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.

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.

Tables

Table 1. Table 1: Overview of new and previous results with respect to the dual parameter  ℓ := n − k assign ℓ 𝑛 𝑘 \ell:=n-k , where  n := | V | assign 𝑛 𝑉 n:=|V| ,  k := | M | assign 𝑘 𝑀 k:=|M| ,  m := | E | assign 𝑚 𝐸 m:=|E| and Δ := max v ∈ V ⁡ | ℒ ​ ( v ) | assign Δ subscript 𝑣 𝑉 ℒ 𝑣 \Delta:=\max_{v\in V}|\mathcal{L}(v)| denotes the maximum list size in  G 𝐺 G . The lower bound result for  CGM assumes the strong exponential time hypothesis (SETH)  [ 19 ] .

	General graphs	Trees
LGM	W[1]-hard [2]	?
LGM, $\Delta =2$	W[1]-hard (Cor. 1)	?
GM	W[1]-hard [2]	$𝒪\left(3^{\ell }\cdot n\right)$ (Thm. 4)
		no poly. kernel (Thm. 6)
CGM	$𝒪\left(2^{\ell }\cdot m\right)$ [2],	$𝒪\left({\sqrt{2}}^{\ell }+n\right)$ (Thm. 8),
	no ${\left(2-ϵ\right)}^{\ell }\cdot n^{𝒪\left(1\right)}$ (Thm. 1)
	no poly. kernel (Thm. 2)	$\left(2\ell +1\right)$-vertex kernel (Thm. 7)

Equations

$$D_v[{\lambda }_1,\dots ,{\lambda }_j]=\{\begin{array}{ll}1& \text{if Tv has a safe subtree containing for each ci, 1≤i≤j,}\\ & \text{exactly occ(Tv,ci)−λi vertices of color ci},\\ 0& \text{otherwise}.\end{array}$$

$$D_v[{\lambda }_1,\dots ,{\lambda }_j]=1\iff {\lambda }_1=\dots ={\lambda }_j=0.$$

$$D_v^i[{\lambda }_1,\dots ,{\lambda }_j]=\{\begin{array}{ll}1& \text{if Tvi has a safe subtree containing for each ci, 1≤i≤j,}\\ & \text{exactly occ(Tv,ci)−λi vertices of color ci},\\ 0& \text{otherwise}.\end{array}$$

$$D_v^1[{\lambda }_1,\dots ,{\lambda }_j]=\{\begin{array}{ll}1& \text{if Du1[λ1,…,λj]=1}\\ 1& \text{if occ(Tu1,ci)=λi and Tu1 contains only abundant colors},\\ 0& \text{otherwise}.\end{array}$$

$$D_v^i[{\lambda }_1,\dots ,{\lambda }_j]=\{\begin{array}{ll}1& \text{if Tui contains only abundant colors and}\\ & D_v^{i-1}[{\lambda }_1-\mathrm{occ}(T_{u_i},c_1),\dots ,{\lambda }_j-\mathrm{occ}(T_{u_i},c_1)]=1,\\ 1& \text{if there is (λ1′,…,λj′) such that}\\ & D_v^{i-1}[{\lambda }_1^{\prime },\dots ,{\lambda }_j^{\prime }]=D_{u_i}[{\lambda }_1-{\lambda }_1^{\prime },\dots ,{\lambda }_j-{\lambda }_j^{\prime }]=1,\\ 0& \text{otherwise.}\end{array}$$

$$\sum _{j=0}^{{\ell }_i}j+1=\sum _{j=1}^{{\ell }_i+1}j=({\ell }_i+2)\cdot ({\ell }_i+1)/2$$

$$\frac{{\prod }_{1\le i\le j}(({\ell }_i+2)\cdot ({\ell }_i+1)/2)}{({\prod }_{1\le i<j}(({\ell }_i+2)\cdot ({\ell }_i+1)/2)\cdot (({\ell }_j+1)({\ell }_j)/2)\cdot 3}=\frac{({\ell }_j+2)({\ell }_j+1)/2}{3({\ell }_j+1)({\ell }_j)/2}=\frac{{\ell }_j+2}{3({\ell }_j)}<1,$$

Full text

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 $\mathcal{O}(\sqrt{2}^{\mskip 1.5mu{}\ell}+|V|)$ time and admits a polynomial-size problem kernel.

1 Introduction

The Subgraph Isomorphism problem is the following pattern matching problem in graphs: given a (typically large) host graph $G$ and a (small) query graph $H$, return one (or all) occurrence(s) of $H$ in $G$, where the term occurrence denotes here a subset $S$ of $V(G)$ such that $G[S]$, the subgraph of $G$ induced by $S$, is isomorphic to $H$. This type of graph mining problem has different applications, notably in biology [25]. Subgraph Isomorphism is a structural graph pattern matching problem, where one looks for similar graph structures between $H$ and $G$. In some biological contexts, however, additional information is provided to the vertices of the graphs, for example their biological function. This can be modeled by labeling each vertex of the graph, for example by giving it one or several colors, each corresponding to an identified function. In the presence of such functional annotation, the structure of a given induced subgraph may be of less importance than the functions it corresponds to. Thus, a new set of functional graph pattern matching problems has emerged, starting with the Graph Motif problem [20], which was introduced in the context of the analysis of metabolic networks. In Graph Motif, the query is a multiset $M$ of colors that represents the functions of interest, and we search for an occurrence of $M$ in the host graph, where the previous demand of being isomorphic to the query is replaced by a connectivity demand.

Graph Motif (GM)

Input: A multiset $M$ built on a set $C$ of colors, an undirected graph $G=(V,E)$, and a coloring $\chi:V\to C$.

Question: Is there a set $S\subseteq V$ such that $G[S]$ is connected and there is a one-to-one mapping $f:S\to M$ such that $f(v)=\chi(v)$ for all $v\in S$?

Many variants of GM have been introduced and studied. In particular, List-Colored Graph Motif (or LGM) is a generalization of GM that is used to identify, in a given protein interaction network, protein complexes that are similar to a given protein complex from a different species [7]. In LGM, the graph $G$ is associated with a list-coloring $\mathcal{L}:V\to 2^{C}$, that is, each vertex $v$ is associated with a set $\mathcal{L}(v)$ of colors, and the question is whether there is a set $S\subseteq V$ such that (i) $G[S]$ is connected and (ii) the one-to-one mapping $f$ from $S$ to $M$ we look for satisfies $\forall v\in S:f(v)\in\mathcal{L}(v)$. The special case of GM in which $M$ is a set is called Colorful Graph Motif (or CGM). Many optimization problems related to GM have received interest, including some that are related to tandem mass spectrometry and where the input graph is directed and edge-weighted [24]. All these problem variants have given rise to a very abundant literature. CGM, GM, and LGM are NP-hard even in very restricted cases [20, 12, 6]. Consequently, many of the above-mentioned studies have focused on (dis)proving fixed-parameter tractability of the problems (see e.g. [26] for an informal survey on the topic). In such cases, very often the parameter $k:=|M|=|S|$ is considered.

In this paper, we study the parameterized complexity of GM, CGM, and LGM, but we differ from the usual viewpoint by focusing on the dual parameter $\ell:=|V|-|S|$, that is, $\ell$ is the number of vertices to be deleted from $G$ to obtain a solution. Although the choice of $\ell$ may be disputable because a priori it may be too large to expect a good behavior in practice, there are several arguments for choosing such a parameter: First, after some initial data reduction, the input may be divided into smaller connected components, where $\ell$ is not much larger than $k$. Second, the algorithms for parameter $k$ rely on algebraic techniques or dynamic programming, and in both cases, the worst-case running time is equivalent to the actual running time. In contrast, for example for CGM, the algorithm for parameter $\ell$ is a search tree algorithm [2], and search tree algorithms can be substantially accelerated via pruning rules. Finally, there are subgraph mining problems where the dual parameter $\ell$ is usually bigger than the parameter $k$ but leads to the current-best algorithm (in terms of performance on real-world instances), see e.g. [18]. Hence, parameterization by $\ell$ may be useful even if $\ell$ is bigger than $k$, and thus deserves to be studied.

Related work and our contribution.

GM is NP-hard, even when $M$ is composed of two colors [12]. Concerning the parameterized complexity for parameter $k:=|M|$, the current-best randomized algorithm has a running time of $2^{k}\cdot n^{\mathcal{O}(1)}$ [3, 23] where $n:=|V|$, and there is evidence that this cannot be improved to a running time of $(2-\epsilon)^{k}\cdot n^{\mathcal{O}(1)}$ [3]. The current-best running time for a deterministic algorithm is $5.22^{k}\cdot n^{\mathcal{O}(1)}$ [22]. GM on trees can be solved in $n^{\mathcal{O}(c)}$ time where $c$ is the number of colors in $M$ [12], but is W[1]-hard with respect to $c$ [12]. Other parameters, essentially related to the structure of the input graph $G$, have been studied by Ganian [17], Bonnet and Sikora [6], and Das et al. [9]. For example, Graph Motif is fixed-parameter tractable when parameterized by the size of a vertex cover of the input graph [17, 6]. Finally, concerning parameter $\ell$, GM has been shown to be W[1]-hard, even when $M$ is composed of two colors [2].

Since CGM is a special case of GM, any above-mentioned positive result for GM also holds for CGM. In addition, CGM is NP-hard even for trees of maximum degree 3 [12], and does not admit a polynomial-size problem kernel with respect to $k$ even if $G$ has diameter two or if $G$ is a comb graph (a special type of tree with maximum degree 3) [1]. Finally, CGM can be solved in $\mathcal{O}(2^{\ell}\cdot m)$ time [2], where $m:=|E|$. The LGM problem is an extension of GM and thus any negative result for GM propagates to LGM. Moreover, LGM is fixed-parameter tractable with respect to $k$, and the current-best algorithm runs in $2^{k}\cdot n^{\mathcal{O}(1)}$ time [23]. Concerning parameter $\ell$, LGM has been shown to be W[1]-hard even when $M$ is a set [2].

As mentioned above, we study GM, LGM and CGM with respect to the dual parameter $\ell:=n-k$. Since many results in general graphs turn out to be negative, we also study the special case where the input graph $G$ is a tree. Our results are summarized in Table 1. In a nutshell, we strengthen previous hardness results for the general case and show that the $\mathcal{O}(2^{\ell}\cdot m)$-time algorithm for CGM is essentially optimal. Then, we show that for GM on trees and for some special cases of LGM on trees, a fixed-parameter algorithm can be achieved. Finally, we show that for CGM on trees, a polynomial-size problem kernel and better running times than for general graphs can be achieved.

Preliminaries.

For an integer $n$, we use $[n]:=\{1,\ldots,n\}$ to denote the set of the integers from $1$ through $n$. Throughout the paper, the input graph for our three problems is $G=(V,E)$, and we let $n:=|V|$ (resp. $m:=|E|$) denote its number of vertices (resp. edges). For a vertex set $S\subseteq V$, we let $G[S]:=(S,\{\{u,v\}\mid u,v\in S\})$ denote the subgraph induced by $S$. The set $S$ of vertices sought for in the three problems is called an occurrence of $M$. If $G$ is vertex-colored, we call a vertex set $S$ colorful if $|S|=|M|$ and all vertices in $S$ have pairwise different colors. A vertex $v$ is called unique if it is assigned a color $c$ that is assigned to no other vertex in $V$. For a multiset $M$ and an element $c$ of $M$, we use $M(c)$ to denote the multiplicity of $c$ in $M$.

To analyze the structure of the coloring constraints for instances of LGM, we consider the following auxiliary graph.

Definition 1**.**

Let $(M,G,\mathcal{L})$ be an instance of LGM. The vertex-color graph $H$ of $(M,G,\mathcal{L})$ is the bipartite graph with vertex set $V\cup C$ and edge set $\{\{v,c\}\mid v\in V,c\in\mathcal{L}(v)\}$.

Observe that GM instances are LGM instances where in the vertex-color graph $H$ each vertex from $V$ has degree one. In other words, $H$ is a disjoint union of stars whose non-leaf is a vertex from $C$. Moreover, an LGM instance where $H$ is a disjoint union of bicliques can be easily replaced by an equivalent GM instance: For each biclique $K$ in $H$, replace the color set $K\cap C$ by one color with multiplicity $\sum_{c\in C}M(c)$ in $M$ and assign this color to all vertices in $K\cap V$.

We briefly recall the relevant notions of parameterized algorithmics [8, 11]. Parameterized algorithmics aims at analyzing the impact of structural input properties on the difficulty of computational problems. Formally, a parameterized problem $L$ is a subset of $\Sigma^{*}\times\mathbb{N}$ where the first component is the input instance and the second component is the parameter. A parameterized problem $L$ is fixed-parameter tractable if every input instance $(I,k)$ can be solved in $f(k)\cdot|I|^{\mathcal{O}(1)}$ time where $f$ is a computable function depending only on $k$. A reduction to a problem kernel, or kernelization, is an algorithm that takes as input an instance $(I,k)$ of a parameterized problem and produces in polynomial time an instance $(I^{\prime},k^{\prime})$ such that

• •

$(I,k)$ is a yes-instance if and only if $(I^{\prime},k^{\prime})$ is a yes-instance and

• •

$|I^{\prime}|\leq g(k)$ where $g$ is a computable function depending only on $k$.

The instance $(I^{\prime},k^{\prime})$ is called problem kernel and $g$ is called the size of the problem kernel. If $g$ is a polynomial function, then the problem admits a polynomial-size problem kernelization. The class W[1] is a basic class of presumed fixed-parameter intractability [8, 11], that is, if a problem is W[1]-hard for parameter $k$, then we assume that it cannot be solved in $f(k)\cdot n^{\mathcal{O}(1)}$ time [8, 11]. The strong exponential time hypothesis (SETH) assumes that, for any $\epsilon>0$, CNF-SAT cannot be solved in time $(2-\epsilon)^{n}\cdot|\Phi|^{\mathcal{O}(1)}$ where $\Phi$ is the input formula and $n$ is the number of variables [19].

This work is structured as follows. In Section 2, we present lower bounds for LGM and CGM on general graphs. These negative results motivate our study of the case when $G$ is a tree; our results for GM on trees and CGM on trees will be presented in Section 3 and Section 4, respectively. We conclude with an outlook of future work in Section 5.

2 Parameterization by Dual in General Graphs: Tight Lower Bounds

CGM can be solved in $\mathcal{O}(2^{\ell}\cdot m)$ time [2]. We show that this running time bound is essentially optimal.

Theorem 1**.**

Colorful Graph Motif* cannot be solved in $(2-\epsilon)^{\ell}\cdot n^{\mathcal{O}(1)}$ time unless the strong exponential time hypothesis (SETH) fails.*

Proof.

We present a polynomial-time reduction from CNF-SAT:

Input: A boolean formula $\Phi$ in conjunctive normal form with clauses ${\cal C}_{1},\ldots,{\cal C}_{q}$ over variable set $X=\{x_{1},\ldots,x_{r}\}$.

Question: Is there an assignment $\beta:X\to\{\texttt{true},\texttt{false}\}$ that satisfies $\Phi$?

The reduction works as follows. First, for each variable $x_{i}\in X$ introduce two variable vertices $v^{t}_{i}$ and $v^{f}_{i}$ and color each of the two vertices with color $\chi^{X}_{i}$. The idea is that (with the final occurrence) we must select exactly one vertex for this color. This selection will correspond to a truth assignment to $X$. Now, introduce for each clause ${\cal C}_{i}$ a clause vertex $u_{i}$, color $u_{i}$ with a unique color $\chi^{\cal C}_{i}$ and make $u_{i}$ adjacent to vertex $v^{t}_{j}$ if $x_{j}$ occurs nonnegated in ${\cal C}_{i}$ and to vertex $v^{f}_{j}$ if $x_{j}$ occurs negated in ${\cal C}_{i}$. Finally, introduce one further vertex $v^{*}$ with a unique color $\chi^{*}$, make $v^{*}$ adjacent to all variable vertices and let $M$ be the set containing each of the introduced colors exactly once. Note that there are exactly $|X|$ colors that appear twice in $G$ and that all other colors appear exactly once. Hence, $\ell=|X|$. We next show the correctness of the reduction. Let $I=(M,G,\chi)$ denote the constructed instance of CGM.

First, assume that $\Phi$ is satisfiable and let $\beta$ be a satisfying assignment of $X$. For the CGM instance $I$ consider the vertex set $S\subseteq V$ that contains all clause vertices, vertex $v^{*}$, and for each variable $x_{i}$ the vertex $v^{t}_{i}$ if $\beta$ sets $x_{i}$ to true and the vertex $v^{f}_{i}$ otherwise. Clearly, $|S|=|M|$ and no two vertices of $S$ have the same color. To show that $I$ is a yes-instance of CGM it remains to show that $G[S]$ is connected. First, the subgraph induced by the variable vertices in $S$ plus $v^{*}$ is a star and thus it is connected. Second, since $\beta$ is a satisfying assignment each clause vertex in $S$ has at least one neighbor in $S$ (which is by construction a variable vertex). Hence, $G[S]$ is connected.

Conversely, assume that $I$ is a yes-instance of CGM, and let $S$ be a colorful vertex set with $|S|=|M|$ such that $G[S]$ is connected. Since $S$ is colorful, the variable vertices in $S$ correspond to a truth assignment of $X$. This assignment satisfies $X$: Indeed, since $G[S]$ is connected, there is a path in $G[S]$ between each clause vertex $u_{i}$ and $v^{*}$, and thus there is a neighbor of $u_{i}$ that is in $S$. If this neighbor is $v^{t}_{j}$ (resp. $v^{f}_{j}$), then by construction, $\beta$ assigns true (resp. false) to $x_{j}$ and thus ${\cal C}_{i}$ is satisfied.

Thus, the two instances are equivalent. Now observe that since $\ell=|X|=r$ and $n=2r+q+1$, any $(2-\epsilon)^{\ell}\cdot n^{\mathcal{O}(1)}$-time algorithm implies a $(2-\epsilon)^{r}\cdot(r+q)^{\mathcal{O}(1)}$-time algorithm for CNF-SAT. This directly contradicts the SETH. ∎

The above reduction also makes the existence of a polynomial-size problem kernel for parameter $\ell$ unlikely. This is implied by the following two facts. First, CNF-SAT parameterized by the number of variables does not admit a polynomial-size problem kernel unless $\textrm{NP}\subseteq\textrm{coNP/poly}$ [10]. Second, the reduction presented in the proof of Theorem 1 is a polynomial parameter transformation [5] from CNF-SAT parameterized by the number of variables to CGM parameterized by $\ell$. More precisely, given an input CNF-SAT formula $\Phi$ on variable set $X$, the reduction produces an instance $I=(M,G,\chi)$ of CGM with $\ell=|X|$. Now, any polynomial-size problem kernelization applied to $I$ produces in polynomial time an equivalent CGM instance $I^{\prime}$ of size $\ell^{\mathcal{O}(1)}=|X|^{\mathcal{O}(1)}$. Since CNF-SAT is NP-hard, we can now transform this CGM instance in polynomial time into an equivalent CNF-SAT instance that has size $\ell^{\mathcal{O}(1)}=|X|^{\mathcal{O}(1)}$. Hence, a polynomial-size problem kernel for CGM parameterized by $\ell$ implies a polynomial-size problem kernel for CNF-SAT parameterized by $|X|$. This implies $\textrm{NP}\subseteq\textrm{coNP/poly}$ [10] (which in turn implies a collapse of the polynomial hierarchy).

Theorem 2**.**

Colorful Graph Motif* parameterized by $\ell$ does not admit a polynomial-size problem kernel unless $\text{\rm NP}\subseteq\text{\rm{coNP/poly}}$.*

We have thus resolved the parameterized complexity of CGM parameterized by $\ell$ on general graphs and now turn to the more general LGM problem, which is W[1]-hard with respect to $\ell$ [2]. Here, it would be desirable to obtain fixed-parameter algorithms for parameter $\ell$ at least for some restricted inputs. In other words, we would like to further exploit the structure of real-world instances to obtain tractability results. A very natural approach here is to consider the size and structure of the list-colorings $\mathcal{L}(v)$ as additional parameter. Unfortunately, the problem remains W[1]-hard even for the following very restricted case of list-colorings. Recall, that the vertex-color graph is the bipartite graph with vertex set $V\cup C$ in which $v\in V$ and $c\in C$ are adjacent if and only if $c\in\mathcal{L}(v)$.

Theorem 3**.**

List-Colored Graph Motif* is W[1]-hard with respect to $\ell$ even if the vertex-color graph is a disjoint union of paths.*

Proof.

We reduce from the Multicolored Independent Set problem:

Input: An undirected graph $H=(W,F)$ and a vertex-labeling $\lambda:W\to\{1,\ldots,k\}$.

Question: Is there a set $S\subseteq W$ such that $|S|=k$, the vertices in $S$ have pairwise different labels and $H[S]$ has no edges?

Multicolored Independent Set has been shown to be W[1]-hard when parameterized by $k$ [13]. We call the colors of the Multicolored Independent Set labels to avoid confusion with the colors of the List-Colored Graph Motif instance. Assume without loss of generality that each label class in $H$ contains the same number $x$ of vertices (this can be achieved by padding smaller classes with additional vertices) and that there is an arbitrary but fixed ordering of the vertices of $H$.

The reduction works as follows. We first describe the input graph $G$ for LGM. We let $V=V_{0}\cup V_{1}\cup\{v^{*}\}$, where $V_{0}=W$ and $V_{1}=\{v_{e}|e\in F(H)\}$. Now construct the edge set $E$ of $G$ as follows. First, make vertex $v^{*}$ adjacent to all vertices of $V_{0}$. Then, for each edge $\{u,w\}$ of $H$ make vertex $v_{\{u,w\}}$ adjacent to $u$ and $w$. This completes the construction of $G$. Now let us describe the coloring of the vertices. We start with the colors given to $V_{0}=W$. For each label $i$ from $\lambda$ do the following: create $x-1$ colors $c^{i}_{1},\ldots,c^{i}_{x-1}$. Now, with respect to the above-mentioned ordering, color the first vertex of label class $i$ with color $c^{i}_{1}$, color any $j$th vertex, $2\leq j\leq x-1$, with the list $\{c^{i}_{j-1},c^{i}_{j}\}$, and finally color the $x$th vertex with color $c^{i}_{x-1}$. Next, color each vertex from $V_{1}\cup\{v^{*}\}$ with a unique color. Let $\mathcal{L}$ denote the list-coloring of $V(G)$ that we just described. We define the motif $M$ as the set containing each color present in $\mathcal{L}$. Clearly, the reduction works in polynomial time. Note that $|V|=kx+|E|+1$ and $|M|=k(x-1)+|E|+1$ and thus $\ell=|V|-|M|=k$. To prove our claim, it thus remains to show the correctness of the reduction.

$(H,k)$ is a yes-instance of Multicolored Independent Set $\Leftrightarrow$ $(M,G,\mathcal{L})$ is a yes-instance of LGM.

($\Rightarrow$) Let $S$ be a size-$k$ independent set with pairwise different vertex labels in $H$. Consider the set $Y:=V\setminus S$ in $G$. First, note that $G[Y]$ is connected: vertex $v^{*}$ is adjacent to all vertices in $Y\cap V_{0}$ and each vertex $v_{\{u,w\}}$ of $Y\cap V_{1}=V_{1}$ has at least one neighbor in $Y\cap V_{0}$, because at most one of the endpoints of $\{u,w\}$ is in the independent set $S$.

It remains to show that we can assign colors to the vertices such that the union of the vertex colors is $M$. All vertices with unique colors are contained in $Y$ and their coloring is clear. All other vertices are in $V_{0}$. Now consider label class $i$ of $V_{0}$. Exactly one vertex $u$ of label class $i$ is contained in $S$. Let $j$ be the number such that $u$ is the $j$th vertex of the label class $i$. Then, color the $q$th vertex of label class $i$ with color $c^{i}_{q}$ if $q<j$ and with color $c^{i}_{q-1}$ if $q>j$. Clearly this coloring assigns $x-1$ different colors to the vertices of each label class. Hence, there is a coloring of the vertices of $Y$ that is equal to $M$.

($\Leftarrow$) Let $Y$ denote an occurrence of $M$ in $G$. First, observe that there are only $x-1$ colors for the $x$ vertices of each label class. Hence, $Y$ contains exactly $x-1$ vertices of each label class. Now let $S$ denote the set containing, for each label class, the only vertex not contained in $Y$. Clearly, $|S|=k$ and the elements of $S$ have pairwise different labels in $H$. Furthermore, $S$ is an independent set in $H$: since $G[Y]$ is connected, there is for each edge vertex $v_{\{u,w\}}$ at least one of its neighbors in $Y$. Hence, at most one of the endpoints of each edge $\{u,w\}$ is in $S$. ∎

We immediately obtain the following.

Corollary 1**.**

List-Colored Graph Motif* is W[1]-hard with respect to $\ell$ even if $|\mathcal{L}(v)|\leq 2$ for every vertex $v$ in $G$.*

3 Graph Motif on Trees

Motivated by these negative results on general graphs, we now study the special case where the input graph is a tree. For LGM, we were not able to resolve the parameterized complexity with respect to $\ell$ for this case. Hence, we focus on the more restricted GM problem. We show that GM is fixed-parameter tractable with respect to $\ell$ if the input graph is a tree. Recall that for general graphs, GM is W[1]-hard for $\ell$ even if the motif $M$ contains only two colors [2]. Hence, our result shows that the tree structure significantly helps when parameterizing by $\ell$. We then show that the fixed-parameter algorithm for GM on trees extends to some special cases of LGM in which the vertex-color graph is also a tree. Finally, we show that a polynomial-size kernel for GM on trees parameterized by $\ell$ is unlikely.

3.1 A Dynamic Programming Algorithm

Call a color of $M$ abundant if it occurs more often in $G$ than in $M$. The abundant colors are exactly the ones that have to be “deleted” to obtain a solution $S$. Let $c_{1},\ldots,c_{j}$ denote the abundant colors of $M$, and let $\ell_{i}$ denote the difference between the number of vertices in $V$ that have color $c_{i}$ and the multiplicity $M(c_{i})$ of $c_{i}$ in $M$. This implies in particular that $\sum_{1\leq i\leq j}\ell_{i}=\ell$.

The algorithm is a dynamic programming algorithm that works on a rooted representation $T$ of $G$. We obtain $T$ by choosing an arbitrary vertex $r\in V$ and rooting $G$ at $r$. As usual for dynamic programming on trees, the idea is to combine partial solutions of subtrees. Our algorithm is somewhat similar to a previous dynamic programming algorithm for GM on graphs of bounded treewidth [12] but the analysis and concrete table setup is different.

In the following, let $T_{v}$ denote the subtree of $T$ rooted at vertex $v$. For each subtree, we let $\operatorname{occ}(T_{v},c)$ denote the number of vertices in $T_{v}$ that have color $c$. If a solution contains vertices from $T_{v}$ and further vertices, then it must contain $v$ and all vertices with nonabundant colors in $T_{v}$. Hence, in the dynamic programming it is sufficient to consider subtrees described in the following definition.

Definition 2**.**

We call a connected subtree $T^{\prime}$ of $T_{v}$ safe if $T^{\prime}$ contains $v$ and if every vertex of $T_{v}$ that is colored by a nonabundant color is contained in $T^{\prime}$.

We fill a family of dynamic programming tables $D_{v}$, one table for each $v\in V$. The entries of $D_{v}$ are defined as follows:

[TABLE]

Assume for now that the table has completely been filled out. Then, it can be easily determined whether $G$ has an occurrence $S$ of $M$.

If $S$ is an occurrence of $M$, then let $v$ denote the root of $T[S]$. Clearly, $T[S]$ is a safe subtree of $T_{v}$. Moreover, every vertex with a nonabundant color is contained in $T[S]$ and for all vertices with an abundant color $c_{i}$, the tree $T[S]$ contains $\operatorname{occ}(T_{v},c_{i})-\lambda_{i}$ vertices with color $c_{i}$ for some $\lambda_{i}\geq 0$. Thus, there is some table entry $D_{v}[\lambda_{1},\ldots,\lambda_{j}]$ whose value is $1$ and where $\operatorname{occ}(T_{v},c_{i})-\lambda_{i}$ is the multiplicity of $c_{i}$ for each $c_{i}$.

Conversely, if there is some entry $D_{v}[\lambda_{1},\ldots,\lambda_{j}]$ with value 1 such that $T_{v}$ contains all vertices with nonabundant colors and for each $c_{i}$, $1\leq i\leq j$, $\operatorname{occ}(T_{v},c_{i})-\lambda_{i}$ is exactly the multiplicity of $c_{i}$ in $M$, then there is at least one safe subtree of $T_{v}$ whose vertex set is an occurrence of $M$.

Hence, one may solve GM by filling table $D$, and then checking for each vertex $v$ whether one of the entries of $D_{v}[\lambda_{1},\ldots,\lambda_{j}]$ with value 1 implies the existence of a solution. For the running time bound, the main observation that we exploit is that if a safe rooted subtree of $T_{v}$ contains all the vertices of $T_{v}$ that are in a solution $S$, then it contains at least $\operatorname{occ}(T_{v},c_{i})-\ell_{i}$ vertices with color $c_{i}$. Consequently, the relevant range of values for $\lambda_{i}$ is in $[0,\ell_{i}]$ and thus bounded in the parameter value $\ell$.

We now describe how to fill in table $D$. To initialize $D$, consider each leaf $v$ of the tree $T$. By the definition of $D$, an entry can have value $1$ only if there is a corresponding safe tree which needs to contain $v$. Thus,

[TABLE]

Now, to compute the entries of $D$ for a nonleaf vertex $v$, we combine the entries of the children of $v$. To this end, fix an arbitrary ordering of the children of $v$ and denote them by $u_{1},\ldots,u_{\deg(v)}$. Now, let $T^{i}_{v}$ denote the subtree rooted at $v$ containing the vertices of each $T_{u_{q}}$, $1\leq q\leq i$, and no vertices from each $T_{u_{q}}$, $q>i$. For increasing $i$, we compute solutions for $T^{i}_{v}$, eventually computing the solutions for $T^{\deg(v)}_{v}=T_{v}$. To compute these solutions, we define an auxiliary table $D^{i}_{v}$. The table entries are defined just as for $D$, that is,

[TABLE]

Observe that, since $T^{\deg(v)}_{v}=T_{v}$, we have $D^{\deg(v)}_{v}=D_{v}$ and thus by computing $D^{\deg(v)}_{v}$ we also compute $D_{v}$. Now, $D^{1}_{v}$ can be computed in a straightforward fashion from the entries of $D_{u_{1}}$.

[TABLE]

The first case corresponds to the case that the safe subtree $T^{\prime}$ of $T^{i}_{v}$ contains at least one vertex of $T_{u_{1}}$, the second case corresponds to the case that $T^{\prime}$ contains only $v$.

To compute $D^{i}_{v}$ for $i>1$, we combine entries of $D^{i-1}_{v}$ with $D_{u_{i}}$.

[TABLE]

The first case corresponds to the situation in which no vertex of $T_{u_{i}}$ is part of the safe subtree, in the second case the safe subtree contains some vertices of $T_{u_{i}}$ and some vertices of $T^{i-1}_{v}$.

This completes the description of the dynamic programming recurrences. The correctness follows from the fact that the recurrence considers all possible cases to “distribute” the vertex deletions. It remains to bound the running time.

Theorem 4**.**

Graph Motif* can be solved in $\mathcal{O}(3^{\ell}\cdot n)$ time if $G$ is a tree.*

Proof.

As described above, the relevant values of each $\lambda_{i}$ are in $[0,\ell_{i}]$. Thus, for each subtable $D^{i}_{v}$ and $D_{v}$, the number of entries to compute is $\prod_{1\leq i\leq j}(\ell_{i}+1)$. The dominating term in the overall running time is the computation of the second term in the recurrence for $D^{i}_{v}[\lambda_{1},\ldots,\lambda_{j}]$ where we consider all $(\lambda^{\prime}_{1},\ldots,\lambda^{\prime}_{j})$ such that $D^{i-1}_{v}[\lambda_{1}^{\prime},\ldots,\lambda_{j}^{\prime}]=D_{u_{i}}[\lambda_{1}-\lambda^{\prime}_{1},\ldots,\lambda_{j}-\lambda^{\prime}_{j}]=1$. The number of possible choices can be computed as follows. For each $\lambda_{i}$, the values of $\lambda^{\prime}_{i}$ range between [math] and $\lambda_{i}$. Overall this gives

[TABLE]

possibilities for choosing $\lambda_{i}$ and $\lambda^{\prime}_{i}$. We now bound this product in $\ell$. Thus, we aim to find the vector $(\ell_{1},\ldots,\ell_{j})$ that maximizes $\prod_{1\leq i\leq j}(\ell_{i}+2)\cdot(\ell_{i}+1)/2$ under the constraint $\sum_{1\leq i\leq j}\ell_{i}=\ell$. We claim that this is the vector with $\ell_{1}=\ldots\ell_{j}=1$.

To this end, consider a vector $(\ell_{1},\ldots,\ell_{j})$ whose maximum entry is at least $2$. Without loss of generality, assume thus $\ell_{j}>1$. Now consider $(\ell_{1},\ldots,\ell_{j}-1,1)$ and observe that $(\sum_{1\leq i<{j}}\ell_{i})+(\ell_{j}-1)+1=\ell$, that is, the new vector also satisfies the summation constraint. Moreover,

[TABLE]

where the inequality follows from $\ell_{j}>1$. Since the new vector has more entries with value 1, we conclude that the maximum value is reached when all entries assume value 1. Consequently, the worst case number of recurrences that need to be evaluated for filling a subtable $D^{i}_{v}$ or $D_{v}$ is $\mathcal{O}(3^{\ell})$. The overall number of subtables to fill is $\mathcal{O}(\sum_{v\in V}\deg(v))=\mathcal{O}(n)$. This implies the overall running time bound. ∎

3.2 An Extension to Subcases of List-Colored Graph Motif on Trees

The fixed-parameter tractability of GM on trees can be extended to give fixed-parameter tractability for LGM when the input graph $G$ is a tree and the vertex-color graph $H$ is a forest with bounded degree.

The first step in our algorithm is to apply the following two data reduction rules which are obviously correct.

Rule 1**.**

If there is a color vertex $v$ in $H$ such that the degree of $v$ in $H$ is smaller than the multiplicity of $v$ in $M$, then return “no”.

Rule 2**.**

If there is a color vertex $v$ in $H$ such that the degree of $v$ in $H$ equals the multiplicity of $v$ in $M$, then set $\mathcal{L}(u)=\{v\}$ for all neighbors $u$ of $v$ in $H$.

With these reduction rules at hand, we can show that the following special case of LGM is fixed-parameter tractable with respect to $\ell$.

Lemma 1**.**

LGM* can be solved in $\mathcal{O}(3^{\ell}\cdot n)$ time if $G$ is a tree and the vertex-color graph $H$ is a forest in which for every color vertex $c$, the difference between the degree of $c$ in $H$ and the multiplicity of $c$ in $M$ is at most one.*

Proof.

We describe a reduction of this special case of LGM on trees to GM on trees. In the following, we assume without loss of generality that every color in the instance has multiplicity at least one in $M$. First, apply Reduction Rules 1 and 2 exhaustively. This can be done in $\mathcal{O}(n)$ time by computing the difference between $M(c)$ and $\deg_{H}(c)$ once for all $c\in C$ and updating this value whenever we delete an edge. Afterwards, for every color vertex $c$ in $M$, we have $M(c)\leq\deg_{H}(c)$, due to Reduction Rule 1, and $M(c)\geq\deg_{H}(c)-1$ since the reduction rules do not increase the degree of a vertex. Moreover, if $M(c)=\deg_{H}(c)$ in $M$, then the connected component of $c$ in $H$ consists of $c$ and $\deg_{H}(c)$ leaf neighbors of $c$. By the above, the vertex-color graph $H$ contains the two types of connected components considered below. For both of them, we show that the constraints of $\mathcal{L}$ can be replaced by simple coloring constraints. Consider a connected component $H^{\prime}$ of $H$.

Case 1: $H^{\prime}$ is a star with a central color vertex $c$ such that $M(c)=\deg_{H}(c)$ in $M$. Replace $c$ by $\deg_{H}(c)$ new colors and assign a different color to each neighbor of $c$ in $H$. This is equivalent as all neighbors of $c$ in $H$ are contained in any occurrence of $M$.

Case 2: $H^{\prime}$ is a tree in which each color vertex $c$ fulfills $M(c)=\deg_{H}(c)-1$. Let $C^{\prime}$ denote the set of color vertices in $H^{\prime}$ and $V^{\prime}$ denote the set of vertices of $H^{\prime}$ that do not correspond to colors. Replace $C^{\prime}$ by one new color $c^{*}$ and set the multiplicity of $c^{*}$ to $|V^{\prime}|-1$. To show correctness of this replacement, we show that for every $v\in V^{\prime}$, there is an assignment $f^{\prime}:V^{\prime}\setminus\{v\}\to C^{\prime}$ such that $f^{\prime}(u)\in\mathcal{L}(u)$ for each $u\in V^{\prime}\setminus\{v\}$ and each color $c\in C^{\prime}$ is assigned exactly $\deg_{H}(c)-1$ vertices by $f^{\prime}$. To see the existence of this assignment consider the version of $H^{\prime}$ that is rooted at $v$. For each color vertex $c$ in $H^{\prime}$, let $V^{\prime}_{c}$ denote the children of $c$ in this rooted tree. For each vertex $u\in V^{\prime}_{c}$ set $f^{\prime}(u):=c$. With this assignment, there are exactly $\deg_{H}(c)-1$ vertices that are assigned to $c$ and every vertex $u\in V^{\prime}\setminus\{v^{\prime}\}$ is assigned to some color $c$ of $C^{\prime}$, namely to its predecessor in $H^{\prime}$.

Applying these replacements exhaustively then results in an equivalent instance of GM on trees which can be solved in the claimed running time due to Theorem 4. ∎

We now show how to use the running time bound of Lemma 1 to obtain a fixed-parameter algorithm for the dual parameter $\ell$ for the special case of LGM when the color-vertex graph is a tree and each color has a bounded multiplicity in $M$. Thus, let $M(C):=\max_{c\in C}M(c)$ denote the largest multiplicity in $M$. We will achieve the algorithm by branching on colors $c$ where the difference between $M(c)$ and $\deg_{H}(c)$ is at least two. We call such a color vertex 2-abundant in the following. The first step of the algorithm is to apply Reduction Rules 1 and 2 exhaustively.

Branching Rule 1**.**

If the vertex-color graph $H$ contains a connected component $H^{\prime}$ with at least one 2-abundant color vertex, then do the following.

• •

Root $H^{\prime}$ arbitrarily.

• •

Choose some 2-abundant vertex $c$ of $H^{\prime}$ such that the subtree of $H^{\prime}$ rooted at $c$ has no further 2-abundant vertex.

• •

Choose a set $V_{c}$ of $M(c)+1$ arbitrary children of $c$.

• •

For each $u\in V_{c}$ branch into the case that $c$ is removed from $\mathcal{L}(u)$.

Proof of correctness..

First, observe that such a 2-abundant color vertex $c$ always exists and that it can be found in linear time by a bottom-up traversal of the rooted tree. Second, observe that since $M(c)\geq\deg_{H}(c)+2$, the vertex $c$ has at least $M(c)+1$ children. Hence, some child $u$ of $c$ in $H^{\prime}$ does not receive the color $c$ in any occurrence of $M$. Thus, if the original instance is a yes-instance, then the branch in which we remove $c$ from $\mathcal{L}(u)$ is a yes-instance. Conversely, any occurrence of $M$ in an instance created by the branching rule is an occurrence of $M$ in the original instance. ∎

If Branching Rule 1 does not apply, then we can solve the instance in $\mathcal{O}(3^{\ell}\cdot n)$ time by Lemma 1. It thus remains to ensure that the rule cannot be applied too often. To this end, we apply one further reduction rule. To formulate the rule we need the following definition. We call a connected component $H^{\prime}$ of the vertex-color graph $H$ costly if $H^{\prime}$ either consists of just one vertex $v\in V$ or $H^{\prime}$ is a tree such that all color vertices $c$ in $H^{\prime}$ have multiplicity exactly $\deg_{H}(c)-1$ in $M$.

Rule 3**.**

If $G$ contains at least $\ell+1$ costly components, then return “no”.

Proof of correctness..

For each costly component at least one vertex is not contained in any occurrence of $M$. This is obvious for those components consisting only of one vertex $v$ from $V$. For the other costly components, this follows from Case 2 in the proof of Lemma 1. ∎

Now it remains to observe that in each instance created by an application of Branching Rule 1, the number of costly components is increased by exactly one. Hence, after at most $\ell+1$ branching steps, Reduction Rule 3 directly reports that we have a “no” instance. Since we branch into $M(c)+1\leq M_{C}+1$ cases in each application of Branching Rule 1, we thus create $\mathcal{O}((M_{C}+1)^{\ell+1}))$ instances that either adhere to the conditions of Lemma 1 or are rejected due to Reduction Rules 1 or 3 and can thus be solved in $\mathcal{O}(3^{\ell}\cdot n)$ time. Altogether, we obtain the following running time.

Theorem 5**.**

If $G$ is a tree, and the color vertex graph $H$ is a forest, then LGM can be solved in $\mathcal{O}((M_{C}+1)^{(\ell+1)}\cdot 3^{\ell}\cdot n)$ time.

When $M$ is a set, the largest multiplicity is one, giving the following running time.

Corollary 2**.**

If $G$ is a tree, $H$ is a forest, and $M$ is a set, then LGM can be solved in $\mathcal{O}(6^{\ell}\cdot n)$  time.

By observing that Branching Rule 1 branches into at most $\deg_{H}(c)-1$ branches, we also obtain the following running time bound in terms of the maximum degree of color vertices in $H$.

Corollary 3**.**

If $G$ is a tree, and $H$ is a tree whose color vertices have degree at most $\Delta_{C}$, then LGM can be solved in $\mathcal{O}((\Delta_{C}-1)^{(\ell+1)}\cdot 3^{\ell}\cdot n)$ time.

3.3 A Kernelization Lower Bound

We now show that GM does not admit a polynomial-size problem kernel with respect to $\ell$, even if $G$ is a tree. The proof is based on a cross-composition [4] from the Multicolored Clique problem.

Multicolored Clique

Input: An undirected graph $H=(W,F)$ and a vertex-labeling $\lambda:W\to\{1,\ldots,k\}$.

Question: Is there a vertex set $S\subseteq W$ such that $|S|=k$, the vertices in $S$ have pairwise different labels and $H[S]$ is a clique?

Multicolored Clique has been shown to be W[1]-hard parameterized by $k$ [13]. We refer to the colors of the Multicolored Clique instance as labels to avoid confusion with the colors of the GM instance. Informally, cross-compositions are reductions that combine many instances of one problem into one instance of another problem. The existence of a cross-composition from an NP-hard problem to a parameterized problem $Q$ implies that $Q$ does not admit a polynomial-size problem kernel (unless $\text{NP}\subseteq\text{{coNP/poly}}$) [4].

Definition 3** ([4]).**

Let $L\subseteq\Sigma^{*}$ be a language, let $R$ be a polynomial equivalence relation on $\Sigma^{*}$, and let $Q\subseteq\Sigma^{*}\times\mathbb{N}$ be a parameterized problem. An or-cross-composition of $L$ into $Q$ (with respect to $R$) is an algorithm that, given $t$ instances $x_{1},x_{2},\ldots,x_{t}\in\Sigma^{*}$ of $L$ belonging to the same equivalence class of $R$, takes time polynomial in $\sum_{i=1}^{t}|x_{i}|+k$ and outputs an instance $(y,k)\in\Sigma^{*}\times\mathbb{N}$ of $Q$ such that

• •

the parameter value $k$ is polynomially bounded in $\max_{i=1}^{t}|x_{i}|+\log t$, and

• •

the instance $(y,k)$ is a yes-instance for $Q$ if and only if at least one instance $x_{i}$ is a yes-instance for $L$.

We present an or-cross composition of Multicolored Clique into GM on trees parameterized by $\ell$. The polynomial equivalence relation $R$ will be simply to assume that all the Multicolored Clique instances have the same number of vertices $n$. The main trick is to encode vertex identities in the graph of the Multicolored Clique instance by numbers of colored vertices in the GM instance; this approach was also followed in previous works on GM [12, 6].

Given $t$ instances $(H_{1}=(W_{1},F_{1}),\lambda_{1}),H_{2}=(W_{2},F_{2}),\lambda_{2}),\ldots,H_{t}=(W_{t},F_{t}),\lambda_{t})$ of Multicolored Clique such that $|W_{i}|=n$ for all $i\in[t]$, we reduce to an instance of GM where the input graph is a tree as follows. Herein, we assume without loss of generality that $t=2^{s}$ for some integer $s$.

The first construction step is to add one vertex $r$ that connects the different parts of the instance and which will be contained in every occurrence of the motif. The vertex $r$ thus receives a unique color that may not be deleted. To this vertex $r$ we attach subtrees corresponding to edges of the input instances. Deleting vertices of such a subtree then corresponds to selecting the endpoints of the corresponding edge.

Instance selection gadget.

The technical difficulty in the construction is to ensure that the solution of GM deletes only vertices in subtrees corresponding to edges of the same graph. To achieve this, we introduce $k\cdot(k-1)\cdot\log t$ instance selection colors $\iota[p,q,\tau]$ where $p\in[k]$, $q\in[k]\setminus\{p\}$, and $\tau\in[\log t]$, and demand that the solution deletes exactly one vertex of each instance selection color. To ensure that exactly one instance is selected, we use two further colors $\iota^{+}$ and $\iota^{-}$.

For each Multicolored Clique instance $(H_{i},\lambda_{i})$, attach an instance selection path $P_{i}$ to $r$ that is constructed based on the number $i$. Let $b(i)$ denote the binary expansion of $i$ and let $b_{\tau}(i)$, $\tau\in[\log t]$, denote the $\tau$th digit of $b(i)$. Construct a path $P_{i}$ containing first a vertex with color $\iota^{+}$, then in arbitrary order exactly one vertex of each color in the color set $I_{i}:=\{\iota[p,q,\tau]:b_{\tau}(i)=1\}$, and then a vertex with color $\iota^{-}$. Attach the path $P_{i}$ to $r$ by making the vertex with color $\iota^{+}$ a neighbor of $r$.

The idea of the construction is that exactly one instance selection path $P_{i}$ is completely deleted and that this will force any solution to delete paths that “complement” $P_{i}$ (that is, paths which contain all $\iota[p,q,\tau]$ such that $b_{\tau}(i)=0$) in the rest of the graph.

Edge selection gadget.

To force deletion of subtrees corresponding to exactly $\binom{k}{2}$ edges with different labels, we introduce $2k(k-1)$ label selection colors $\lambda[p,q]^{+}$ and $\lambda[p,q]^{-}$ where $p\in[k]$ and $q\in[k]\setminus\{p\}$. These colors will ensure that, for each pair of labels $p$ and $q$, the solution deletes exactly one path corresponding to the ordered pair $(p,q)$ and one path corresponding to the pair $(q,p)$.

There are two further sets of colors. One set is used for ensuring vertex consistency of the chosen edges, that is, to make sure that all the selected edges with label pair $(p,\cdot)$ correspond to the same vertex with label $p$. More precisely, we introduce a color $\omega[p,q]$ for each $p\in[k]$ and each $q\in[k]\setminus\{p\}$, except for the biggest $q\in[k]\setminus\{p\}$.

The final color set is used to check that the edges selected for label pair $(p,q)$ and for label pair $(q,p)$ are the same. To this end, we introduce a set of colors $\varepsilon[p,q]$ for each $p\in[k]$ and each $q\in[k]\setminus\{p\}$ such that $q>p$. To perform the checks of vertex and edge consistency, we encode the identities of vertices and edges into path lengths. More precisely, we assign each vertex $v\in W_{i}$ a unique (with respect to the vertices of $W_{i}$) number $\#(v)\in[n]$.

Now, for each label pair $(p,q)$ and each instance $i$, attach one path $P_{i}(u,v)$ to $r$ for each edge $\{u,v\}$ where $u$ has color $p$ and $v$ has color $q\neq p$. The path $P_{i}(u,v)$

• •

starts with a vertex with color $\lambda[p,q]^{+}$ that is made adjacent to $r$,

• •

then contains exactly one vertex of each color in $\{\iota[p,q,\tau]:\iota[p,q,\tau]\notin I_{i}\}$,

• •

then contains $\#(u)$ vertices of color $\varepsilon[p,q]$ if $p<q$ and $n-\#(v)$ vertices of color $\varepsilon[q,p]$ if $p>q$,

• •

then, if $q$ is not the biggest label in $[k]\setminus p$, contains $\#(u)$ vertices with color $\omega[p,q]$,

• •

then, if $q$ is not the smallest label in $[k]\setminus p$, contains $n-\#(u)$ vertices with color $\omega[p,q^{\prime}]$, where $q^{\prime}$ is the next-smaller label in $[k]\setminus p$ (if $p=q-1$, then $q^{\prime}=q-2$; otherwise $q^{\prime}=q-1$), and

• •

ends with a vertex with color $\lambda[p,q]^{-}$.

Let ${\cal C}$ denote the multiset containing all the vertex colors of all vertices added during the construction with their respective multiplicities. In the correctness proof it will be easier to argue about the colors that are not contained in $M$. Hence, the construction is completed by setting the multiset $D$ of colors to “delete” to contain each color exactly once except

• •

the color of $r$ which is not contained in $D$,

• •

the vertex consistency colors $\omega[p,q]$ each of which is contained with multiplicity $n$, and

• •

the edge selection colors $\varepsilon[p,q]$ each of which is contained with multiplicity $n$.

The motif $M$ is defined as $M:={\cal C}\setminus D$. It remains to show the correctness.

Theorem 6**.**

Graph Motif* does not admit a polynomial-size problem kernel with respect to $\ell$ even if $G$ is a tree unless $\text{\rm NP}\subseteq\text{\rm{coNP/poly}}$.*

Proof.

To complete the proof we need to show that the construction fulfills the properties of cross-compositions. First, the construction clearly runs in polynomial time. Second, the number of introduced colors is polynomial in $k+\log t$ and thus the value of $\ell=|D|$ is bounded polynomial in $n+\log t$. Thus, it remains to show that the composition is an or-cross composition, that is:

At least one $(H_{i},\lambda_{i})$ is a yes-instance of Multicolored Clique $\Leftrightarrow$ $(M,G,\mathcal{L})$ is a yes-instance of GM.

($\Rightarrow$) Let $S\in W_{i}$ be a vertex set of size $k$ such that $H_{i}[S]$ is a clique and the vertices in $S$ have pairwise different labels. Consider the induced subgraph $G^{\prime}$ of $G$ obtained by completely deleting the path $P_{i}$ and, for each $\{u,v\}\in H_{i}[S]$, the paths $P_{i}(u,v)$ and $P_{i}(v,u)$. Since only complete paths are deleted and since each path in $G$ is attached to $r$, the graph $G^{\prime}$ is connected. It remains to show that the multiset of deleted colors is $D$. First, $\iota^{+}$ and $\iota^{-}$ are deleted once and contained once in $D$. Second, each instance selection color $\iota[p,q,\tau]$ is deleted once as required by $D$: If $\iota[p,q,\tau]$ is contained in $P_{i}$, then it is not contained in any $P_{i}(u,v)$. Conversely, if $\iota[p,q,\tau]$ is not contained in $P_{i}$ then it is contained in each $P_{i}(u,v)$ where $u$ has color $p$ and $v$ has color $q$. Third, exactly $n$ vertices of each vertex consistency color $\omega[p,q]$ are deleted: these vertices are contained only in two paths $P_{i}(u,v)$, namely if $u$ has label $p$ and $v$ has either label $q$ or label $q+1$. Since all the deleted paths with label pair $(p,\cdot)$ correspond to the same vertex $u$, the number of vertices with color $\omega[p,q]$ is $\#(v)$ if $v$ has label $q$ and $n-\#(v)$ if $v$ has label $p$. Hence, exactly $n$ vertices with this color are deleted, as required by $D$. Finally, we show that exactly $n$ vertices of each edge selection color $\varepsilon[p,q]$, $p<q$, are deleted: Let $u$ and $v$ be the vertices of $S$ with label $p$ and $q$, respectively. Then, the deleted path $P_{i}(u,v)$ contains $\#(u)$ vertices with color $\varepsilon[p,q]$ and the deleted path $P_{i}(v,u)$ contains $n-\#(u)$ vertices with this color. Altogether, the multiset of colors in $G^{\prime}$ is exactly ${\cal C}\setminus D=M$.

($\Leftarrow$) Let $G^{\prime}$ be a connected subgraph of $G$ whose multiset of vertex colors is exactly $M$. Let $V_{D}:=V(G)\setminus V(G^{\prime})$ denote the set of deleted vertices, that is, vertices not in $G^{\prime}$. The color multiset of the vertex colors of $V_{D}$ is exactly $D$. Thus, exactly one vertex with color $\iota^{+}$ and color $\iota^{-}$ is deleted. Consequently exactly one path $P_{i}$ is completely deleted from $G^{\prime}$: deleting $\iota^{+}$ in some $P_{i}$ implies that the $\iota^{-}$ in $P_{i}$ is also deleted. Thus, no further vertices from any $P_{j}$, $j\neq i$, may be deleted.

Moreover, since each label selection color $\lambda[p,q]^{+}$ or $\lambda[p,q]^{-}$ is contained exactly once in $D$, the set $V_{D}$ also contains exactly one path $P_{j}(u,v)$ where $u$ has label $p$ and $v$ has label $q$. Moreover, we have $j=i$ by the assignment of the instance selection colors: If $j\neq i$, then there is some $\tau\in[\log t]$ such that $b_{\tau}(i)\neq b_{\tau}(j)$. Then, however $\iota[p,q,\tau]$ is either not contained in the colors of $V_{D}$ or it is contained twice in the colors of $V_{D}$. In either case, the set of deleted colors is different from $D$.

Thus, all the deleted paths in the edge selection gadgets correspond to the same instance $i$. Now consider the paths for label pairs $(p,\cdot)$. These label pairs correspond to the same vertex: Otherwise, there is some $P_{i}(u,v)$ and some $P_{i}(u^{\prime},v^{\prime})$ such that $u\neq u^{\prime}$, $v$ has label $q$, and $v^{\prime}$ has label $q+1$. Then, however, the number of vertices with color $\omega[p,q]$ does not equal $n$ since $P_{i}(u,v)$ contains $\#(u)$ vertices of this color, $P_{i}(u^{\prime},v^{\prime})$ contains $n-\#(u^{\prime})$ vertices of this color and $\#(u)\neq\#(u^{\prime})$. Hence, the deleted paths correspond to a vertex set $S$ with $k$ different labels in some $H_{i}$. It remains to show that the graph $H_{i}[S]$ is a clique.

Consider an arbitrary pair of labels $p$ and $q$ where $p<q$. Moreover, let $u\in S$ and $v\in S$ have label $p$ and $q$, respectively. Let $P_{i}(u,v^{\prime})$ be the path for $u$ that is deleted with this color pair and let $P_{i}(v,u^{\prime})$ be the path for $v$ that is deleted for this color pair. Then, $P_{i}(u,v^{\prime})$ contains exactly $\#(u)$ vertices with color $\varepsilon[p,q]$, $P_{i}(v,u^{\prime})$ contains exactly $n-\#(u^{\prime})$ vertices of this color. Since $D$ contains exactly $n$ vertices of this color, this implies $u=u^{\prime}$. By construction, this implies that $u$ and $v$ are neighbors in $H_{i}$. ∎

4 Colorful Graph Motif on Trees

For the combination of vertex-colored trees as input graphs and motifs that are sets, the problem becomes considerably easier. First, we show that in this case CGM admits a linear-vertex problem kernel that can be computed in linear time. The idea for the problem kernelization is based on two simple observations. First, we observe that the number of vertices that are not unique is bounded in CGM.

Lemma 2**.**

Let $(M,G,\chi)$ be an instance of Colorful Graph Motif. Then at most $2\ell$ vertices in $G$ are not unique.

Proof.

Let $C^{+}$ denote the set of colors that occur more than once in $G$ and let $\operatorname{occ}(c)$ denote the number of occurrences of a color $c$ in $G$. We denote $c^{+}:=|C^{+}|$, $n^{+}:=\sum_{c\in C^{+}}\operatorname{occ}(c)$, and $n^{-}$ the number of unique vertices in $G$. By definition, no color is repeated in $M$, thus $|M|=c^{+}+n^{-}$ ; moreover, $|V|=n^{+}+n^{-}$. Hence, the number $\ell=|V|-|M|$ of vertices to delete satisfies $\ell=n^{+}-c^{+}$. By definition $n^{+}\geq 2c^{+}$, and thus we conclude that $\ell\geq n^{+}/2$. ∎

Second, if there are two vertices that are unique, then the uniquely determined path between these vertices is contained in every occurrence of the motif. The kernelization accordingly removes all the vertices that lie on these paths. More precisely, these vertices are “contracted” into the root $r$. Afterwards, in a second phase some further vertices are removed because their colors have been used during the contraction. Eventually, this results in an instance which has at most one unique vertex and thus, by Lemma 2, bounded size. For an example of the kernelization, see Figure 1. Below, we give a more detailed description.

Theorem 7**.**

Colorful Graph Motif* on trees admits a problem kernel with at most $2\ell+1$ vertices that can be computed in $\mathcal{O}(n)$ time.*

Proof.

We first describe the kernelization algorithm, then we show its correctness and finally bound its running time. By Lemma 2, the size bound holds if the instance has no unique vertex. Thus, we assume that there is a unique vertex in the following.

Given an instance $(M,G,\chi)$ of CGM, first root the input tree $G$ at an arbitrary unique vertex $r$. Now call a subtree with root $v$ pendant if it contains all descendants of $v$ in $G$. Then, compute in a bottom-up fashion maximal pendant subtrees such that no vertex in this subtree is unique. Call these subtrees the pendant non-unique subtrees. By Lemma 2, the total number of vertices in pendant non-unique subtrees is at most $2\ell$. Now the algorithm removes vertices in two phases.

Phase I. Remove from $G$ all vertices except $r$ that are not contained in a pendant non-unique subtree. Remove all colors of removed vertices from $M$. If there is a color $c$ such that two vertices with color $c$ are removed in this step, then return “no”. Make $r$ adjacent to the root of each pendant non-unique subtree.

Phase II. In the first step of this phase, for each color $c$ where at least one vertex has been removed in Phase I, remove all vertices from $G$ that have color $c$. In the second step of this phase, remove all descendants of these vertices. Finally, let $M^{\prime}$ denote the set of colors that are contained in the remaining instance. This completes the kernelization algorithm; the resulting instance has at most $2\ell+1$ vertices since all vertices except $r$ are unique. To show correctness, we first observe the following.

Claim: every occurrence of $M$ in $G$ contains no vertex $v$ that is removed during Phase II of the kernelization. This can be seen as follows. First, every occurrence of $M$ in $G$ contains all vertices removed during Phase I: these vertices are either unique or lie on the uniquely determined path between two unique vertices. Now consider a vertex $v$ removed during Phase II. If $v$ is removed in the first step of Phase II, then $v$ has the same color $c$ as a vertex $u$ removed during Phase I. Consequently, $v$ is not contained in an occurrence of $M$: By the above, the occurrence contains $u$ and it contains no other vertex with color $c$. Otherwise, $v$ is removed in the second step of Phase II, because $v$ is not connected to $r$. Since every occurrence of $M$ contains $r$, it thus cannot contain $v$.

We now show the correctness of the kernelization, that is, the equivalence of the original instance $(M,G,\chi)$ and the resulting instance $(M^{\prime},G^{\prime},\chi^{\prime})$. First, assume that $(M,G,\chi)$ is a yes-instance. Let $S_{T}$ be an occurrence of $M$ in $G$, and let $T$ denote $G[S_{T}]$; by the above claim, $T$ contains only vertices that are removed during Phase I or that are contained in $G^{\prime}$. Consider the subtree $T^{\prime}$ of $G$ that contains all vertices of $T$ that are not removed during the kernelization. We show that $T^{\prime}$ is connected in $G^{\prime}$ and contains all colors of $M^{\prime}$. Connectivity can be seen as follows. First, observe that $T$ and $T^{\prime}$ contain $r$. Second, any vertex $v\neq r$ of $T^{\prime}$ is contained in some pendant non-unique subtree of $G$. Thus, $v$ is in $T$ connected to $r$ via a path that first visits only vertices of $T^{\prime}$, including the root of the pendant non-unique subtree. The root of the pendant non-unique subtree is in $G^{\prime}$ adjacent to $r$. Thus, each vertex $v\neq r$ has in $T^{\prime}$ a path to $r$ which implies that $T^{\prime}$ is connected. It remains to prove that $T^{\prime}$ contains all colors of $M^{\prime}$. Consider a color $c\in M^{\prime}$. Since $c\in M^{\prime}$, none of the vertices with color $c$ are removed in Phase I of the kernelization. Moreover, since no vertex of $T$ is removed in Phase II of the kernelization, we have that the vertex of $T$ with color $c$ is contained in $T^{\prime}$. Thus, $T^{\prime}$ contains each color of $M^{\prime}$. Finally, $T^{\prime}$ contains each color at most once since $T$ does.

Now assume that $(M^{\prime},G^{\prime},\chi^{\prime})$ is a yes-instance and let $S_{T^{\prime}}$ be an occurrence of $M^{\prime}$ in $G^{\prime}$. Let $T$ denote $G[S_{T^{\prime}}\cup V_{I}]$, where $V_{I}$ is the set of vertices removed during Phase I of the kernelization. We show that $T$ is connected and contains every color of $G$ exactly once. To see that $T$ is connected observe the following: Clearly, $G[\{r\}\cup V_{I}]$ is connected. Moreover, each vertex $v\neq r$ of $T^{\prime}$ has in $T^{\prime}$ a path to $r$. This path contains a subpath from $v$ to the root $r^{\prime}$ of the pendant non-unique subtree containing $v$. In $G$, $r^{\prime}$ is adjacent to some vertex of $\{r\}\cup V_{I}$. Therefore, $r^{\prime}$ is connected to $r$ in $T$ and thus $T$ is connected. It remains to show that $T$ contains every color of $G$ exactly once. Clearly, $T^{\prime}$ contains at least one vertex of each color $c\in M^{\prime}$. Moreover, it also contains at least one vertex of each color $c\in M\setminus M^{\prime}$ since it contains all vertices of $V_{I}$. Besides, it contains each color only once: The vertices of $T^{\prime}$ have pairwise different colors and different colors than those of the vertices of $V_{I}$. Finally, the vertices of $V_{I}$ have different pairwise colors since the kernelization did not return “no”.

The running time can be seen as follows. Determining the pendant non-unique subtrees can be done by a standard bottom-up procedure in linear time. Removing all vertices during Phase I can also be achieved in linear time. After removing a vertex with color $c$ in Phase I, we label $c$ as occupied. When we remove a vertex with an occupied color during Phase I, we immediately return “no”. After the removal of vertices during Phase I, we can construct $M^{\prime}$ from $M$ in linear time by removing each occupied color. Finally, we can in linear time add an edge between $r$ and every root of a pendant non-unique subtree and then remove all remaining vertices that have an occupied color. The final graph $G^{\prime}$ is obtained by performing a depth-first search from $r$, in order to include only those vertices still reachable from $r$. ∎

Now, let us turn to developing fast(er) FPT algorithms for CGM. It can be seen that it is possible to solve CGM in trees in time $1.62^{\ell}\cdot n^{\mathcal{O}(1)}$, by ’branching on colors with the most occurrences’ until every color appears at most twice. More precisely, for a color $c$ that appears at least three times and some vertex $v$ with color $c$, we can branch into the two cases to either delete $v$ or to delete the at least two other vertices that have color $c$. The branching vector222For an introduction to the analysis of branching vectors, refer to [8, 16]. for this branching rule is $(1,2)$ or better. Now, if every color appears at most twice, then CGM on trees can be solved in polynomial time [12, Lemma 2]. By a different branching approach, the above running time can be further improved.

Branching Rule 2**.**

If there is a color $c$ such that there are two vertices $u$ and $v$ with color $c$ that are both not leaves of the tree $G$, then branch into the case to delete from $G$ either

• •

the maximal subtree containing $u$ and all vertices $w$ such that the path from $v$ to $w$ contains $u$, or

• •

the maximal subtree containing $v$ and all vertices $w$ such that the path from $u$ to $w$ contains $v$.

Proof of correctness..

No occurrence may contain vertices of both subtrees, since in this case it contains $u$ and $v$ which have the same color. ∎

If the rule does not apply, then one can solve the problem in linear time; here, let $\operatorname{occ}(c)$ denote the number of occurrences of a color $c$ in $G$.

Lemma 3**.**

Let $(M,G,\chi)$ be an instance of Colorful Graph Motif such that $G$ is a tree and for each color $c$ with $\operatorname{occ}(c)>1$ at least $\operatorname{occ}(c)-1$ occurrences of $c$ are leaves of $G$, then $(M,G,\chi)$ can be solved in $\mathcal{O}(n)$ time.

Proof.

For each color $c$ with $\operatorname{occ}(c)>1$, the algorithm simply deletes $\operatorname{occ}(c)-1$ leaves with color $c$. This can be done in linear time by visiting all leaves via depth-first search, checking for each leaf in $\mathcal{O}(1)$ time whether $\operatorname{occ}(c)>1$ and deleting the leaf in $\mathcal{O}(1)$ time if this is the case. The resulting graph contains each color exactly once, and it is connected since a tree cannot be made disconnected by deleting leaves. ∎

Altogether, we arrive at the following running time.

Theorem 8**.**

Colorful Graph Motif* can be solved in $\mathcal{O}(\sqrt{2}^{\mskip 1.5mu{}\ell}+n)$ time if $G$ is a tree.*

Proof.

The algorithm is as follows. First, reduce the input instance in $\mathcal{O}(n)$ time to an equivalent one with $\mathcal{O}(\ell)$ vertices using the kernelization of Theorem 7. Now, apply Branching Rule 2. If this rule is no longer applicable, then solve the instance in $\mathcal{O}(\ell)$ time (by applying the algorithm behind Lemma 3). Since the graph has $\mathcal{O}(\ell)$ vertices, applicability of Branching Rule 2 can be tested in $\mathcal{O}(\ell)$ time. Thus, the overall running time is $\mathcal{O}(\ell)$ times the number of search tree nodes. Since each application of Branching Rule 2 creates two branches and reduces $\ell$ by at least two in each branch, the search tree has size $\mathcal{O}(2^{\ell/2})=\mathcal{O}(\sqrt{2}^{\mskip 1.5mu{}\ell})$. The resulting running time is $\mathcal{O}(\sqrt{2}^{\mskip 1.5mu{}\ell}\cdot\ell+n)$. Furthermore, the factor of $\ell$ in the running time can be removed by interleaving search tree and kernelization [21], that is, by applying the kernelization algorithm of Theorem 7 in each search tree node. ∎

5 Conclusion

In this paper, we have studied the Graph Motif, List-Colored Graph Motif and Colorful Graph Motif problems, and in particular their behavior in terms of parameterized complexity, when the parameter is $\ell=|V|-|M|$, i.e. the number of vertices of $G$ that are not kept in a solution.

We left open the parameterized complexity for parameter $\ell$ for List-Colored Graph Motif on trees, even when the vertex-color graph is a forest.

As mentioned in the introduction, parameterization by $\ell$ may be interesting not only from a theoretic, but also from an applied point of view. Unfortunately, for the practically relevant case of List-Colored Graph Motif we have obtained W[1]-hardness even for very restricted color lists $\mathcal{L}$. Moreover, as noted by Fertin et al. [14], a reduction of Rauf et al. [24] shows that the variant of Colorful Graph Motif where $G$ is directed and has edge weights is W[1]-hard with respect to $\ell$. However, the combination of $\ell$ with further structure related to the colors of $C$ led to tractability results [14, 15]. It would be interesting to identify such color-related structure also for List-Colored Graph Motif.