Tree Projections and Structural Decomposition Methods: Minimality and Game-Theoretic Characterization

by   Gianluigi Greco, et al.

Tree projections provide a mathematical framework that encompasses all the various (purely) structural decomposition methods that have been proposed in the literature to single out classes of nearly-acyclic (hyper)graphs, such as the tree decomposition method, which is the most powerful decomposition method on graphs, and the (generalized) hypertree decomposition method, which is its natural counterpart on arbitrary hypergraphs. The paper analyzes this framework, by focusing in particular on "minimal" tree projections, that is, on tree projections without useless redundancies. First, it is shown that minimal tree projections enjoy a number of properties that are usually required for normal form decompositions in various structural decomposition methods. In particular, they enjoy the same kind of connection properties as (minimal) tree decompositions of graphs, with the result being tight in the light of the negative answer that is provided to the open question about whether they enjoy a slightly stronger notion of connection property, defined to speed-up the computation of hypertree decompositions. Second, it is shown that tree projections admit a natural game-theoretic characterization in terms of the Captain and Robber game. In this game, as for the Robber and Cops game characterizing tree decompositions, the existence of winning strategies implies the existence of monotone ones. As a special case, the Captain and Robber game can be used to characterize the generalized hypertree decomposition method, where such a game-theoretic characterization was missing and asked for. Besides their theoretical interest, these results have immediate algorithmic applications both for the general setting and for structural decomposition methods that can be recast in terms of tree projections.


page 1

page 2

page 3

page 4


Lean tree-cut decompositions: obstructions and algorithms

The notion of tree-cut width has been introduced by Wollan [The structur...

Optimal Tree Decompositions Revisited: A Simpler Linear-Time FPT Algorithm

In 1996, Bodlaender showed the celebrated result that an optimal tree de...

Treewidth versus clique number. III. Tree-independence number of graphs with a forbidden structure

We continue the study of (tw,ω)-bounded graph classes, that is, heredita...

On The Power of Tree Projections: Structural Tractability of Enumerating CSP Solutions

The problem of deciding whether CSP instances admit solutions has been d...

Incremental Updates of Generalized Hypertree Decompositions

Structural decomposition methods, such as generalized hypertree decompos...

Minimal Unimodal Decompositions on Trees

The decomposition of a density function on a domain into a minimal sum o...

The enumeration of meet-irreducible elements based on hierarchical decompositions of implicational bases

We study the well-known problem of translating between two representatio...

Please sign up or login with your details

Forgot password? Click here to reset