On the Kernelization Complexity of Colorful Motifs

Abstract

The \sc Colorful Motif} problem asks if, given a vertex-colored graph G, there exists a subset S of vertices of G such that the graph induced by G on S is connected and contains every color in the graph exactly once. The problem is motivated by applications in computational biology and is also well-studied from the theoretical point of view. In particular, it is known to be NP-complete even on trees of maximum degree three~[Fellows et al, ICALP 2007]. In their pioneering paper that introduced the color-coding technique, Alon et al.~[STOC 1995] show, {\em inter alia}, that the problem is FPT on general graphs. More recently, Cygan et al.~[WG 2010] showed that {\sc Colorful Motif} is NP-complete on {\em comb graphs}, a special subclass of the set of trees of maximum degree three. They also showed that the problem is not likely to admit polynomial kernels on forests. We continue the study of the kernelization complexity of the {\sc Colorful Motif problem restricted to simple graph classes. Surprisingly, the infeasibility of polynomial kernelization persists even when the input is restricted to comb graphs. We demonstrate this by showing a simple but novel composition algorithm. Further, we show that the problem restricted to comb graphs admits polynomially many polynomial kernels. To our knowledge, there are very few examples of problems with many polynomial kernels known in the literature. We also show hardness of polynomial kernelization for certain variants of the problem on trees; this rules out a general class of approaches for showing many polynomial kernels for the problem restricted to trees. Finally, we show that the problem is unlikely to admit polynomial kernels on another simple graph class, namely the set of all graphs of diameter two. As an application of our results, we settle the classical complexity of \cds{} on graphs of diameter two --- specifically, we show that it is \NPC

Deterministic Parameterized Algorithms for the Graph Motif Problem

“…The paper [3] also shows that RGM is NP-hard when M is a set and G is a tree. Even seemingly simpler cases of RGM are known to be NP-hard (see [7,8,9]). Moreover, a natural optimization version of RGM D , minimizing the number of deletions from M , is hard to approximate within factor |V | 1 3 − [10].…”

Turing Kernelization for Finding Long Paths in Graph Classes Excluding a Topological Minor

“…Consequently, each torso again has degree bounded by ∆ and cannot contain a path of length at least k. By Theorem 5.6, if n denotes the size of such a bag, then n ε /4 + 2 < k, where ε = 1/ log 2 (max(425, α, d + 2 d ) − 1). Thus the size of each bag is n < (4(k − 2)) 1/ε = k O(d+log 2 α) = k O H (1) . Adding the ≤ |V (H)| deleted vertices back to every bag, we obtain a tree decomposition of G * root as claimed.…”

Algorithmic Aspects of Disjunctive Domination in Graphs

“…It is known that domination problem is NP-complete for split graphs [4] and for graphs with diameter two [2]. But disjunctive domination problem can be easily solved in these graph classes.…”