Approximating Rank-Width and Clique-Width Quickly

Abstract

Rank-width was defined by Oum and Seymour [2006. Approximating clique-width and branchwidth. J. Combin. Theory Ser. B 96, 4, 514-528] to investigate clique-width. They constructed an algorithm that either outputs a rank-decomposition of width at most f (k) for some function f or confirms that rank-width is larger than k in time O(|V | 9 log |V |) for an input graph G = (V, E) and a fixed k. We develop three separate algorithms of this kind with faster running time. We construct an O(|V | 4 )-time algorithm with f (k) = 3k + 1 by constructing a subroutine for the previous algorithm; we avoid generic algorithms minimizing submodular functions used by Oum and Seymour. Another one is an O(|V | 3 )-time algorithm with f (k) = 24k by giving a reduction from graphs to binary matroids; then we use an approximation algorithm for matroid branch-width by Hliněný [2005 INTRODUCTIONGraph complexity measures such as tree-width and branch-width are important for algorithmic purposes and for understanding the structure of families of graphs. One of them is the clique-width, defined by Courcelle and Olariu [2000]. We discuss its definition in the next section. Many NP-hard graph problems are solvable in polynomial time if a tree-like decomposition corresponding to clique-width, called Permission to make digital/hard copy of all or part of this material without fee for personal or classroom use provided that the copies are not made or distributed for profit or commercial advantage, the ACM copyright/server notice, the title of the publication, and its date appear, and notice is given that copying is by permission of the ACM, Inc. To copy otherwise, to republish, to post on servers, or to redistribute to lists requires prior specific permission and/or a fee. c 202008 ACM 0000-0000/202008/0000-0001 $5.00 2003] are not expressible in monadic second-order logic but have nevertheless polynomial-time algorithms on graphs of bounded clique-width if the input graph is given with a k-expression. Therefore we hope to have, for each fixed k, a polynomial-time algorithm to find a k-expression of an input graph if the input graph has clique-width at most k. This problem is still open when k > 3. Corneil et al. [2000] solved this problem when k = 3.Instead, Oum and Seymour [2006] found an "approximation" algorithm that either outputs a (2 3k+2 − 1)-expression or confirms that the clique-width of G is larger than k. This can be combined with algorithms requiring a k-expression and therefore those algorithms no longer have to require a k-expression as an input to be polynomial-time algorithms. To obtain this approximation algorithm, they defined another graph width parameter, called the rank-width and showed that rank-width is at most clique-width and clique-width is at most 2 1+rank-width − 1. In addition, they showed a polynomial-time algorithm to find a rank-decomposition of width 3k + 1 or to confirm that the rank-width is larger than k. (Rank-width is defined as the minimum possible width of all rank-decompositions. We will discuss its...

Connection Matrices for MSOL-Definable Structural Invariants

“…It was shown in [CO00] that T W (k) ⊆ CW (2 k+1 + 1). Using the main results of [CMR01,Mak04] combined with [Oum05] we get from Proposition 6 the following complexity result.…”

On quasi-planar graphs: Clique-width and logical description

“…Rank-width [26,38,39,40] is a graph complexity measure that is equivalent to clique-width in the sense that the same graph classes have bounded rankwidth and bounded clique-width. It provides a polynomial-time approximation algorithm for computing clique-width and clique-width terms [40].…”

Practical algorithms for MSO model-checking on tree-decomposable graphs

“…When we use the O(|V | 3 ) time approximation algorithm by Oum [145] to compute a corresponding f (w)-expression, we get the following general result for graphs of bounded clique-width:…”

“…Theorem 35 (Courcelle, Makowsky, Rotics [48], Oum [145]). Let P be an linear EMSO 1 -definable optimization problem, w ∈ N an integer and f : N → N. Then one can solve P on graphs G = (V, E) of order n := |V | and clique-width at most w in time O(n 3 ) and in time O(n) if an f (w)-expression is part of the input.…”

“…In order to design an approximation algorithm for clique-width with an approximation ratio independent of |V |, Oum and Seymour introduced in [146] a new width measure named rank-width. The O(|V | 3 ) time bound given in Theorem 35 is due to the running time of the rankwidth approximation algorithm [145], which also approximates the clique-width. 6.1.2.…”