Space Saving by Dynamic Algebraization

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 effi-cient dynamic programming algorithm based on tree decompositions in poly-nomial space. We show how to construct a tree decomposition and extend the algebraic techniques of Lokshtanov and Nederlof [18] such that the dy-namic programming algorithm runs in time O∗(2h), 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 [21]. We apply our algorithm to the problem of counting perfect matchings on grids and show that it outperforms other polynomial-space so-lutions. We also apply the algorithm to other set covering and partitioning problems