Extended Formulation for CSP that is Compact
for Instances of Bounded Treewidth^††thanks: This research was partially supported by the project 14-10003S of GA ČR.

An instance $Q=(V,\mathcal{D},\mathcal{H},\mathcal{C})$ of CSP consists of

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  a set of variables $z_{v}$, one for each $v\in V$; without loss of generality we assume that $V=\{1,\ldots,n\}$,

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  a set $\mathcal{D}$ of finite domains $D_{v}\subseteq\mathbb{R}$ (also denoted $D(v)$), one for each $v\in V$,

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  a set of hard constraints $\mathcal{H}\subseteq\{C_{U}\ |\ U\subseteq V\}$ where each hard constraint $C_{U}\in\mathcal{H}$ with $U=\{i_{1},i_{2},\dots,i_{k}\}$ and $i_{1}<i_{2}<\cdots<i_{k}$, is a $|U|$-ary relation $C_{U}\subseteq D_{i_{1}}\times D_{i_{2}}\times\cdots\times D_{i_{k}}$,

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  a set of soft constraints $\mathcal{C}\subseteq\{C_{U}\ |\ U\subseteq V\}$ where each soft constraint $C_{U}\in\mathcal{C}$ with $U=\{i_{1},i_{2},\dots,i_{k}\}$ and $i_{1}<i_{2}<\cdots<i_{k}$, is a $|U|$-ary relation $C_{U}\subseteq D_{i_{1}}\times D_{i_{2}}\times\cdots\times D_{i_{k}}$.

The constraint graph of $Q$ is defined as $G=(V,E)$ where $E=\{\{u,v\}\ |\ \exists C_{U}\in\mathcal{C}\cup\mathcal{H}\textrm{ s.t. }\{u,v\}\subseteq U\}$. We say that a CSP instance $Q$ has bounded treewidth if the constraint graph of $Q$ has bounded treewidth. In binary CSP, every hard and soft relation is a unary or binary relation, and in boolean CSP, the domain of every variable is $\{0,1\}$. We use $D$ to denote the maximal size of all domains, that is, $D=\max_{u\in V}|D_{u}|$.

For a vector $z=(z_{1},z_{2},\ldots,z_{n})$ and $U=\{i_{1},i_{2},\dots,i_{k}\}\subseteq V$ with $i_{1}<i_{2}<\cdots<i_{k}$, we define the projection of $z$ on $U$ as $z|_{U}=(z_{i_{1}},z_{i_{2}},\ldots,z_{i_{k}})$. A vector $z\in\mathbb{R}^{n}$ satisfies the constraint $C_{U}\in\mathcal{C}\cup\mathcal{H}$ if and only if $z|_{U}\in C_{U}$. We say that a vector $z^{\star}=(z^{\star}_{1},\ldots,z^{\star}_{n})$ is a feasible assignment for $Q$ if $z^{\star}\in D_{1}\times D_{2}\times\ldots\times D_{n}$ and $z^{\star}$ satisfies every hard constraint $C\in\mathcal{H}$. For a given feasible assignment $z^{\star}$ we define an extended feasible assignment ex$(z^{\star})=(z^{\star},h^{\star})\in\mathbb{R}^{n+|\mathcal{C}|}$ as follows: the coordinates of $h^{\star}$ are indexed by the soft constraints from $\mathcal{C}$ (to be more precise: by the subsets $U$ of $V$ used as lower indices of the soft constraints) and for each $C_{U}\in\mathcal{C}$, we have $h^{\star}_{U}=1$ if and only if $z^{\star}|_{U}\in C_{U}$, and $h^{\star}_{U}=0$ otherwise. We denote by $\mathcal{F}(Q)$ the set of all feasible assignments for $Q$, by $\mathcal{F}^{ex}(Q)=\{$ex$(z^{\star})\ |\ z^{\star}\in\mathcal{F}(Q)\}$ the set of all extended feasible assignments for $Q$. For every instance $Q$ we define two polytopes: $CSP(Q)$ is the convex hull of $\mathcal{F}^{ex}(Q)$ and $CSP^{\prime}(Q)$ the convex hull of $\mathcal{F}(Q)$. We also define three trivial linear projections:

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  $\textrm{proj}_{V}(z,h)=z$,       $\textrm{proj}_{E}(z,h)=h$,       $\textrm{proj}_{id}(z,h)=(z,h)$

where $z\in\mathbb{R}^{n}$ and $h\in\mathbb{R}^{|\mathcal{C}|}$, and observe that $\textrm{proj}_{V}(CSP(Q))=CSP^{\prime}(Q)$.

In the decision version of CSP, the set $\mathcal{C}$ of soft constraints is empty and the task is to decide whether there exists a feasible assignment. In the maximization (minimization, resp.) version of the problem, the task is to find a feasible assignment that maximizes (minimizes, resp.) the number of satisfied (unsatisfied, resp.) soft constraints. Note that there is no difference between maximization and minimization versions of the problem with respect to optimal solutions but the two versions differ significantly from an approximation perspective.

In the weighted version of CSP we are also given a weight function $w:\mathcal{C}\rightarrow\mathbb{R}$ that specifies for each soft constraint $C\in\mathcal{C}$ its weight $w(C)$. The goal is to find a feasible assignment that maximizes (minimizes, resp.) the total weight of satisfied (unsatisfied, resp.) constraints. The unweighted version of CSP is equivalent to the weighted version with $w(C)=1$ for all $C\in\mathcal{C}$.

Even more generally, the relations in the soft constraints can be replaced by bounded real valued payoff functions: a soft constraint $C_{U}\in\cal C$ with $U=\{i_{1},i_{2},\dots,i_{k}\}$ is not a $|U|$-ary relation but a function $w:D_{i_{1}}\times D_{i_{2}}\times\ldots\times D_{i_{k}}\rightarrow\mathbb{R}$ and the payoff of the soft constraint $C_{U}$ for a feasible assignment $z^{\star}$ is $w(z^{\star}|_{U})$; the objective is to maximize (minimize, resp.) the total payoff. For the sake of simplicity of the presentation we do not consider the problem in this generality although the techniques used in this paper apply in the general setting as well.

For notions related to the treewidth of a graph, we stick to the standard terminology as given in the book by Kloks [10]).

Our main result is summarized as the following theorem.

As a corollary we obtain upper bounds on the extension complexity for several NP-hard problems on the class of graphs with bounded treewidth; as far as we know, these results have not been known.

We start by introducing the terms and notation that we use throughout this section. We assume that $Q=(V,\mathcal{D},\mathcal{H},\mathcal{C})$ is a given instance of CSP. For every subset $W\subseteq V$ we define the set of all configurations of $W$ as

where $\lambda$ is a symbol not appearing in any of the domains $D_{u}$, $u\in V$. For a configuration $K\in\mathcal{K}(U)$ and $v\in V$, we use the notation $K(v)$ to refer to the $v$-th element of $K$. Also, for a configuration $K\in\mathcal{K}(U)$, $v\in V\setminus U$ and $\alpha\in D_{v}$, we use the notation $K[v\leftarrow\alpha]$ to denote the configuration $K^{\prime}\in\mathcal{K}(U\cup\{v\})$ such that $K^{\prime}(v)=\alpha$ and $K^{\prime}(u)=K(u)$ for every $u\neq v$.

For an $n$-dimensional vector $K=(\alpha_{1},\dots,\alpha_{n})$ and a subset of variables $U\subseteq V$ we denote by $K\hskip-2.84526pt\upharpoonright_{U}$ the restriction of $K$ to $U$ that is defined as an $n$-dimensional vector with $K\hskip-2.84526pt\upharpoonright_{U}(i)=K(i)$ for $i\in U$ and $K\hskip-2.84526pt\upharpoonright_{U}(i)=\lambda$ for $i\not\in U$ (i.e., we set to $\lambda$ all coordinates of $K$ outside of $U$). We denote by $\Lambda$ the configuration $(\lambda,\dots,\lambda)\in\mathcal{K}(\emptyset)$; note that for $\alpha\in D_{v}$, $\Lambda[v\leftarrow\alpha]$ is the configuration from $\mathcal{K}(\{v\})$ with exactly one non-$\lambda$ element, namely the $v$-th element, equaling $\alpha$.

In our linear program, for every index $v\in V$ and every $i\in D_{v}$, we introduce a binary variable $y_{v}^{i}$. The task of the variable $y_{v}^{i}$ is to encode the value of the CSP-variable $z_{v}$: the variable $y_{v}^{i}$ is set to one if and only if $z_{v}=i$. Since in every solution each variable assumes a unique value, we enforce the constraint $\sum_{i\in D(v)}y_{v}^{i}=1$ for each $v\in V$.

For every configuration $K\in\bigcup_{U:C_{U}\in\mathcal{C}\cup\mathcal{H}}\mathcal{K}(U)$ we introduce a binary variable $g(K)$. The intended meaning of the variable $g(K)$, for $K\in\mathcal{K}(U)$ and $U\subseteq V$, is to provide information about the values of the CSP-variables $z_{u}$ for $u\in U$ in the following way: $g(K)=1$ if and only if for every $u\in U$, $z_{u}=K(u)$. To ensure consistency between the $y$ and $g$ variables, for every $C_{U}\in\mathcal{C}\cup\mathcal{H}$ and for every $v\in U$, we enforce the constraint $\sum_{K\in\mathcal{K}(U):K(v)=i}g(K)=y_{v}^{i}$. Note that for binary CSP, the $g$ variables capture the values of CSP-variables $z$ for pairs of elements from $V$ that correspond to edges of the constraint graph.

Relaxing the integrality constraints we obtain the following initial LP relaxation of the CSP problem $Q=(V,\mathcal{D},\mathcal{H},\mathcal{C})$:

Note that there is a one to one correspondence between the (extended) feasible assignments of $Q$ and integral solutions of (1) - (3); from now on we denote by $\textrm{proj}_{1}$ the linear projection of the convex hull of integral solutions of (1) - (3) to $CSP(Q)$. Also observe that the total weight of CSP-constraints satisfied by an integral vector $(\bm{y},\bm{g})$ satisfying (1) - (3) is¹¹1In the case of general payoff functions, the total weight is given by $\sum_{C_{U}\in\mathcal{C}}\sum_{K\in\mathcal{K}(U):K|_{U}\in C_{U}}w(K|_{U})g(K)$

Unfortunately, even for CSP problems whose constraint graph is series-parallel, the polytope given by the LP (1) - (3) is not integral (consider, e.g., the instance of CSP corresponding to the independent set problem on $K_{3}$). The weakness of the formulation is that no global consistency among the $\bm{y}$ variables is guaranteed. To strengthen the relaxation, we introduce new variables and constraints derived from a tree decomposition of the constraint graph of $Q$.

Here we describe, for every CSP instance $Q=(V,\mathcal{D},\mathcal{H},\mathcal{C})$, a polytope $P(Q)$, and in the next subsection we prove that $P(Q)$ is an extended formulation of $CSP(Q)$ and $CSP^{\prime}(Q)$. The set of variables in the given LP description of $P(Q)$ is substantially different from the set of variables used in the LP (1) - (3), and the set of new constraints is completely different from the the set of constraints in the LP (1) - (3). Whereas in the previous subsection, there is (roughly) a variable $g(K)$ for every feasible assignment of every subset of CSP variables corresponding to a soft or hard constraint, here we have a variable for every feasible assignment of every subset of CSP variables corresponding to a bag in a given tree decomposition of the constraint graph. Nevertheless, as we show after defining $P(Q)$, there exists a simple linear projection of $P(Q)$ to the convex hull of all integral points in the polytope given by the LP (1) - (3).

Let $T=(V_{T},E_{T})$ be a fixed nice tree decomposition [10] of the constraint graph of $Q$ and for every node $a\in V_{T}$, let $B(a)\subseteq V$ denote the corresponding bag. Let $\mathcal{B}=\{B(a)\ |\ a\in V_{T}\}$ denote the set of all bags of $T$. Let $\mathcal{K}_{\mathcal{B}}=\bigcup_{B\in\mathcal{B}}\mathcal{K}(B)$ be the set of all configurations of all bags in $T$. We use $V_{I}\subseteq V_{T}$ to denote the subset of all introduce nodes in $T$ and $V_{F}\subseteq V_{T}$ to denote the subset of all forget nodes in $T$.

For every configuration $K\in\mathcal{K}_{\mathcal{B}}$ we introduce a binary variable $f(K)$. As in the previous subsection, the intended meaning of the variable $K\in\mathcal{K}(B)$, for $B\in\mathcal{B}$, is to provide information about the values of the CSP-variables $z_{u}$ for $u\in B$ in the following way: $f(K)=1$ if and only if for every $u\in B$, $z_{u}=K(u)$. To ensure consistency among variables indexed by the configurations of the same bag, namely to ensure that for every $B\in\mathcal{B}$ there exists exactly one configuration $K\in\mathcal{K}(B)$ with $f(K)=1$, we introduce for every $B\in\mathcal{B}$ the LP constraint $\sum_{K\in\mathcal{K}(B)}f(K)=1$.

For every introduce node $c\in V_{T}$ with a child $b\in V_{T}$ and for every configuration $K\in\mathcal{K}(B(b))$ we have the constraint $\sum_{K^{\prime}\in\mathcal{K}(B(c)):K^{\prime}\ \hskip-2.84526pt\upharpoonright_{B(b)}=K}f(K^{\prime})=f(K)$, and symmetrically, for every forget node $c\in V_{T}$ with a child $b\in V_{T}$ and for every configuration $K\in\mathcal{K}(B(c))$ we have the constraint $\sum_{K^{\prime}\in\mathcal{K}(B(b)):K^{\prime}\ \hskip-2.84526pt\upharpoonright_{B(c)}=K}f(K^{\prime})=f(K)$.

Relaxing the integrality constraints and putting all these additional constraints together, we obtain:

For the given binary CSP instance $Q$, we denote the polytope associated with the LP (4) - (7), as $P(Q)$.

Consider now a vector $\bm{f}\in P(Q)$ and the following set of linear equations:

It is just a technical exercise to check that for a given $\bm{f}\in P(Q)$, there always exists a unique solution $(\bm{y},\bm{g})$ of this LP and that the unique $(\bm{y},\bm{g})$ is a linear projection of $\bm{f}$. Moreover, such a vector $(\bm{y},\bm{g})$ also satisfies the LP constraints (1) - (3). The point is that there exists a linear projection of $P(Q)$ into the polytope defined by the LP (1) - (3); moreover, an integral point from $P(Q)$ is mapped on an integral point. From now on we denote this projection $\textrm{proj}_{2}$.

As in the previous subsections, we assume that $Q=(V,\mathcal{D},\mathcal{H},\mathcal{C})$ is a given instance of CSP, $G=(V,E)$ is the constraint graph of $Q$ and $T=(V_{T},E_{T})$ a fixed nice tree decomposition of $G$. We start by introducing several notions that will help us dealing with tree decompositions and our linear program.

For a node $a\in V_{T}$, let $T(a)=(V_{a},E_{a})$ be the subtree of $T$ rooted in $a$; the configurations relevant to $T(a)$ are those in the set $\mathcal{R}(a)=\bigcup_{b\in V_{a}}\mathcal{K}(B(b))$, and the variables relevant to $T(a)$ are those $f(K)$ for which $K\in\mathcal{R}(a)$. For succinctness of notation, we denote the projection $\bm{f}|_{\mathcal{R}(a)}$ of the vector $\bm{f}$ on the set of variables relevant to $T(a)$ also by $\bm{f}|_{a}$. The constraints relevant to $T(a)$ are those containing only the variables relevant to $T(a)$. We say that a vector $I\in\{0,1\}^{\mathcal{R}(a)}$ agrees with the configuration $K\in\mathcal{R}(a)$ if $I(K)=1$.

Let $\bm{f}$ be a fixed solution of the LP (4) - (7) that corresponds to a vertex of the polytope $P(Q)$. Our main tool is the following lemma.