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

Abstract

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 treeor 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 −) d • log O(1) n for VERTEX COVER, (3 −) d • log O(1) n for 3-COLORING and (3 −) d • log O(1) n for DOMINATING SET for any > 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 d O(d 2) • n and uses space O(d 3 log d + d log n) and a hybrid of branching and dynamic programming that achieves a running time of O(3 d log d • n) while using O(2 d d log d + d log n) space.

Lower Bounds on Dynamic Programming for Maximum Weight Independent Set

“…As formulas can be thought of as bounded space analogies of circuits, Theorem 4 gives further evidence (in addition to e.g. [9,24,31]) supporting that while treewidth is the right parameter for CSP-like problems when equipped with unlimited space, treedepth is the right parameter when dealing with bounded space.…”

A Polynomial Excluded-Minor Approximation of Treedepth

“…(These properties are also noted in [10,13].) The first lemma establishes that greedy rooted tree decompositions are, in fact, tree decompositions in the sense of Definition 2.1.…”

“…The same notion appears at least twice in the literature: in [13] under the name good treedepth decomposition and in [10] under the name reduced separation forest. An even "greedier" class of tree decompositions appears in [15] under the name minimal rooted trees.…”

“…It appears in the literature under various names including vertex ranking number [30], ordered chromatic number [17], and minimum elimination tree height [23], before being systematically studied under the name treedepth by Ossona de Mendes and Nešetřil [20]. Bounded treedepth graphs play an important role in areas such as the theory of sparse graph classes [21,22], parameterized complexity theory [13,15,24], and model theory [28,29].…”

Vertex Deletion Parameterized by Elimination Distance and Even Less

“…We either return a minimum odd cycle transversal that respects the annotations, or conclude that they cannot lead to an optimal solution for the initial graph. The idea of branching on a top-most vertex of an elimination forest has been used by, e.g., Chen et al [22] to show that Dominating set is FPT when parameterized by treedepth. They also provide branching strategies for q-coloring and Vertex cover.…”