Approximating Treewidth, Pathwidth, Frontsize, and Shortest Elimination Tree

Inapproximability of Treewidth and Related Problems

“…As mentioned earlier, determining the exact Treewidth of a graph and producing an associated optimal tree decomposition (see Definition 2.1) is known to be NP-hard (Arnborg et al, 1987), and a central open problem is to determine whether or not there exists a polynomial time constant factor approximation algorithm for Treewidth (see e.g., Bodlaender et al, 1995;Feige et al, 2005;Bodlaender, 2005). The current best polynomial time approximation algortihm for Treewidth (Feige et al, 2005), computes the Treewidth tw(G) within a factor O( log tw(G)).…”

“…The current best polynomial time approximation algortihm for Treewidth (Feige et al, 2005), computes the Treewidth tw(G) within a factor O( log tw(G)). On the other hand, the only hardness result to date for Treewidth shows that it is NP-hard to compute Treewidth within an additive error of n for some > 0 (Bodlaender et al, 1995). No hardness of approximation is known and not even the possibility of a polynomial-time approximation scheme for Treewidth has been ruled out.…”

“…The Treewidth of a graph (Robertson & Seymour, 1984) is a fundamental parameter of a graph that measures how close the graph is to being a tree. Treewidth is very closely related to the other notions in machine learning such as Branch-width, Clique-width and Elimination-width (for an overview of Treewidth and related notions, see Bodlaender, Gilbert, Hafsteinsson, & Kloks, 1995). On graphs with small Treewidth and where the tree decomposition is known, a dynamic programming algorithm yields a polynomial-time algorithm.…”

“…However, Bodlaender et al (1995) obtained an O(log n) factor approximation algorithm for Treewidth. In fact, they actually show that if there is a factor c approximation algorithm for vertex separator, then there is an O(c) approximation algorithm for Treewidth.…”

On the parameterized complexity of monotone and antimonotone weighted circuit satisfiability

“…First, using a similar proof to that of Lemma 4.6, we can show that a circuit whose genus is n o(1) and having at most 2k critical gates has treewidth n o (1) . Using the result of Bodlaender et al [5], we can compute a tree decomposition of the underling graph of the circuit of treewidth O(n o(1) · log n).…”