Treewidth and Pathwidth parameterized by the vertex cover number

Chapelle, Mathieu; Liedloff, Mathieu; Todinca, Ioan; Villanger, Yngve

doi:10.1016/j.dam.2014.12.012

Cited by 3 publications

(1 citation statement)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For fractional hypertreewidth there are parameterized algorithms whose parameters depend on the sizes of intersections of hyperedges [13]. For treewidth and chordal sandwich, different techniques have been used to obtain an O * (3 vc ) time algorithm for treewidth [8] and an O * (2 vc ) time algorithm for chordal sandwich, where vc is the size of a minimum vertex cover of the admissible edge set [23].…”

Section: Related Workmentioning

confidence: 99%

Finding Optimal Triangulations Parameterized by Edge Clique Cover

Korhonen

2022

Algorithmica

View full text Add to dashboard Cite

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 ($$\texttt {cc}$$ cc ) to such problems. This parameterization occurs naturally in many problems in this setting, e.g., in the perfect phylogeny problem $$\texttt {cc}$$ cc is at most the number of taxa, in fractional hypertreewidth $$\texttt {cc}$$ cc is at most the number of hyperedges, and in treewidth of Bayesian networks $$\texttt {cc}$$ cc is at most the number of non-root nodes. We show that the number of minimal separators of graphs is at most $$2^\texttt {cc}$$ 2 cc , the number of potential maximal cliques is at most $$3^\texttt {cc}$$ 3 cc , and these objects can be listed in times $$O^*(2^\texttt {cc})$$ O ∗ ( 2 cc ) and $$O^*(3^\texttt {cc})$$ O ∗ ( 3 cc ) , respectively, even when no edge clique cover is given as input; the $$O^*(\cdot )$$ O ∗ ( · ) notation omits factors polynomial in the input size. These enumeration algorithms imply $$O^*(3^\texttt {cc})$$ 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 $$O^*(4^m)$$ O ∗ ( 4 m ) time and $$O^*(3^m)$$ O ∗ ( 3 m ) time algorithms, respectively, where m is the number of hyperedges. When an edge clique cover of size $$\texttt {cc}'$$ cc ′ is given as a part of the input we give $$O^*(2^{\texttt {cc}'})$$ O ∗ ( 2 cc ′ ) time algorithms for treewidth, minimum fill-in, and chordal sandwich. This implies an $$O^*(2^n)$$ O ∗ ( 2 n ) time algorithm for perfect phylogeny, where n is the number of taxa. We also give polynomial space algorithms with time complexities $$O^*(9^{\texttt {cc}'})$$ O ∗ ( 9 cc ′ ) and $$O^*(9^{\texttt {cc}+ O(\log ^2 \texttt {cc})})$$ O ∗ ( 9 cc + O ( log 2 cc ) ) for problems in this framework.

show abstract

Section: Related Workmentioning

confidence: 99%