Exact Algorithms for Treewidth and Minimum Fill-In

Abstract

We show that the treewidth and the minimum fill-in of an n-vertex graph can be computed in time O(1.8899 n ). Our results are based on combinatorial proofs that an n-vertex graph has O(1.7087 n ) minimal separators and O(1.8135 n ) potential maximal cliques. We also show that for the class of AT-free graphs the running time of our algorithms can be reduced to O(1.4142 n ).

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

“…E.g.,Treewidth, Minimum Fill-in [12], their weighted versions [2,18] and several problems related to phylogeny [18], or Treelength [20]. Pipelined with our main combinatorial result, we deduce that all these problems can be solved in time O * (4 vc ) or O * (1.7347 mw ).…”

“…Recall that the O * notation suppresses polynomial factors in the size of the input, i.e., O * (f (k)) should be read as f (k) · n O (1) where n is the number of vertices of the input graph. Minimal separators and potential maximal cliques have been used for solving several classical optimization problems, e.g., Treewidth, Minimum Fill-In [12], Longest Induced Path, Feedback Vertex Set or Independent Cycle Packing [13]. Pipelined with our combinatorial bounds, we obtain a series of algorithmic consequences in the area of FPT algorithms parameterized by the vertex cover and the modular width of the input graph.…”

A Parameterized Algorithm for Chordal Sandwich

“…It arises in particular in sparse matrix computations [16] and in perfect phylogeny since it has the problem of triangulating colored graphs as a special case [1,2]. It can also be seen as a generalization of the problems of adding or deleting edges in a minimum or minimal way in an arbitrary input graph to obtain a chordal graph, which have attracted considerable attention [13,14,19,20,23,24,26,27]. The NP-completeness of the problem follows from the results of several papers [1,6,32].…”

“…The construction is simple, and can for instance be found in [13]. It is thus interesting to see that the set F of admissible edges implies a strong enough restriction to be able to bound the number of such objects that are useful in a solution to a function that is only exponential in the size of the minimum vertex cover of F .…”

“…In particular, we show that the number of minimal separators and potential maximal cliques that can be useful to achieve a solution is bounded by O(2 k n 2 ). Combining this with a similar approach to that of [13] in such a way that we only consider potential maximal cliques that can lead to a solution, gives the running time O(2 k n 5 ).…”