassung: | 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 2-matching 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 the first polynomial-time algorithm for the weighted
T
-free 2-matching 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. |
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ISSN: | 0025-5610 1436-4646 |
DOI: | 10.1007/s10107-021-01661-y |