Tree decompositions of graphs: Saving memory in dynamic programming

AbstractWe propose a simple and effective heuristic to save memory in dynamic programming on tree decompositions when solving graph optimization problems. The introduced “anchor technique” is based on a tree-like set covering problem. We substantiate our findings by experimental results. Our strategy has negligible computational overhead concerning running time but achieves memory savings for nice tree decompositions and path decompositions between 60% and 98%

Technical Communications of ICLP

Abstract Dynamic programming (DP) on tree decompositions is a well studied approach for solving hard problems efficiently. State-of-the-art implementations usually rely on tables for storing information, and algorithms specify how the tuples are manipulated during traversal of the decomposition. However, a major bottleneck of such table-based algorithms is relatively high memory consumption. The goal of the doctoral thesis herein discussed is to mitigate performance and memory shortcomings of such algorithms. The idea is to replace tables with an efficient data structure that no longer requires to enumerate intermediate results explicitly during the computation. To this end, Binary Decision Diagrams (BDDs) and related concepts are studied with respect to their applicability in this setting. Besides native support for efficient storage, from a conceptual point of view BDDs give rise to an alternative approach of how DP algorithms are specified. Instead of tuple-based manipulation operations, the algorithms are specified on a logical level, where sets of models can be conjointly updated. The goal of the thesis is to provide a general tool-set for problems that can be solved efficiently via DP on tree decompositions

Algorithms for propositional model counting

AbstractWe present algorithms for the propositional model counting problem #SAT. The algorithms utilize tree decompositions of certain graphs associated with the given CNF formula; in particular we consider primal, dual, and incidence graphs. We describe the algorithms coherently for a direct comparison and with sufficient detail for making an actual implementation reasonably easy. We discuss several aspects of the algorithms including worst-case time and space requirements

Memory Requirements for Table Computations in Partial k-Tree Algorithms

This paper addresses memory requirement issues arising in implementations of algorithms on graphs of bounded treewidth. Such dynamic programming algorithms require a large data table for each vertex of a treedecomposition T of the input graph. We give a linear-time algorithm that finds the traversal order of T minimizing the number of tables stored simultaneously. We show that this minimum value is lower-bounded by the pathwidth of T plus one, and upper bounded by twice the pathwidth of T plus one. We also give a linear-time algorithm finding the depth-first traversal order minimizing the sum of the sizes of tables stored simultaneously