Commit fe97fd5d by Daniel Gonçalves

### New section "Parameters"

Replacing the tree-decomposition by clique-sum operations.
Introduction of clique-sum extendable representation.
Replacing treewidth by k-trees
parent d42d6198
 ... @@ -20,6 +20,7 @@ ... @@ -20,6 +20,7 @@ \newcommand{\bb}[1]{\mathbb{#1}} \newcommand{\bb}[1]{\mathbb{#1}} \newcommand{\brm}[1]{\operatorname{#1}} \newcommand{\brm}[1]{\operatorname{#1}} \newcommand{\cbdim}{\brm{dim}_{cb}} \newcommand{\cbdim}{\brm{dim}_{cb}} \newcommand{\ecbdim}{\brm{dim}^{ext}_{cb}} \newcommand{\tw}{\brm{tw}} \newcommand{\tw}{\brm{tw}} \newcommand{\vol}{\brm{vol}} \newcommand{\vol}{\brm{vol}} %%%%% %%%%% ... @@ -33,11 +34,12 @@ ... @@ -33,11 +34,12 @@ \newtheorem{proposition}[theorem]{Proposition} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{observation}[theorem]{Observation} \newtheorem{observation}[theorem]{Observation} \newtheorem{question}[theorem]{Question} \newtheorem{question}[theorem]{Question} \newtheorem{definition}[theorem]{Definition} \title{On comparable box dimension} \title{On comparable box dimension} \author{Zden\v{e}k Dvo\v{r}\'ak\thanks{Computer Science Institute, Charles University, Prague, Czech Republic. E-mail: {\tt rakdver@iuuk.mff.cuni.cz}. \author{Zden\v{e}k Dvo\v{r}\'ak\thanks{Computer Science Institute, Charles University, Prague, Czech Republic. E-mail: {\tt rakdver@iuuk.mff.cuni.cz}. Supported by the ERC-CZ project LL2005 (Algorithms and complexity within and beyond bounded expansion) of the Ministry of Education of Czech Republic.}\and Supported by the ERC-CZ project LL2005 (Algorithms and complexity within and beyond bounded expansion) of the Ministry of Education of Czech Republic.}\and Daniel Gon\c{c}alves\thanks{...}\and Daniel Gon\c{c}alves\thanks{LIRMM, Université de Montpellier, CNRS, Montpellier, France. E-mail: {\tt goncalves@lirmm.fr}. Supported by the ANR grant GATO ANR-16-CE40-0009.}\and Abhiruk Lahiri\thanks{...}\and Abhiruk Lahiri\thanks{...}\and Jane Tan\thanks{Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK. E-mail: {\tt jane.tan@maths.ox.ac.uk}}\and Jane Tan\thanks{Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK. E-mail: {\tt jane.tan@maths.ox.ac.uk}}\and Torsten Ueckerdt\thanks{Karlsruhe Institute of Technology. E-mail: {\tt torsten.ueckerdt@kit.edu}}} Torsten Ueckerdt\thanks{Karlsruhe Institute of Technology. E-mail: {\tt torsten.ueckerdt@kit.edu}}} ... @@ -47,8 +49,12 @@ Torsten Ueckerdt\thanks{Karlsruhe Institute of Technology. E-mail: {\tt torsten ... @@ -47,8 +49,12 @@ Torsten Ueckerdt\thanks{Karlsruhe Institute of Technology. E-mail: {\tt torsten \maketitle \maketitle \begin{abstract} \begin{abstract} Two boxes in $\mathbb{R}^d$ are comparable if one of them is a subset of a translation of the other. The comparable box dimension of a graph $G$ is the minimum integer $d$ such that $G$ can be represented as a touching graph of comparable axis-aligned boxes in $\mathbb{R}^d$. We show that proper minor-closed classes have bounded Two boxes in $\mathbb{R}^d$ are comparable if one of them is a subset comparable box dimension and explore further properties of this notion. of a translation of the other. The comparable box dimension of a graph $G$ is the minimum integer $d$ such that $G$ can be represented as a touching graph of comparable axis-aligned boxes in $\mathbb{R}^d$. We show that proper minor-closed classes have bounded comparable box dimension and explore further properties of this notion. \end{abstract} \end{abstract} \section{Introduction} \section{Introduction} ... @@ -98,237 +104,40 @@ This gives arbitrarily precise approximation algorithms for all monotone maximiz ... @@ -98,237 +104,40 @@ This gives arbitrarily precise approximation algorithms for all monotone maximiz expressible in terms of distances between the solution vertices and tractable on graphs of bounded treewidth~\cite{distapx} expressible in terms of distances between the solution vertices and tractable on graphs of bounded treewidth~\cite{distapx} or expressible in the first-order logic~\cite{logapx}. or expressible in the first-order logic~\cite{logapx}. \section{Operations} Let us start with a simple lemma saying that the addition of a vertex increases the comparable box dimension by at most one. \section{Parameters} In particular, this implies that $\cbdim(G)\le |V(G)|$. \begin{lemma}\label{lemma-apex} For any graph $G$ and $v\in V(G)$, we have $\cbdim(G)\le \cbdim(G-v)+1$. \end{lemma} \begin{proof} Let $f$ be a touching representation of $G-v$ by comparable boxes in $\mathbb{R}^d$, where $d=\cbdim(G-v)$. We define a representation $h$ of $G$ as follows. For each $u\in V(G)\setminus\{v\}$, let $h(u)=[0,1]\times f(u)$ if $uv\in E(G)$ and $h(u)=[1/2,3/2]\times f(u)$ if $uv\not\in E(G)$. Let $h(v)=[-1,0]\times [-M,M] \times \cdots \times [-M,M]$, where $M$ is chosen large enough so that $f(u)\subseteq [-M,M] \times \cdots \times [-M,M]$ for every $u\in V(G)\setminus\{v\}$. Then $h$ is a touching representation of $G$ by comparable boxes in $\mathbb{R}^{d+1}$. \end{proof} We need a bound on the clique number in terms of the comparable box dimension. Let us first bound the clique number $\omega(G)$ in terms of For a box $B=I_1\times \cdots \times I_d$ and $i\in\{1,\ldots,d\}$, let $B[i]=I_i$. $\cbdim(G)$. \begin{lemma}\label{lemma-cliq} \begin{lemma}\label{lemma-cliq} If $G$ has a touching representation $f$ by comparable boxes in $\mathbb{R}^d$, then $\omega(G)\le 2^d$. For any graph $G$, then $\omega(G)\le 2^{\cbdim(G)}$. \end{lemma} \begin{proof} To represent any clique $A = \{a_1,\ldots,a_w\}$ in $G$, the corresponding boxes $f(a_1),\ldots,f(a_w)$ have pairwise non-empty intersection. Since axis-aligned boxes have the Helly property, there is a point $p \in \mathbb{R}^d$ contained in $f(a_1) \cap \cdots \cap f(a_w)$. As each box is full-dimensional, its interior intersects at least one of the $2^d$ orthants at $p$. Since $f$ is a touching representation, $f(a_1),\ldots,f(a_d)$ have pairwise disjoint interiors and hence $w \leq 2^d$. \end{proof} A \emph{tree decomposition} of a graph $G$ is a pair $(T,\beta)$, where $T$ is a rooted tree and $\beta:V(T)\to 2^{V(G)}$ assigns a \emph{bag} to each of its nodes, such that \begin{itemize} \item for each $uv\in E(G)$, there exists $x\in V(T)$ such that $u,v\in\beta(x)$, and \item for each $v\in V(G)$, the set $\{x\in V(T):v\in\beta(x)\}$ is non-empty and induces a connected subtree of $T$. \end{itemize} For nodes $x,y\in V(T)$, we write $x\preceq y$ if $x=y$ or $x$ is a descendant of $y$ in $T$. For each vertex $v\in V(G)$, let $p(v)$ be the node $x\in V(T)$ such that $v\in \beta(x)$ and $x$ is nearest to the root of $T$. The \emph{adhesion} of the tree decomposition is the maximum of $|\beta(x)\cap\beta(y)|$ over distinct $x,y\in V(T)$, and its \emph{width} is the maximum of the sizes of the bags minus $1$. The \emph{treewidth} of a graph is the minimum of the widths of its tree decompositions. We will need to know that graphs of bounded treewidth have bounded comparable box dimension. In fact, we will prove the following stronger fact. \note{(TODO: Was this published somewhere before? I am only aware of the upper bound of $t+2$ on the boxicity of $G$~\cite{box-treewidth}.)} \begin{lemma}\label{lemma-tw} Let $(T,\beta)$ be a tree decomposition of a graph $G$ of width $t$. Then $G$ has a touching representation $h$ by axis-aligned hypercubes in $\mathbb{R}^{t+1}$ such that for $u,v\in V(G)$, if $p(u)\prec p(v)$, then $h(u)\sqsubset h(v)$. Moreover, the representation can be chosen so that no two hypercubes have the same size. \end{lemma} \begin{proof} Without loss of generality, we can assume that the root has a bag of size one and that for each $x\in V(T)$ with parent $y$, we have $|\beta(x)\setminus\beta(y)|=1$ (if $\beta(x)\subseteq \beta(y)$, we can contract the edge $xy$; if $|\beta(x)\setminus\beta(y)|>1$, we can subdivide the edge $xy$, introducing $|\beta(x)\setminus\beta(y)|-1$ new vertices, and set their bags appropriately). It is now natural to relabel the vertices of $G$ so that $V(G)=V(T)$, by giving the unique vertex in $\beta(x)\setminus\beta(y)$ the label $x$. In particular, $p(x)=x$. Furthermore, switching to this new labeling, we can assume that $y\in\beta(x)$. Otherwise, if $y$ is not the root of $T$, we can replace the edge $xy$ by the edge from $x$ to the parent of $y$. If $y$ is the root of $T$, then the subtree rooted in $x$ induces a union of connected components in $G$, and we can process this subtree separately from the rest of the graph (being careful to only use hypercubes smaller than the one representing $y$ and of different sizes from those used on the rest of the graph). Let us now greedily color $G$ by giving $x$ a color different from the colors of all other vertices in $\beta(x)$; such a coloring $\varphi$ uses only colors $\{1,\ldots,t+1\}$. Let $D=4\Delta(T)+1$. Let $V(G)=V(T)=\{x_1,x_2,\ldots, x_n\}$, where for every $ij$. We greedily color the vertices in order, giving $v_i$ the smallest i.e., so that $f(v_i)\sqsubseteq f(v_j)$ whenever $i>j$. We greedily color the vertices in order, giving $v_i$ the smallest color different from the colors of all vertices $v_j$ such that $jj$ color different from the colors of all vertices $v_j$ such that $jj$ ... @@ -346,9 +155,100 @@ is at most $(3^d-1)^2$. ... @@ -346,9 +155,100 @@ is at most $(3^d-1)^2$. Consequently, when choosing the color of $v_i$ greedily, we only need to avoid colors of at most Consequently, when choosing the color of $v_i$ greedily, we only need to avoid colors of at most $$(5^d-1) + (3^d-1) + (3^d-1)^2<5^d+9^d<2\cdot 9^d$$ $$(5^d-1) + (3^d-1) + (3^d-1)^2<5^d+9^d<2\cdot 9^d$$ vertices. vertices. We proceed similarly to bound the chromatic number. \end{proof} \end{proof} \section{Operations} It is clear that given a touching representation of a graph $G$, one easily obtains a touching representation with boxes of an induced subgraph $H$ of $G$ by simply deleting the boxes corresponding to the vertices in $V(G)\setminus V(H)$. In this section we are going to consider other basic operations on graphs. \subsection{Vertex addition} Let us start with a simple lemma saying that the addition of a vertex increases the comparable box dimension by at most one. In particular, this implies that $\cbdim(G)\le |V(G)|$. \begin{lemma}\label{lemma-apex} For any graph $G$ and $v\in V(G)$, we have $\cbdim(G)\le \cbdim(G-v)+1$. \end{lemma} \begin{proof} Let $f$ be a touching representation of $G-v$ by comparable boxes in $\mathbb{R}^d$, where $d=\cbdim(G-v)$. We define a representation $h$ of $G$ as follows. For each $u\in V(G)\setminus\{v\}$, let $h(u)=[0,1]\times f(u)$ if $uv\in E(G)$ and $h(u)=[1/2,3/2]\times f(u)$ if $uv\not\in E(G)$. Let $h(v)=[-1,0]\times [-M,M] \times \cdots \times [-M,M]$, where $M$ is chosen large enough so that $f(u)\subseteq [-M,M] \times \cdots \times [-M,M]$ for every $u\in V(G)\setminus\{v\}$. Then $h$ is a touching representation of $G$ by comparable boxes in $\mathbb{R}^{d+1}$. \end{proof} \subsection{Strong product} Let $G\boxtimes H$ denote the \emph{strong product} of the graphs $G$ and $H$, i.e., the graph with vertex set $V(G)\times V(H)$ and with distinct vertices $(u_1,v_1)$ and $(u_2,v_2)$ adjacent if and only if either $u_1=u_2$ or $u_1u_2\in E(G)$ and either $v_1=v_2$ or $v_1v_2\in E(G)$. To obtain a touching representation of $G\boxtimes H$ it suffice to take a product of representations of $G$ and $H$, but the obtained representation may contain uncomparable boxes. Thus, bounding $\cbdim(G\boxtimes H)$ in terms of $\cbdim(G)$ and $\cbdim(H)$ seems to be a complicated task. In the following lemma we overcome this issue, by constraining one of the representations. \begin{lemma}\label{lemma-sp} Consider a graph $H$ having a touching representation $h$ in $\mathbb{R}^{d_H}$ with hypercubes of unit size. Then for any graph $G$, the strong product of these graphs is such that $\cbdim(G\boxtimes H) \le \cbdim(G) + d_H$. \end{lemma} \begin{proof} The proof simply consists in taking a product of the two representations. Indeed, consider a touching respresentation with comparable boxes $g$ of $G$ in $\mathbb{R}^{d_G}$, with $d_G=\cbdim(G)$, and the depresentation $h$ of $H$. Let us define a representation $f$ of $G\boxtimes H$ in $\mathbb{R}^{d_G+d_H}$ as follows. $$f((u,v))[i]=\begin{cases} g(u)[i]&\text{ if i\le d_G}\\ h(u)[i-d_G]&\text{ if i > d_G} \end{cases}$$ Notice first that the boxes of $f$ are comparable as $f((u,v)) \sqsubset f((u',v'))$ if and only if $g(u) \sqsubset g(u')$. Now let us observe that for any two vertices $u, u'$ of $G$, there is an hyperplane separating the interiors of $g(u)$ and $g(u')$, and similarly for $h$ and $H$. This implies that the boxes in $f$ are interior disjoint. Indeed, the same hyperplane that separates $g(u)$ and $g(u')$, when extended to $\mathbb{R}^{d_G+d_H}$, now separates any two boxes $f((u,v))$ and $f((u',v'))$. This implies that $f$ is a touching representation of a subgraph of $G\boxtimes H$. Similarly, one can also observe that there is a point $p$ in the intersection of $f((u,v))$ and $f((u',v'))$, if and only if there is a point $p_G$ in the intersection of $g(u)$ and $g(u')$, and a point $p_H$ in the intersection of $h(v)$ and $h(v')$. Indeed, one can obtain $p_G$ and $p_H$ by projecting $p$ in $\mathbb{R}^{d_G}$ or in $\mathbb{R}^{d_G}$ respectively, and conversely $p$ can be obtained by taking the product of $p_G$ and $p_H$. Thus $f$ is indeed a touching representation of $G\boxtimes H$. \end{proof} \subsection{Subgraph} Examples show that the comparable box dimension of a graph $G$ may be larger than the one a subgraph $H$ of $G$. However we show that the comparable box dimension of a subgraph is at most exponential in the comparable box dimension of the whole graph. This is essentially Corollary~25 in~\cite{subconvex}, but since the setting is somewhat different and the construction of~\cite{subconvex} uses rotated boxes, we provide details of the argument. Next, let us show a bound on the comparable box dimension of subgraphs. Next, let us show a bound on the comparable box dimension of subgraphs. \begin{lemma}\label{lemma-subg} \begin{lemma}\label{lemma-subg} ... @@ -387,65 +287,344 @@ If $G$ is a subgraph of a graph $G'$, then $\cbdim(G)\le \cbdim(G')+4\cdot 81^{\ ... @@ -387,65 +287,344 @@ If$G$is a subgraph of a graph$G'$, then$\cbdim(G)\le \cbdim(G')+4\cdot 81^{\ Let us remark that an exponential increase in the dimension is unavoidable: We have $\cbdim{K_{2^d}}=d$, Let us remark that an exponential increase in the dimension is unavoidable: We have $\cbdim{K_{2^d}}=d$, but the graph obtained from $K_{2^d}$ by deleting a perfect matching has comparable box dimension $2^{d-1}$. but the graph obtained from $K_{2^d}$ by deleting a perfect matching has comparable box dimension $2^{d-1}$. Corollaries~\ref{cor-cs} and~\ref{cor-subg} now give the main result of this section. \begin{corollary}\label{cor-comb} If $G$ is obtained from graphs in a class $\GG$ by clique-sums, then $\cbdim(G)\le 5\cdot 81^{6^{\cbdim(\GG)}}$. \end{corollary} \section{The product structure and minor-closed classes} \subsection{Clique-sums} A \emph{clique-sum} of two graphs $G_1$ and $G_2$ is obtained from their disjoint union by identifying vertices of a clique in $G_1$ and a clique of the same size in $G_2$ and possibly deleting some of the edges of the resulting clique. A \emph{full clique-sum} is a clique-sum in which we keep all the edges of the resulting clique. The main issue to overcome in obtaining a representation for a (full) clique-sum is that the representations of $G_1$ and $G_2$ can be degenerate''. Consider e.g.\ the case that $G_1$ is represented by unit squares arranged in a grid; in this case, there is no space to attach $G_2$ at the cliques formed by four squares intersecting in a single corner. This can be avoided by increasing the dimension, but we need to be careful so that the dimension stays bounded even after an arbitrary number of clique-sums. We thus introduce the notion of \emph{clique-sum extendable} representations. \begin{definition} Consider a graph $G$ with a distinguished clique $C^*$, called the \emph{root clique} of $G$. A touching representation (with comparable boxes or not) $h$ of $G$ in $\mathbb{R}^d$ is called \emph{$C^*$-clique-sum extendable} if the following conditions hold. \begin{itemize} \item[{\bf(vertices)}] There are $|V(C^*)|$ dimensions, denoted $d_u$ for each vertex $u\in V(C^*)$, such that: \begin{itemize} \item[(v1)] for each vertex $u\in V(C^*)$, $h(u)[d_u] = [-1 ,0]$ and $h(u)[i] = [0,1]$, for any dimension $i\neq d_u$, and \item[(v2)] for any vertex $v\notin V(C^*)$, $h(v) \subset [0,1)^d$. \end{itemize} \item[{\bf(cliques)}] For every clique $C$ of $G$ we define a point $p(C)\in I_C \cap [0,1)^d$, where $I_C =\cap_{v\in V(C)} h(v)$, and we define the box $h^\epsilon(C)$, for any $\epsilon > 0$, by $h^\epsilon(C)[i] = [p(C)[i],p(C)[i]+\epsilon]$, for every dimension $i$. Furthermore, for a sufficiently small $\epsilon > 0$ these \emph{clique boxes} verify the following conditions. \begin{itemize} \item[(c1)] For any two cliques $C_1\neq C_2$, $h^\epsilon(C_1) \cap h^\epsilon(C_2) = \emptyset$ (i.e. $p(C_1) \neq p(C_2)$). \item[(c2)] A box $h(v)$ intersects $h^\epsilon(C)$ if and only if $v\in V(C)$, and in that case their intersection is a facet of $h^\epsilon(C)$ incident to $p(C)$ (i.e. if we denote this intersection $I$, then $I[i] = \{p(C)[i] \}$ for some dimension $i$, and $I[j] = [p(C)[j],p(C)[j]+\epsilon]$ for the other dimensions $j\neq i$). \end{itemize} \end{itemize} \end{definition} Note that we may consider that the root clique is empty, that is the empty subgraph with no vertices. In that case the clique is denoted $\emptyset$. Let $\ecbdim(G)$ be the minimum dimension such that $G$ has a $\emptyset$-clique-sum extendable touching representation with comparable boxes. The following lemma ensures that clique-sum extendable representations behave well with respect to full clique-sums. \begin{lemma}\label{lem-cs} Consider two graphs $G_1$ and $G_2$, given with a $C^*_1$- and a $C^*_2$-clique-sum extendable representations with comparable boxes $h_1$ and $h_2$, in $\mathbb{R}^{d_1}$ and $\mathbb{R}^{d_2}$ respectively. Let $G$ be the graph obtained after performing a full clique-sum of these two graphs on any clique $C_1$ of $G_1$, and on the root clique $C^*_2$ of $G_2$. Then $G$ admits a $C^*_1$-clique sum extendable representation with comparable boxes $h$ in $\mathbb{R}^{\max(d_1,d_2)}$. Furthermore, we have that the aspect ratio of $h(v)$ is the same as the one of $h_1(v)$, if $v\in V(G_1)$, or the same as $h_2(v)$ if $v\in V(G_2)\setminus V(C^*_2)$. \note{ shall we remove the aspect ratio thing ? only needed for the hypercubes of the k-trees, but those are not really needed...} \end{lemma} \begin{proof} The idea is to translate (allowing also exchanges of dimensions) and scale $h_2$ to fit in $h_1^\epsilon(C_1)$. Consider an $\epsilon >0$ sufficiently small so that, $h_1^\epsilon(C_1)$ verifies all the (cliques) conditions, and such that $h_1^\epsilon(C_1) \sqsubseteq h_1(v)$ for any vertex $v\in V(G_1)$. Without loss of generality, let us assume that $V(C_1)=\{v_1,\ldots,v_k\}$, and we also assume that $h_1(v_i)$ and $h_1^\epsilon(C_1)$ touch in dimension $i$ (i.e. $h_1(v_i)[i] \cap h_1^\epsilon(C_1)[i] = \{p(C_1)[i]\}$, and $h_1(v_i)[j] \cap h_1^\epsilon(C_1)[j] = [p(C_1)[j],p(C_1)[j]+\epsilon]$ for $j\neq i$. Now let us consider $G_2$ and its representation $h_2$. Here the vertices of $C^*_2$ are also denoted $v_1,\ldots,v_k$, and let us denote $d_{v_i}$ the dimension in $h_2$ that fulfills condition (v1) with respect to $v_i$. Let $d=\max(d_1,d_2)$. We are now ready for defining $h$. For the vertices of $G_1$ it is almost the same representation as $h_1$, as we set $h(v)[i] = h_1(v)[i]$ for any dimension $i\le d_1$. If $d=d_2 > d_1$, then for any dimension $i>d_1$ we set $h(v)[i] = [0,1]$ if $v\in V(C^*_1)$, and $h(v)[i] = [0,\frac12]$ if $v\in V(G_1)\setminus V(C^*_1)$. Similarly the clique points $p_1(C)$ become $p(C)$ by setting $p(C)[i] = p_1(C)[i]$ for $i\le d_1$, and $p(C)[i] = \frac14$ for $i> d_1$. For $h_2$ and the vertices in $V(G_2) \setminus \{v_1,\ldots,v_k\}$ we have to consider a mapping $\sigma$ from $\{1,\ldots,d_2\}$ to $\{1,\ldots,d\}$ such that $\sigma(d_{v_i}) = i$. This mapping describes the changes of dimension we have to perform. We also have to perform a scaling in order to make $h_2$ fit inside $h_1^\epsilon(C_1)$. This is ensured by multiplying the coordinates by $\epsilon$. More formally, for any vertex \$v\in V(G_2) \setminus