Speeding up Dynamic Programming for Some NP-Hard Graph Recoloring Problems

Abstract

Abstract. A vertex coloring of a tree is called convex if each color induces a connected component. The NP-hard Convex Recoloring problem on vertex-colored trees asks for a minimum-weight change of colors to achieve a convex coloring. For the non-uniformly weighted model, where the cost of changing a vertex v to color c depends on both v and c, we improve the running time on trees from O(∆ κ ·κn) to O(3 κ ·κn), where ∆ is the maximum vertex degree of the input tree T , κ is the number of colors, and n is the number of vertices in T . In the uniformly weighted case, where costs depend only on the vertex to be recolored, one can instead parameterize on the number of bad colors β ≤ κ, which is the number of colors that do not already induce a connected component. Here, we improve the running time from O(∆ β · βn) to O(3 β · βn). For the case where the weights are integers bounded by M , using fast subset convolution, we further improve the running time with respect to the exponential part to O(2 κ · κ 4 n 2 M log 2 (nM )) and O(2 β · β 4 n 2 M log 2 (nM )), respectively. Finally, we use fast subset convolution to improve the exponential part of the running time of the related 1-Connected Coloring Completion problem.

Fast Polynomial-Space Algorithms Using Inclusion-Exclusion

“…In this section we solve a generalization of the CONVEX RECOLORING problem studies in [19] and improve upon one of their results. Let T = (V , E) be a tree and…”

“…It is worth mentioning that in [19] the slightly different CONVEX TREE RECOLORING (CTR) was studied, but that CTR is a special case of CTC.…”

“…(A special case of) the CONVEX TREE COLORING problem was studied in [19], where a O (2 k c) time and O (c) space algorithm was given. We will continue this study by emphasizing the space usage aspect.…”

Reconciling taxonomy and phylogenetic inference: formalism and algorithms for describing discord and inferring taxonomic roots

“…Note that this sort of bound is different than those described above, as OPT can get large even when the total number of bad colors is small. The work for the general case culminated in the work of Ponta, Hüffner, and Niedermeier [ 9 ], who use the childwise iterative approach to dynamic programing to construct an algorithm of complexity O ( n τ 3 τ ).…”

Quadratic Kernelization for Convex Recoloring of Trees

“…Further improvements on the running time and faster algorithms for weighted variants, including a dynamic programming algorithm that uses linear time for fixed k, were obtained by Ponta et al [24]. A different proof that the problem can be solved in linear time for fixed k can be found in [6].…”