Minimum Size Tree-decompositions

Abstract

Abstract. Tree-Decompositions are the corner-stone of many dynamic programming algorithms for solving graph problems. Since the complexity of such algorithms generally depends exponentially on the width (size of the bags) of the decomposition, much work has been devoted to compute treedecompositions with small width. However, practical algorithms computing tree-decompositions only exist for graphs with treewidth less than 4. In such graphs, the time-complexity of dynamic programming algorithms based on tree-decompositions is dominated by the size (number of bags) of the treedecompositions. It is then interesting to try to minimize the size of the tree-decompositions. In this extended abstract, we consider the problem of computing a tree-decomposition of a graph with width at most k and minimum size. More precisely, we focus on the following problem: given a fixed k ≥ 1, what is the complexity of computing a tree-decomposition of width at most k with minimum size in the class of graphs with treewidth at most k? We prove that the problem is NP-complete for any fixed k ≥ 4 and polynomial for k ≤ 2. On going work also suggests it is polynomial for k = 3.

On Tradeoffs Between Width- and Fill-like Graph Parameters

“…The treewidth counterpart of the 'length' minimization problem can be defined two-fold. It can be seen as minimizing, besides of the width of a tree decomposition, the number of its bags [27]. On the other hand, minimization of width and the diameter of the underlying tree-structure of the decomposition has been studied in [6,9].…”

“…Not much is known in terms of two-criteria optimization in the graph searching games. To mention some, examples, there is an analysis of simultaneous minimization of time (number of 'parallel' steps) and the number of searchers for the visible variant [14] and for the inert one [27] of the node search. Also, some results on tradeoffs between the cost and the number of searchers for the edge search game can be found in [13].…”

Subexponential Time Algorithms for Finding Small Tree and Path Decompositions

“…Also, it is well known that the problem to decide if the treewidth or pathwidth of a graph is bounded by a given number k is fixed parameter tractable. However, recent results show that if we ask to minimize the number of bags of the tree or path decomposition of width at most k, then the problem becomes para-NP-complete (i.e., NP-complete for some fixed k) as shown in [6,9].…”

“…Li et al [9] introduced the MSTD k problem: given a graph G and integer , does G have a tree decomposition of width at most k and with at most bags. They show the problem to be NP-complete for k ≥ 4 and for k ≥ 5 for connected graphs, with a proof similar to that of Dereniowski et al [6] for the pathwidth case, and show that the problem can be solved in polynomial time when k ≤ 2.…”