Space Saving by Dynamic Algebraization

TL;DR

This paper introduces a polynomial-space dynamic programming algorithm based on algebraic techniques and tree decompositions, optimizing space complexity while maintaining efficient computation for problems like counting perfect matchings.

Contribution

It presents a novel method to construct tree decompositions and extend algebraic techniques, achieving polynomial space complexity with competitive time bounds.

Findings

Outperforms existing polynomial-space algorithms for counting perfect matchings on grids.

Applies to various set covering and partitioning problems with improved efficiency.

Runs in time $O^*(2^h)$, where $h$ relates to the graph's tree-depth.

Abstract

Dynamic programming is widely used for exact computations based on tree decompositions of graphs. However, the space complexity is usually exponential in the treewidth. We study the problem of designing efficient dynamic programming algorithm based on tree decompositions in polynomial space. We show how to construct a tree decomposition and extend the algebraic techniques of Lokshtanov and Nederlof such that the dynamic programming algorithm runs in time $O^*(2^h)$, where $h$ is the maximum number of vertices in the union of bags on the root to leaf paths on a given tree decomposition, which is a parameter closely related to the tree-depth of a graph. We apply our algorithm to the problem of counting perfect matchings on grids and show that it outperforms other polynomial-space solutions. We also apply the algorithm to other set covering and partitioning problems.