Computing hypergraph width measures exactly

Hypergraph width measures are a class of hypergraph invariants important in studying the complexity of constraint satisfaction problems (CSPs). We present a general exact exponential algorithm for a large variety of these measures. A connection between these and tree decompositions is established. This enables us to almost seamlessly adapt the combinatorial and algorithmic results known for tree decompositions of graphs to the case of hypergraphs and obtain fast exact algorithms. As a consequence, we provide algorithms which, given a hypergraph H on n vertices and m hyperedges, compute the generalized hypertree-width of H in time O*(2^n) and compute the fractional hypertree-width of H in time O(m*1.734601^n).Comment: 12 pages, 1 figur

Computing hypergraph width measures exactly

Hypergraph width measures are a class of hypergraph invariants important in studying the complexity of constraint satisfaction problems (CSPs). We present a general exact exponential algorithm for a large variety of these measures. A connection between these and tree decompositions is established. This enables us to almost seamlessly adapt the combinatorial and algorithmic results known for tree decompositions of graphs to the case of hypergraphs and obtain fast exact algorithms. As a consequence, we provide algorithms which, given a hypergraph H on n vertices and m hyperedges, compute the generalized hypertree-width of H in time O*(2n) and compute the fractional hypertree-width of H in time O(1.734601n.m).

Solving Graph Problems via Potential Maximal Cliques : An Experimental Evaluation of the Bouchitté-Todinca Algorithm

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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