A Relation between the Protocol Partition Number and the Quasi-Additive Bound

In this note, we show that the linear programming for computing the quasi-additive bound of the formula size of a Boolean function presented by Ueno [MFCS'10] is equivalent to the dual problem of the linear programming relaxation of an integer programming for computing the protocol partition number. Together with the result of Ueno [MFCS'10], our results imply that there exists no gap between our integer programming for computing the protocol partition number and its linear programming relaxation

A Matroid Generalization of the Super-Stable Matching Problem

A super-stable matching, which was introduced by Irving, is a solution concept in a variant of the stable matching problem in which the preferences may contain ties. Irving proposed a polynomial-time algorithm for the problem of finding a super-stable matching if a super-stable matching exists. In this paper, we consider a matroid generalization of a super-stable matching. We call our generalization of a super-stable matching a super-stable common independent set. This can be considered as a generalization of the matroid generalization of a stable matching for strict preferences proposed by Fleiner. We propose a polynomial-time algorithm for the problem of finding a super-stable common independent set if a super-stable common independent set exists

On the Complexity of Stable Fractional Hypergraph Matching

In this paper, we consider the complexity of the problem of finding a stable fractional matching in a hypergraphic preference system. Aharoni and Fleiner proved that there exists a stable fractional matching in every hypergraphic preference system. Furthermore, Kintali, Poplawski, Rajaraman, Sundaram, and Teng proved that the problem of finding a stable fractional matching in a hypergraphic preference system is PPAD-complete. In this paper, we consider the complexity of the problem of finding a stable fractional matching in a hypergraphic preference system whose maximum degree is bounded by some constant. The proof by Kintali, Poplawski, Rajaraman, Sundaram, and Teng implies the PPAD-completeness of the problem of finding a stable fractional matching in a hypergraphic preference system whose maximum degree is 5. In this paper, we prove that (i) this problem is PPAD-complete even if the maximum degree is 3, and (ii) if the maximum degree is 2, then this problem can be solved in polynomial time. Furthermore, we prove that the problem of finding an approximate stable fractional matching in a hypergraphic preference system is PPAD-complete

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

A parameterized view to the robust recoverable base problem of matroids under structural uncertainty

We study a robust recoverable version of the matroid base problem where the uncertainty is imposed on combinatorial structures rather than on weights as studied in the literature. We prove that the problem is NP-hard even when a given matroid is uniform or graphic. On the other hand, we prove that the problem is fixed-parameter tractable with respect to the number of scenarios

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

Hardness of Finding Combinatorial Shortest Paths on Graph Associahedra

We prove that the computation of a combinatorial shortest path between two vertices of a graph associahedron, introduced by Carr and Devadoss, is NP-hard. This resolves an open problem raised by Cardinal. A graph associahedron is a generalization of the well-known associahedron. The associahedron is obtained as the graph associahedron of a path. It is a tantalizing and important open problem in theoretical computer science whether the computation of a combinatorial shortest path between two vertices of the associahedron can be done in polynomial time, which is identical to the computation of the flip distance between two triangulations of a convex polygon, and the rotation distance between two rooted binary trees. Our result shows that a certain generalized approach to tackling this open problem is not promising. As a corollary of our theorem, we prove that the computation of a combinatorial shortest path between two vertices of a polymatroid base polytope cannot be done in polynomial time unless P = NP. Since a combinatorial shortest path on the matroid base polytope can be computed in polynomial time, our result reveals an unexpected contrast between matroids and polymatroids