Polynomial-Delay Enumeration of Large Maximal Common Independent Sets in Two Matroids

Finding a maximum cardinality common independent set in two matroids (also known as Matroid Intersection) is a classical combinatorial optimization problem, which generalizes several well-known problems, such as finding a maximum bipartite matching, a maximum colorful forest, and an arborescence in directed graphs. Enumerating all maximal common independent sets in two (or more) matroids is a classical enumeration problem. In this paper, we address an ``intersection'' of these problems: Given two matroids and a threshold $\tau$, the goal is to enumerate all maximal common independent sets in the matroids with cardinality at least $\tau$. We show that this problem can be solved in polynomial delay and polynomial space. We also discuss how to enumerate all maximal common independent sets of two matroids in non-increasing order of their cardinalities

Finding Optimal Triangulations Parameterized by Edge Clique Cover

Publisher Copyright: © 2022, The Author(s).We consider problems that can be formulated as a task of finding an optimal triangulation of a graph w.r.t. some notion of optimality. We present algorithms parameterized by the size of a minimum edge clique cover (cc) to such problems. This parameterization occurs naturally in many problems in this setting, e.g., in the perfect phylogeny problem cc is at most the number of taxa, in fractional hypertreewidth cc is at most the number of hyperedges, and in treewidth of Bayesian networks cc is at most the number of non-root nodes. We show that the number of minimal separators of graphs is at most 2 cc, the number of potential maximal cliques is at most 3 cc, and these objects can be listed in times O∗(2 cc) and O∗(3 cc) , respectively, even when no edge clique cover is given as input; the O∗(·) notation omits factors polynomial in the input size. These enumeration algorithms imply O∗(3 cc) time algorithms for problems such as treewidth, weighted minimum fill-in, and feedback vertex set. For generalized and fractional hypertreewidth we give O∗(4 m) time and O∗(3 m) time algorithms, respectively, where m is the number of hyperedges. When an edge clique cover of size cc′ is given as a part of the input we give O∗(2cc′) time algorithms for treewidth, minimum fill-in, and chordal sandwich. This implies an O∗(2 n) time algorithm for perfect phylogeny, where n is the number of taxa. We also give polynomial space algorithms with time complexities O∗(9cc′) and O∗(9cc+O(log2cc)) for problems in this framework.Peer reviewe