Width, Depth, and Space: Tradeoffs between Branching and Dynamic Programming

• Published in: Algorithms Vol. 11; no. 7; p. 98
• Main Authors: Chen, Li-Hsuan, Reidl, Felix, Rossmanith, Peter, Sánchez Villaamil, Fernando
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.07.2018
• Subjects: Algorithms; branching algorithm; Coloring; Computer science; Decomposition; Dynamic programming; Graphs; Run time (computers); space lower bound; Theory; treedepth; treewidth; Denmark; Aachen Germany; United States > US; Germany
• DOI: 10.3390/a11070098
• ISSN: 1999-4893

Treedepth is a well-established width measure which has recently seen a resurgence of interest. Since graphs of bounded treedepth are more restricted than graphs of bounded tree- or pathwidth, we are interested in the algorithmic utility of this additional structure. On the negative side, we show with a novel approach that the space consumption of any (single-pass) dynamic programming algorithm on treedepth decompositions of depth d cannot be bounded by (2−ϵ)d·logO(1)n for Vertex Cover, (3−ϵ)d·logO(1)n for 3-Coloring and (3−ϵ)d·logO(1)n for Dominating Set for any ϵ>0. This formalizes the common intuition that dynamic programming algorithms on graph decompositions necessarily consume a lot of space and complements known results of the time-complexity of problems restricted to low-treewidth classes. We then show that treedepth lends itself to the design of branching algorithms. Specifically, we design two novel algorithms for Dominating Set on graphs of treedepth d: A pure branching algorithm that runs in time dO(d2)·n and uses space O(d3logd+dlogn) and a hybrid of branching and dynamic programming that achieves a running time of O(3dlogd·n) while using O(2ddlogd+dlogn) space.