On Cutwidth Parameterized by Vertex Cover

Abstract

We study the CUTWIDTH problem, where the input is a graph G, and the objective is find a linear layout of the vertices that minimizes the maximum number of edges intersected by any vertical line inserted between two consecutive vertices. We give an algorithm for CUTWIDTH with running time O(2 k n O(1) ). Here k is the size of a minimum vertex cover of the input graph G, and n is the number of vertices in G. Our algorithm gives an O(2 n/2 n O(1) ) time algorithm for CUTWIDTH on bipartite graphs as a corollary. This is the first non-trivial exact exponential time algorithm for CUTWIDTH on a graph class where the problem remains NP-complete. Additionally, we show that CUTWIDTH parameterized by the size of the minimum vertex cover of the input graph does not admit a polynomial kernel unless NP ⊆ coNP/poly. Our ker- (ICALP, Springer, Berlin, 2011; SWAT, Springer, Berlin, 2012) that both TREEWIDTH and PATHWIDTH parameterized by vertex cover do admit polynomial kernels.

Computing Tree-Depth Faster Than 2 n

“…On the other hand, for several other vertex permutation problems no such algorithms are known. The two natural problems to attack are (i) the computation of cutwidth, and (ii) the Minimum Feedback Arc Set in Digraph problem; see [3,5] for definitions and details. It is known that the cutwidth of a graph can be computed in time O * (2 t ), where t is the size of a vertex cover in the graph [5]; thus the problem is solvable in time O * (2 n/2 ) on bipartite graphs.…”

Algorithms Parameterized by Vertex Cover and Modular Width, Through Potential Maximal Cliques

“…Recently Chapelle et al [6] provided an algorithm solving Treewidth and Pathwidth in O * (3 vc ), but those completely different techniques do not seem to work for Minimum Fill-in or Treelength. The interested reader may also refer., e.g., to [10,11] for more (layout) problems parameterized by vertex cover.…”

Treewidth and Pathwidth Parameterized by the Vertex Cover Number

“…Using the size of the vertex cover as a parameter when analyzing algorithms and solving problems is not a new idea. Some examples from the literature are an O * (2 k ) algorithm for CUTWIDTH parameterized by vertex cover [9], an O * (2 k ) algorithm for CHORDAL GRAPH sandwich parameterized by the vertex cover of an edge set [15], and different variants of graph layout problems parameterized by vertex cover [11].…”