Computing Pathwidth Faster Than 2 n

Abstract

Abstract. Computing the Pathwidth of a graph is the problem of finding a tree decomposition of minimum width, where the decomposition tree is a path. It can be easily computed in O * (2 n ) time by using dynamic programming over all vertex subsets. For some time now there has been an open problem if there exists an algorithm computing Pathwidth with running time O * (c n ) for c < 2 † . In this paper we show that such an algorithm with c = 1.9657 exists, and that there also exists an approximation algorithm and a constant τ such that an opt + τ approximation can be obtained in O * (1.89 n ) time.

A Note on Exact Algorithms for Vertex Ordering Problems on Graphs

“…In each case, the O * (2 n ) algorithm resembles the classic Held-Karp algorithm for TSP [13], and the O * (4 n ) its variant by Gurevich and Shelah [12]. Note that for TREEWIDTH, MINIMUM FILL-IN and PATHWIDTH faster algorithms with exponential space are known [8,16], and for TREEWIDTH a faster algorithm with polynomial space is known [9,10].…”

“…Very recently, Suchan and Villanger [16] obtained a faster exact algorithm for PATH-WIDTH, i.e., using O(1.9657 n ) time and exponential space. It is open if this can be used for a faster algorithm with polynomial space.…”

On Cutwidth Parameterized by Vertex Cover

“…Most graph layout problems (with the exception of BANDWIDTH) admit an O(2 n n O (1) ) time dynamic programming algorithm [2,21]. For several of these problems, faster algorithms with running time below O(2 n ) have been found [3,16,30], a stellar example is the recent algorithm by Björklund [3] for HAMILTONIAN PATH. The CUTWIDTH problem is perhaps the best known graph layout problem for which a O(2 n n O (1) ) time algorithm is known, yet no better algorithm has been found.…”

Some Reduction Procedure for Computing Pathwidth of Undirected Graphs

“…On the other hand, nontrivial exponential algorithms are proposed. Suchan et al [14] designed an algorithm that runs in 1.9657 n n O(1) time and space, and afterwards Kitsunai et al [9] gave an algorithm that runs in 1.89 n n O (1) time and space. Robertson and Seymour [13] led to a consequence of the graph minor theorem that for a fixed k there is a polynomial time algorithm that decides whether the pathwidth of a given graph is at most k. Nagamochi [10] proposed a branching algorithm for finding a linear layout with a bounded width that minimizes a given cost function, based on which it takes O(kmn 2k ) time and O(m + n) space to test whether the pathwidth of a given digraph with n vertices and m edges is at most k.…”