On the Enumeration of Tree Decompositions

Many intractable computational problems on graphs admit tractable algorithms when applied to trees or forests. Tree decomposition extracts a tree structure from a graph by grouping nodes into bags, where each bag corresponds to a single node in of the tree. The corresponding operation on hypergraphs is that of a generalized hypertree decomposition [10], which entails a tree decomposition of the primal graph (which has the same set of nodes, and an edge between every two nodes that co-occur in a hyperedge) and an assignment of a hyperedge cover to each bag [11]. Tree decomposition and generalized hypertree decomposition have a plethora of applications, including join optimization in databases [7, 10, 21], constraint-satisfaction problems [17], computation of Nash equilibria in games [10], analysis of probabilistic graphical models [18], and weighted model counting [16,19]. Past research has focused on obtaining a “good” tree decomposition for the given graph, where goodness is typically measured by means of the width—the maximal cardinality of a bag. Nevertheless, finding a tree decomposition of a minimal width is NP-hard [2]. Moreover, in various applications the measure of goodness is different from (though related to) the width [11,16]. Abseher et al. [1] empirically showed that the execution cost of dynamic programming algorithms over a tree decomposition is highly sensitive to features of the tree decomposition other than mere width; in particular, tree decompositions of the same width may entail highly diverging running times on the same problem instance. In this paper, we describe our ongoing effort on the task of enumerating all (or a subset of) the tree decompositions of a graph. Such algorithms have been proposed in the past for small graphs (representing database queries), without complexity guarantees [15, 21]. Our main result so far is an enumeration algorithm that runs in incremental polynomial time, and our current efforts are on a practical and effective implementation.