5 research outputs found
Shortest Reconfiguration of Perfect Matchings via Alternating Cycles
Motivated by adjacency in perfect matching polytopes, we study the shortest reconfiguration problem of perfect matchings via alternating cycles. Namely, we want to find a shortest sequence of perfect matchings which transforms one given perfect matching to another given perfect matching such that the symmetric difference of each pair of consecutive perfect matchings is a single cycle. The problem is equivalent to the combinatorial shortest path problem in perfect matching polytopes. We prove that the problem is NP-hard even when a given graph is planar or bipartite, but it can be solved in polynomial time when the graph is outerplanar
Shortest Reconfiguration of Matchings
Imagine that unlabelled tokens are placed on the edges of a graph, such that
no two tokens are placed on incident edges. A token can jump to another edge if
the edges having tokens remain independent. We study the problem of determining
the distance between two token configurations (resp., the corresponding
matchings), which is given by the length of a shortest transformation. We give
a polynomial-time algorithm for the case that at least one of the two
configurations is not inclusion-wise maximal and show that otherwise, the
problem admits no polynomial-time sublogarithmic-factor approximation unless P
= NP. Furthermore, we show that the distance of two configurations in bipartite
graphs is fixed-parameter tractable parameterized by the size of the
symmetric difference of the source and target configurations, and obtain a
-factor approximation algorithm for every if
additionally the configurations correspond to maximum matchings. Our two main
technical tools are the Edmonds-Gallai decomposition and a close relation to
the Directed Steiner Tree problem. Using the former, we also characterize those
graphs whose corresponding configuration graphs are connected. Finally, we show
that deciding if the distance between two configurations is equal to a given
number is complete for the class , and deciding if the diameter of
the graph of configurations is equal to is -hard.Comment: 31 pages, 3 figure
On the complexity of computing the diameter of a polytope
Abstract: "In this paper, some results on the complexity of computing the combinatorial diameter of a polytope are presented. We show that it is D[superscript P]-hard to determine the diameter of a polytope specified by linear inequalities with integer data. Our result partially resolves a long-term open question.