552 research outputs found
Exact and fixed-parameter algorithms for metro-line crossing minimization problems
A metro-line crossing minimization problem is to draw multiple lines on an
underlying graph that models stations and rail tracks so that the number of
crossings of lines becomes minimum. It has several variations by adding
restrictions on how lines are drawn. Among those, there is one with a
restriction that line terminals have to be drawn at a verge of a station, and
it is known to be NP-hard even when underlying graphs are paths. This paper
studies the problem in this setting, and propose new exact algorithms. We first
show that a problem to decide if lines can be drawn without crossings is solved
in polynomial time, and propose a fast exponential algorithm to solve a
crossing minimization problem. We then propose a fixed-parameter algorithm with
respect to the multiplicity of lines, which implies that the problem is FPT.Comment: 19 pages, 15 figure
Local topology of the free complex of a two-dimensional generalized convex shelling
AbstractA generalized convex shelling was introduced by Kashiwabara et al. for their representation theorem of convex geometries. Motivated by the work by Edelman and Reiner, we study local topology of the free complex of a two-dimensional separable generalized convex shelling. As a result, we prove a deletion of an element from such a complex is homotopy equivalent to a single point or two distinct points, depending on the dependency of the element to be deleted. Our result resolves an open problem by Edelman and Reiner for this case, and it can be seen as a first step toward the complete resolution from the viewpoint of a representation theorem for convex geometries by Kashiwabara et al
Submodularity of some classes of the combinatorial optimization games
Abstract.: Submodularity (or concavity) is considered as an important property in the field of cooperative game theory. In this article, we characterize submodular minimum coloring games and submodular minimum vertex cover games. These characterizations immediately show that it can be decided in polynomial time that the minimum coloring game or the minimum vertex cover game on a given graph is submodular or not. Related to these results, the Shapley values are also investigate
Not All Saturated 3-Forests Are Tight
A basic statement in graph theory is that every inclusion-maximal forest is
connected, i.e. a tree. Using a definiton for higher dimensional forests by
Graham and Lovasz and the connectivity-related notion of tightness for
hypergraphs introduced by Arocha, Bracho and Neumann-Lara in, we provide an
example of a saturated, i.e. inclusion-maximal 3-forest that is not tight. This
resolves an open problem posed by Strausz
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
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