Lower Bounds for the Happy Coloring Problems

TL;DR

This paper investigates the computational complexity of the Maximum Happy Vertices and Edges problems, establishing lower bounds and presenting algorithms that match these bounds, thus clarifying their computational limits.

Contribution

It provides new lower bounds on the complexity of the problems and introduces algorithms with proven optimality under current complexity assumptions.

Findings

NP-hardness of guarantee parameterization

Kernelization lower bounds

Exponential lower bounds under Set Cover and ETH

Abstract

In this paper, we study the Maximum Happy Vertices and the Maximum Happy Edges problems (MHV and MHE for short). Very recently, the problems attracted a lot of attention and were studied in Agrawal '17, Aravind et al. '16, Choudhari and Reddy '18, Misra and Reddy '17. Main focus of our work is lower bounds on the computational complexity of these problems. Established lower bounds can be divided into the following groups: NP-hardness of the above guarantee parameterization, kernelization lower bounds (answering questions of Misra and Reddy '17), exponential lower bounds under the Set Cover Conjecture and the Exponential Time Hypothesis, and inapproximability results. Moreover, we present an $\mathcal{O}^*(\ell^k)$ randomized algorithm for MHV and an $\mathcal{O}^*(2^k)$ algorithm for MHE, where $\ell$ is the number of colors used and $k$ is the number of required happy vertices or edges. These algorithms cannot be improved to subexponential taking proved lower bounds into account.

Equations

$${\mathcal{H}}_i(G,p)=\{v\in \mathcal{H}(G,p)\mid N[v]\cap p^{-1}(i)\ne \varnothing \}.$$

Full text

1 Introduction

In this paper, we study Maximum Happy Vertices and Maximum Happy Edges. The problems are motivated by a study of algorithmic aspects of homophyly law in large networks and were introduced by Zhang and Li in 2015 [zhang2015algorithmic]. The law states that in social networks people are more likely to connect with people they like. Social network is represented by a graph, where each vertex corresponds to a person of the network, and an edge between two vertices denotes that the corresponding persons are connected within the network. Furthermore, we let vertices have a color assigned. The color of a vertex indicates type, character or affiliation of the corresponding person in the network. An edge is called happy if its endpoints are colored with the same color. A vertex is called happy if all its neighbours are colored with the same color as the vertex itself. Equivalently, a vertex is happy if all edges incident to it are happy. Formal definition of Maximum Happy Vertices and Maximum Happy Edges is the following.

{problemx}

{problemx}

Recently, MHV and MHE have attracted a lot of attention and were studied from parameterized [Agrawal2018, Aravind2016, Aravind2017, Choudhari2018, Misra2018] and approximation [zhang2015algorithmic, zhang2018improved, zhang2015improved, xu2016submodular] points of view as well as from experimental perspective [lewis2019finding].

NP-hardness of MHVand MHE was proved by Zhang and Li even in case when only three colors are used. Later, Misra and Reddy [Misra2018] proved \NP-hardness of both MHV and MHE on split and on bipartite graphs. However, MHV is polynomially time solvable on cographs and trees [Misra2018, Aravind2016]. Approximation results for MHV are presented in Zhang et al. [zhang2018improved]. They showed that MHV can be approximated within $\frac{1}{\Delta+1}$, where $\Delta$ is the maximum degree of the input graph, and MHE can be approximated within $\frac{1}{2}+\frac{\sqrt{2}}{4}f(\ell)$, where $f(\ell)=\frac{(1-1/\ell)\sqrt{\ell(\ell-1)}+1/\sqrt{2}}{\ell-1+1/2\ell}$. From parameterized point of view the following parameters were studied: pathwidth [Agrawal2018, Aravind2017], treewidth [Agrawal2018, Aravind2017], neighbourhood diversity [Aravind2017], vertex cover [Misra2018], distance to clique [Misra2018], distance to threshold graphs [Choudhari2018]. Kernelization questions were studied in works [Agrawal2018, gao2018kernelization]. Agrawal [Agrawal2018] provided a $\mathcal{O}(k^{2}\ell^{2})$ kernel for MHV where $\ell$ is the number of colors used and $k$ is the number of desired happy vertices. Independently, Gao and Gao [gao2018kernelization] present $2^{k\ell+k}+k\ell+k+\ell$ kernel for general case and $7(k\ell+k)+\ell-10$ in case of planar graphs.

Short summary of our results can be found below.

No polynomial kernels:

If $\NP\not\subseteq\coNP/\poly$ then there are no polynomial kernels for MHV parameterized by vertex cover, and no polynomial kernels for MHE under the following parametrizations: number of uncolored vertices, number of happy edges, and distance to almost any reasonable graph class. Moreover, under $\NP\not\subseteq\coNP/\poly$, there is no $\mathcal{O}((k^{d}\ell)^{2-\epsilon})$ and no $\mathcal{O}((k^{d}h)^{2-\epsilon})$ bitsize kernel for MHV. Note that these results answer question from [Misra2018]: ”Do the Maximum Happy Vertices and Maximum Happy Edges problems admit polynomial kernels when parameterized by either the vertex cover or the distance to clique parameters?”

Above guarantee:

Above-greedy versions of MHV and MHE are NP-complete even for budget equal $1$.

Exponential lower bounds:

Assuming the Set Cover Conjecture, MHV and MHE do not admit $\mathcal{O}^{*}((2-\epsilon)^{n^{\prime}})$ algorithms, where $n^{\prime}$ is the number of uncolored vertices in the input graph. Even with $\ell=3$, there is no $2^{o(n+m)}$ algorithm for MHV and MHE, unless ETH fails.

Innaproximability:

Unless $\P=\NP$, MHV does not admit approximation algorithm with factors $\mathcal{O}(n^{\frac{1}{2}-\epsilon})$, $\mathcal{O}(m^{\frac{1}{2}-\epsilon})$, $\mathcal{O}(h^{1-\epsilon})$, $\mathcal{O}({\ell}^{1-\epsilon})$, for any $\epsilon>0$.

Algorithms:

We present $\mathcal{O}^{*}(\ell^{k})$ randomized algorithm for MHV and $\mathcal{O}^{*}(2^{k})$ algorithm for MHE. Running time of this algorithms match with the corresponding lower bounds. We should note that an algorithm with the running time of $\mathcal{O}^{*}(2^{k})$ for MHE was also presented by Aravind et al. in [Aravind2017].

2 Preliminaries

Basic notation. We denote the set of positive integer numbers by $\mathbb{N}$. For each positive integer $k$, by $[k]$ we denote the set of all positive integers not exceeding $k$, $\{1,2,\ldots,k\}$. We use $\sqcup$ for the disjoint union operator, i.e. $A\sqcup B$ equals $A\cup B$, with an additional constraint that $A$ and $B$ are disjoint.

We use traditional $\mathcal{O}$-notation for asymptotical upper bounds. We additionally use $\mathcal{O}^{*}$-notation that hides polynomial factors. Many of our results concern the parameterized complexity of the problems, including fixed-parameter tractable algorithms, kernelization algorithms, and some hardness results for certain parameters. For detailed survey in parameterized algorithms we refer to the book of Cygan et al. [cygan2015parameterized].

Throughout the paper, we use standard graph notation and terminology, following the book of Diestel [diestel2018graph]. All graphs in our work are undirected simple graphs. We may refer to the distance to $\mathcal{G}$ parameter, where $\mathcal{G}$ is an arbitrary graph class. For a graph $G$, we say that a vertex subset $S\subseteq V(G)$ is a $\mathcal{G}$ modulator of $G$, if $G$ becomes a member of $\mathcal{G}$ after deletion of $S$, i.e. $G\setminus S\in\mathcal{G}$. Then, the distance to $\mathcal{G}$ parameter of $G$ is defined as the size of its smallest $\mathcal{G}$ modulator.

Graph colorings. When dealing with instances of Maximum Happy Vertices or Maximum Happy Edges, we use a notion of colorings. A coloring of a graph $G$ is a function that maps vertices of the graph to the set of colors. If this function is partial, we call such coloring partial. If not stated otherwise, we use $\ell$ for the number of distinct colors, and assume that colors are integers in $[\ell]$. A partial coloring $p$ is always given as a part of the input for both problems, along with graph $G$. We also call $p$ a precoloring of the graph $G$, and use $(G,p)$ to denote the graph along with the precoloring. The goal of both problems is to extend this partial coloring to a specific coloring $c$ that maps each vertex to a color. We call $c$ a full coloring (or simply, a coloring) of $G$ that extends $p$. We may also say that $c$ is a coloring of $(G,p)$. For convenience, introduce the notion of potentially happy vertices, both for full and partial colorings.

Definition 1**.**

We call a vertex $v$ of $(G,p)$ potentially happy, if there exists a coloring $c$ of $(G,p)$ such that $v$ is happy with respect to $c$. In other words, if $u$ and $w$ are precolored neighbours of $v$, then $p(u)=p(w)$. We denote the set of all potentially happy vertices in $(G,p)$ by $\mathcal{H}(G,p)$.

By $\mathcal{H}_{i}(G,p)$ we denote the set of all potentially happy vertices in $(G,p)$ such that they are either precolored with color $i$ or have a neighbour precolored with color $i$:

[TABLE]

In other words, if a vertex $v\in\mathcal{H}_{i}(G,p)$ is happy with respect to some coloring $c$ of $(G,p)$, then necessarily $c(v)=i$.

For a graph with precoloring $(G,p)$, by $h=|\mathcal{H}(G,p)|$ we denote the number of potentially happy vertices in $(G,p)$. Note that if $c$ is a full coloring of a graph $G$, then $|\mathcal{H}(G,c)|$ is equal to the number of vertices in $G$ that are happy with respect to $c$.

3 Polynomial kernels for structural graph parameters

In this section, we study existence of polynomial kernels for MHV or MHE under several parameterizations. We start with proving lower bounds for structural graph parameters. We provide reductions to both MHV and MHE from the following problem.

{problemx}

Theorem 1** ([Dom2014]).**

Bounded Rank Disjoint Sets* parameterized by $kd$ does not admit a polynomial compression even if every set $S\in\mathcal{F}$ consists of exactly $d$ elements and $|U|=kd$, unless $\NP\subseteq\coNP/\poly$. *

The following two theorems answer open questions posed in [Misra2018].

Theorem 2**.**

Maximum Happy Vertices* parameterized by the vertex cover number does not admit a polynomial compression, unless $\NP\subseteq\coNP/\poly$. *

Proof*.*

We give a polynomial reduction from the Set Packing problem, such that the vertex cover number of the constructed instance of MHV is at most the size of the universe of the initial instance of Set Packing plus one. Since Bounded Rank Disjoint Sets is a special case of Set Packing, from Theorem 1 the theorem statement will then follow. The reduction is as follows.

Given an instance $(U=[n],\mathcal{F}=\{S_{1},S_{2},\ldots,S_{m}\},k)$ of Set Packing, construct an instance $(G,p,k)$ of MHV. For each $i\in U$, introduce vertex $u_{i}$ in $G$ and left it uncolored. For each set $S_{j}\in\mathcal{F}$, introduce a vertex $s_{j}$ in $G$ and precolor it with color $j$, i.e. $p(s_{j})=j$. Thus, the set of colors used in precoloring $p$ is exactly $[m]$. Then, for each $i\in[n]$ and $j\in[m]$ such that $i\in S_{j}$, introduce an edge between $u_{i}$ and $s_{j}$ in $G$. Additionally, introduce two vertices $t_{1}$ and $t_{2}$ to $G$ and precolor them with colors $1$ and $2$ respectively. Then, introduce an edge $(t_{1},t_{2})$ to $G$ and for every $i\in[n]$ and $j\in[2]$, introduce an edge $(u_{i},t_{j})$ in $G$. Thus, vertices $t_{1}$ and $t_{2}$ never become happy and ensure that $u_{i}$ never become happy for any $i\in[n]$. Finally, set the number of required happy vertices to $k$. Observe that $\{u_{1},\ldots,u_{n}\}\cup\{t_{1}\}$ forms a vertex cover of $G$, hence the vertex cover number of $G$ is at most $n+1$.

We now claim that $(U,\mathcal{F},k)$ is a yes-instance of Set Packing if and only if $(G,p,k)$ is a yes-instance of MHV. Let $S_{i_{1}},S_{i_{2}},\ldots,S_{i_{k}}$ be the answer to $(U,\mathcal{F},k)$, i.e. $S_{i_{p}}\cap S_{i_{q}}=\emptyset$ for every distinct $p,q\in[k]$. Since $S_{i_{1}},S_{i_{2}},\ldots,S_{i_{k}}$ are disjoint, $s_{i_{1}},s_{i_{2}},\ldots,s_{i_{k}}$ do not have any common neighbours in $G$. Hence, we can extend coloring $p$ to coloring $c$ in a way that $s_{i_{1}},s_{i_{2}},\ldots,s_{i_{k}}$ are happy with respect to $c$ ($c(u_{i})$ is then, in fact, the index of the set containing $u_{i}$, i.e. $u_{i}\in S_{c(u_{i})}$). At least $k$ vertices become happy in $G$, hence $(G,p,k)$ is a yes-instance of MHV.

In the other direction, let $c$ be a coloring of $G$ extending $p$ so that at least $k$ vertices in $G$ are happy with respect to $c$. Only vertices that can be happy in $(G,p)$ are vertices of type $s_{i}$, hence there are vertices $s_{i_{1}},s_{i_{2}},s_{i_{3}},\ldots,s_{i_{k}}$ that are happy in $G$ with respect to $c$. Since these vertices are precolored with pairwise distinct colors and are simultaneously happy, they may have no common neighbours in $G$. This implies that the corresponding sets of the initial instance $S_{i_{1}},S_{i_{2}},\ldots,S_{i_{k}}$ are pairwise disjoint. Hence, they form an answer to the initial instance $(U,\mathcal{F},k)$ of Set Packing. This completes the proof. $\Box$

Theorem 3**.**

Maximum Happy Edges* parameterized by the number of uncolored vertices or by the number of happy edges does not admit a polynomial compression, unless $\NP\subseteq\coNP/\poly$. *

Proof*.*

As in the proof of Theorem 2, we again provide a polynomial reduction from Bounded Rank Disjoint Sets and then use Theorem 1. In this proof though, we will use the restricted version Bounded Rank Disjoints Sets problem itself (and not the Set Packing problem), formulated in Theorem 1. That is, we will use the constraint that all sets in the given instance are of the same size $d$, and the size of the universe $|U|$ is equal to $kd$. We note that the following reduction has very much in common with the reduction described in the proof of Theorem 2.

Given an instance $([n],\mathcal{F}=\{S_{1},S_{2},\ldots,S_{m}\},k)$ of Bounded Rank Disjoint Sets with $n=kd$ and $|S_{i}|=d$ for every $i\in[m]$, we construct an instance $(G,p,k^{\prime})$ of MHE. We assume that each element of the universe $[n]$ is contained in at least one set, otherwise the given instance is a no-instance. Firstly, as in the proof of Theorem 2, for each element of the universe $i\in[n]$, introduce a corresponding vertex $u_{i}$ in $G$. For each set $S_{j}$, $j\in[m]$, introduce not just one, but $n$ corresponding vertices $s_{j,1},s_{j,2},\ldots,s_{j,n}$. Then again, similarly to the proof of Theorem 1, for each $i,j$ such that $i\in S_{j}$, introduce edges between $u_{i}$ and each vertex $s_{j,t}$ corresponding to the set $S_{j}$, i.e. $n$ edges in total. To finish the construction of $G$, introduce every possible edge $(u_{i},u_{j})$ in $G$.

Thus, $V(G)=\{u_{i}\mid i\in[n]\}\cup\{s_{j,t}\mid j\in[m],t\in[n]\}$ and $E(G)=\{(u_{i},s_{j,t})\mid i\in S_{j},t\in[n]\cup\{(u_{i},u_{j})\mid i,j\in[n],i\neq j\}$. Then, precolor the vertices of $G$ in the usual way, i.e. set $p(s_{j,t})=j$ for every $j\in[m]$ and $t\in[n]$, and leave each vertex $u_{i}$ uncolored. Finally, we set the number of required happy edges to $k^{\prime}=n^{2}+k\binom{d}{2}=(kd)^{2}+k\binom{d}{2}$. Construction of $(G,p,k^{\prime})$ is done in polynomial time. Observe that the number of uncolored vertices in $(G,p,k)$ equals the size of the universe $n$, and the number of required happy edges is polynomial of $n$. Hence, existence of a polynomial kernel respectively to any of these two parameters for MHE contradicts the statement of Theorem 1. We argue that the initial instance is a yes-instance if and only if $(G,p,k^{\prime})$ is a yes-instance of MHE.

We prove first that if $([n],\mathcal{F},k)$ is a yes-instance, then $(G,p,k^{\prime})$ is a yes-instance. Let $([n],\mathcal{F},k)$ be a yes-instance of the restricted version of Bounded Rank Disjoint Sets, and let $S_{i_{1}},S_{i_{2}},\ldots,S_{i_{k}}$ be the instance solution. As usual, extend $p$ to a coloring $c$ of $G$ by setting $c(u_{i})$ to the index of the set in the solution containing $u_{i}$, i.e. $c(u_{i})=i_{t}$ for some $t\in k$ and $i\in S_{c(u_{i})}$. Since $S_{i_{1}},S_{i_{2}},\ldots,S_{i_{k}}$ are disjoint, and their total size equals the size of the universe, such coloring $c$ always exists uniquely for a fixed solution of $([n],\mathcal{F},k)$. We claim that there are exactly $k^{\prime}$ happy edges in $G$ with respect to $c$.

All edges in $G$ are either of type $(u_{i},s_{j,t})$ or of type $(u_{i},u_{j})$. Consider edges of type $(u_{i},s_{j,t})$ for a fixed $i\in[n]$. Happy edges among them are those with $c(s_{j,t})=c(u_{i})$. Since $c(s_{j,t})=p(s_{j,t})=j$ and $i\in S_{c(u_{i})}$, these edges are exacly $(u_{i},s_{c(u_{i}),t})$. Hence, there are $n$ happy edges of this type for a fixed $i\in[n]$ and $n^{2}$ happy edges of this type in total. It is left to count the number of happy edges of the clique, i.e. edges of type $(u_{i},u_{j})$. Observe that each $u_{i}$ is colored with a color corresponding to a containing set of the answer. Since each set is of size $d$, the vertices $u_{i}$ are split by color into $k$ groups of size $d$. Each group contributes exactly $\binom{d}{2}$ happy edges, and no edge connecting vertices from different groups is happy. Thus, there are exactly $k\binom{d}{2}$ happy edges of type $(u_{i},u_{j})$ in $G$ with respect to $c$. We get that exactly $n^{2}+k\binom{d}{2}$ edges of $G$ are happy with respect to $c$, hence $(G,p,k^{\prime})$ is a yes-instance of MHE.

In the other direction, let $(G,p,k^{\prime})$ be a yes-instance of MHE, and let $c$ be an optimal coloring of $G$ extending $p$. At least $k^{\prime}$ edges are happy in $G$ with respect to $c$. Let us show that exactly $k^{\prime}$ edges are happy in $G$ with respect to $c$.

Claim 1**.**

In any optimal coloring $c$ of $G$ extending $p$, $i\in S_{c(u_{i})}$ for each $i\in[n]$.

Proof of Claim 1*.*

Suppose it is not true, and $c$ is an optimal coloring of $(G,p)$ and $i\notin S_{c(u_{i})}$ for some $i\in[n]$. For each $j$ with $i\in S_{j}$, $u_{i}$ is adjacent to $n$ vertices $s_{j,t}$, which are precolored with color $j$. None of edges $(u_{i},s_{j,t})$ are happy with respect to $c$, since $j\neq c(u_{i})$. The only other edges incident to $u_{i}$ are $n-1$ edges of the clique. Thus, $u_{i}$ is incident to at most $n-1$ happy edges.

Choose arbitrary $j$ with $i\in S_{j}$, and put $c(u_{i})=j$. $u_{i}$ becomes incident to at least $n$ happy edges. Happiness of edges not incident with $u_{i}$ has not changed. Thus, the change yields at least one more happy edge. A contradiction with the optimality of $c$. $\Box$

Claim 2**.**

In any optimal coloring $c$ of $G$ extending $p$, there are at most $k\binom{d}{2}$ happy edges of type $(u_{i},u_{j})$ in $G$ with respect to $c$.

Proof of Claim 2*.*

The vertices $u_{i}$ are split into groups containing vertices of the same color by $c$, so the happy edges of type $(u_{i},u_{j})$ are exactly the edges inside the groups. By Claim 1, each $u_{i}$ is colored with a color corresponding to a set containing $i$ in $c$. Hence, each group contains vertices corresponding to elements of the same set, and thus contains at most $d$ vertices. So each $u_{i}$ is incident to at most $d-1$ happy edges of type $(u_{i},u_{j})$, and in total there are at most $n\cdot(d-1)/2=k\binom{d}{2}$ such happy edges in $G$ with respect to $c$. $\Box$

From Claims 1 and 2 follows that at most $n^{2}+k\binom{d}{2}=k^{\prime}$ edges are happy in $G$ with respect to $c$. And as seen in the proof of Claim 2, the only way that yields exactly $k^{\prime}$ happy edges is when $u_{i}$ are split by color into disjoint groups of size $d$, each containing vertices corresponding to a set of the initial instance. Hence, if $c$ yields $k^{\prime}$ happy edges in $G$, $\{S_{c_{(}u_{i})}\mid i\in[n]\}$ is a solution to $([n],\mathcal{F},k)$. Thus, $([n],\mathcal{F},k)$ is a yes-instance of Bounded Rank Disjoint Sets. This finishes the whole proof. $\Box$

Definition 2**.**

We call a graph family $\mathcal{G}$ uniformly polynomially instantiable, if there is an algorithm that, given positive integer $n$ as input, outputs a graph $G$, such that $|V(G)|\geq n$ and $G\in\mathcal{G}$, in $\poly(n)$ time.

Corollary 1**.**

*For any uniformly polynomially instantiable graph family $\mathcal{G}$, Maximum Happy Edges, parameterized by the distance to graphs in $\mathcal{G}$, does not admit a polynomial compression, unless $\NP\subseteq\coNP/\poly$. *

Proof*.*

Suppose it is not true and there is a uniformly polynomially instantiable graph family $\mathcal{G}$, such that MHE parameterized by the distance to graphs in $\mathcal{G}$ admits a polynomial compression. We show how to reduce an instance of MHE with $d$ uncolored vertices to an instance of MHE with the distance to graph in $\mathcal{G}$ being at most $d$, and then get a contradiction with Theorem 3.

Let $(G,p,k)$ be an instance of MHE with $d$ uncolored vertices. Denote the set of all uncolored vertices in $(G,p)$ by $U$ and the set of all precolored vertices by $P$, so $U\sqcup P=V(G)$. Assume that $G$ has no edge between vertices in $P$, otherwise delete it and decrease $k$ by one if its endpoints are of the same color in $(G,p)$. Construct an instance $(G^{\prime},p^{\prime},k^{\prime})$ as follows. Use the algorithm that output instances of $\mathcal{G}$, with $|P|$ as input. The algorithm gives a graph $G^{\prime\prime}$, such that $|V(G^{\prime\prime})|\geq|P|$ and $G^{\prime\prime}\in\mathcal{G}$. Take an arbitrary subset $P^{\prime}\subseteq V(G^{\prime\prime})$ of size $|P|$, and identify its vertices with vertices in $P$. Construct $G^{\prime}$ by introducing $|U|$ new vertices to $G^{\prime\prime}$, that are identified with the vertices of $U$. Denote the set of these vertices by $U^{\prime}$. Then, add an edge between vertices in $U^{\prime}$ or between vertices in $U^{\prime}$ and $P^{\prime}$ if there is an edge between corresponding vertices in $G$. Finally, construct $p^{\prime}$ by precoloring vertices in $P^{\prime}$ with the color of their corresponding vertices in $(G,p)$, leave the vertices of $U^{\prime}$ uncolored, and precolor all remaining vertices arbitrarily. There may be some happy edges between precolored vertices in $(G^{\prime},p^{\prime})$, let their number be $h$. Set $k^{\prime}=k+h$.

One may easily show that the constructed instance $(G^{\prime},p^{\prime},k^{\prime})$ is a yes-instanse of MHE if and only if the initial instance $(G,p,k)$ is a yes-instance of MHE. Moreover, deletion of $U^{\prime}$ from $G^{\prime}$ yields $G^{\prime\prime}\in\mathcal{G}$. Hence, $G^{\prime}$ has the distance to graphs in $\mathcal{G}$ being at most $|U^{\prime}|=|U|=d$. We therefore obtain the required polynomial reduction that leads to the desired contradiction. $\Box$

In the rest of the section we study kernel bitsize lower bounds for MHV, parameterized by either $k+\ell$ or $k+h$, where $h$ is the number of potentially happy vertices. This relates to the result of Agrawal in [Agrawal2018], where the author showed that MHV admits a polynomial kernel with $\mathcal{O}(k^{2}\ell^{2})$ vertices. We show that, for any $d>0$ and any $\epsilon>0$, there is no kernel of bitsize $\mathcal{O}(k^{d}\cdot\ell^{2-\epsilon})$ for MHV. Similarly, we show that there is no kernel of bitsize $\mathcal{O}(k^{d}\cdot h^{2-\epsilon})$ for MHV. To prove these lower bounds, we refer to the framework of weak cross-compositions, that originates from works of Dell and van Mekelbeek [DellMekelbeek2014], Dell and Marx [Dell2012] and Hermelin and Wu [Hermelin2012]. These results are finely summarized by Cygan et al. in the chapter on lower bounds for kernelization [Cygan2015]. We recall the notion of weak cross-compositions.

Definition 3** ([Cygan2015, Dell2012, Hermelin2012]).**

Let $L\subseteq\Sigma^{*}$ be a language and $Q\subseteq\Sigma^{*}\times\mathbb{N}$ be a parameterized language. We say that $L$ weakly-cross-composes into $Q$ if there exists a real constant $d\geq 1$, called the dimension, a polynomial equivalence relation $\mathcal{R}$, and an algorithm $\mathcal{A}$, called the weak cross-composition, satisfying the following conditions. The algorithm $\mathcal{A}$ takes as input a sequence of $x_{1},x_{2},\ldots,x_{t}\in\Sigma^{*}$ that are equivalent with respect to $\mathcal{R}$, runs in time polynomial in $\sum^{t}_{i=1}|x_{i}|$, and outputs one instance $(y,k)\in\Sigma^{*}\times\mathbb{N}$ such that:

1. (a)

for every $\delta>0$ there exists a polynomial $p(\cdot)$ such that for every choice of $t$ and input strings $x_{1},x_{2},\ldots,x_{t}$ it holds that $k\leq p(\max^{t}_{i=1}|x_{i}|)\cdot t^{\frac{1}{d}+\delta}$, and 2. (b)

$(y,k)\in Q$ if and only if there exists at least one index i such that $x_{i}\in L$.

The framework of weak cross-compositions is used for proving conditional lower bounds on polynomial compression bitsize. This is formulated in the following theorem.

Theorem 4** ([Cygan2015, Dell2012, Hermelin2012]).**

*If an \NP-hard language $L$ admits a weak cross-composition of dimension $d$ into a parameterized language $Q$. Then for any $\epsilon>0$, $Q$ does not admit a polynomial compression with bitsize $\mathcal{O}(k^{d-\epsilon})$, unless $\NP\subseteq\coNP/\poly$. *

Dell and Marx [Dell2012] use this framework to show that the Vertex Cover problem parameterized by the solution size does not admit a kernel with subquadratic bitsize. Their result is the following.

Lemma 1** ([Dell2012, Cygan2015]).**

There exists a weak cross composition of dimension $2$ from an \NP-hard problem Multicolored Biclique into the Vertex Cover problem parameterized by the solution size. In fact, this weak cross-composition $\mathcal{A}$, given instances $x_{1},x_{2},\ldots,x_{t}$ of Multicolored Biclique as input, outputs an instance $(G,k^{\prime})$ of Vertex Cover satisfying

• •

$|V(G)|\leq p(\max^{t}_{i=1}|x_{i}|)\cdot\sqrt{t}$, and

• •

$|V(G)|-k^{\prime}\leq q(\max^{t}_{i=1}|x_{i}|),$**

*for some polynomials $p$ and $q$. *

The bound for $|V(G)|-k^{\prime}$ is given because one can look at an instance $(G,k^{\prime})$ of Vertex Cover as at an instance $(G,|V(G)|-k^{\prime})$ of Independent Set. Then, the solution parameter of Independent Set is bounded with polynomial of the maximum input size, independently of the number of instances $t$. We are ready to prove the theorem.