# Graph Motif Problems Parameterized by Dual

## TL;DR

This paper investigates the computational complexity of graph motif problems parameterized by the dual parameter, providing tight bounds and hardness results for various problem variants on general graphs and trees.

## Contribution

It establishes tight complexity bounds for GM, CGM, and LGM problems parameterized by the dual, including optimal algorithms and hardness results, and analyzes kernelization on trees.

## Key findings

- CGM has no $(2-\epsilon)^ ext{ell} imes |V|^{O(1)}$-time algorithm assuming ETH.
- LGM is W[1]-hard with respect to $ ext{ell}$ even with lists of size two.
- GM and CGM are solvable in fixed-parameter time on trees with different complexities.

## Abstract

Let $G=(V,E)$ be a vertex-colored graph, where $C$ is the set of colors used to color $V$. The Graph Motif (or GM) problem takes as input $G$, a multiset $M$ of colors built from $C$, and asks whether there is a subset $S\subseteq V$ such that (i) $G[S]$ is connected and (ii) the multiset of colors obtained from $S$ equals $M$. The Colorful Graph Motif (or CGM) problem is the special case of GM in which $M$ is a set, and the List-Colored Graph Motif (or LGM) problem is the extension of GM in which each vertex $v$ of $V$ may choose its color from a list $\mathcal{L}(v)\subseteq C$ of colors. We study the three problems GM, CGM, and LGM, parameterized by the dual parameter $\ell:=|V|-|M|$. For general graphs, we show that, assuming the strong exponential time hypothesis, CGM has no $(2-\epsilon)^\ell\cdot |V|^{\mathcal{O}(1)}$-time algorithm, which implies that a previous algorithm, running in $\mathcal{O}(2^\ell\cdot |E|)$ time is optimal [Betzler et al., IEEE/ACM TCBB 2011]. We also prove that LGM is W[1]-hard with respect to $\ell$ even if we restrict ourselves to lists of at most two colors. If we constrain the input graph to be a tree, then we show that GM can be solved in $\mathcal{O}(3^\ell\cdot |V|)$ time but admits no polynomial-size problem kernel, while CGM can be solved in $\mathcal{O}(\sqrt{2}^{\ell} + |V|)$ time and admits a polynomial-size problem kernel.