Hypertree width and related hypergraph invariants

Tree-width of graphs is a well studied notion, which plays an important role in structural graph theory and has many algorithmic applications. Various other graph invariants are known to be the same or within a constant factor of tree-width, for example, the bramble number or tangle number of a graph [4, 5], the branch-width [5], the linkedness [4], and the number of cops required to win the robber and cops game on the graph [6]. Several of these notions may be viewed as measures for the global connectivity of a graph. The various equivalent characterisations of tree-width show that it is a natural and robust notion. Formally, let us call two graph or hypergraph invariants I and J equivalent if they are within a constant factor of each other, that is, if there are constants c, d > 0 such that for all graphs or hypergraphs G we have c · I(G) ≤ J(G) ≤ d · I(G). Tree decompositions and tree-width can be generalized to hypergraphs in a straightforward manner; the tree-width of a hypergraph is equal to the tree-width of its primal graph. Motivated by algorithmic problems from database theory and artificial intelligence, Gottlob, Leone, and Scarcello [2] introduced the hypertree-width of a hypergraph. Hypertree-width is based on the same tree decompositions as treewidth, but the width is measured differently. Essentially, the hypertree-width is the minimum number of hyperedges needed to cover all blocks of a tree decomposition. The blocks (or bags, parts) of a tree decomposition (T, (Bt)t∈V (T )) of a hypergraph H = (V,E) are the sets Bt ⊆ V for the tree-nodes t, and a block Bt is covered by the hyperedges e1, . . . , ek ∈ E if Bt ⊆ ⋃k i=1 ei. Hypertree-width is always †Georg Gottlob’s work was supported by the Austrian Science Fund (FWF) under Project N. P17222-N04 ‘Complementary Approaches to Constraint Satisfaction’.