Width, depth and space

The width measure treedepth, also known as vertex ranking, centered coloring and elimination tree height, is a well-established notion which has recently seen a resurgence of interest. Since graphs of bounded treedepth are more restricted than graphs of bounded treeor pathwidth, we are interested in the algorithmic utility of this additional structure. On the negative side, we show with a novel approach that every dynamic programming algorithm on treedepth decompositions of depth t cannot solve Dominating Set with O ( (3 − ) · logn ) space for any > 0. This result implies the same space lower bound for dynamic programming algorithms on tree and path decompositions. We supplement this result by showing a space lower bound of O ( (3 − ) · logn ) for 3-Coloring and O ( (2 − ) · logn ) for Vertex Cover. 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 [16]. We then show that treedepth lends itself to the design of branching algorithms. This class of algorithms has in general distinct advantages over dynamic programming algorithms:a) They use less space than algorithms based on dynamic programming, b) they are easy to parallelize and c) they provide possible solutions before terminating, i.e. they can be used to derive heuristics by using preliminary solutions. Specifically, we design two novel algorithms for Dominating Set on graphs of treedepth t: a pure branching algorithm that runs in time t ) ·n and uses space O(t log t+t logn) and a hybrid of branching and dynamic programming that achieves a running time of O(3 log t · n) while using O(2t log t+ t logn) space. Algorithms for 3-Coloring and Vertex Cover with space complexity O(t · logn) and time complexity O(3 · n) and O(2 · n), respectively, are included for