An ordering constraint satisfaction problem (OCSP) is defined by a family $\mathcal{F}$ of predicates mapping permutations on $\{1,\ldots,k\}$ to $\{0,1\}$. An instance of $Max-OCSP($\mathcal{F}$)$ on $n$ variables consists of a list of constraints, each consisting of a predicate from $\mathcal{F}$ applied on $k$ distinct variables. The goal is to find an ordering of the $n$ variables that maximizes the number of constraints for which the induced ordering on the $k$ variables satisfies the predicate. OCSPs capture well-studied problems including `maximum acyclic subgraph' (MAS) and ``maximum betweenness''.

In this work, we consider the task of approximating the maximum number of satisfiable constraints in the (single-pass) streaming setting, when an instance is presented as a stream of constraints. We show that for every $\mathcal{F}$, $Max-OCSP($\mathcal{F}$)$ is approximation-resistant to $o(n)$-space streaming algorithms, i.e., algorithms using $o(n)$ space cannot distinguish streams where almost every constraint is satisfiable from streams where no ordering beats the random ordering by a noticeable amount. This space bound is tight up to polylogarithmic factors. In the case of MAS our result shows that for every $\epsilon>0$, MAS is not $(1/2+\epsilon)$-approximable in $o(n)$ space. The previous best inapproximability result, due to Guruswami and Tao (APPROX'19), only ruled out $3/4$-approximations in $o(\sqrt n)$ space.

Our results build on recent works of Chou, Golovnev, Sudan, and Velusamy (FOCS'21, J.ACM'24) and Chou, Golovnev, Sudan, Velingker, and Velusamy (STOC'22), who provide a tight, linear-space inapproximability theorem for a broad class of ``standard'' (i.e., non-ordering) constraint satisfaction problems (CSPs) over arbitrary (finite) alphabets.
Our results are obtained by building a family of appropriate standard CSPs (one for every alphabet size $q$) from any given OCSP and applying their theorem to this family of CSPs. To convert the resulting hardness results for standard CSPs back to our OCSP, we show that the hard instances from this earlier theorem have the following ``partition expansion'' property with high probability: For every partition of the $n$ variables into small blocks, for most of the constraints, all variables are in distinct blocks.

Upgraded to $\Omega(n)$ space lower bound. Appeared in APPROX'21 and Computational Complexity.

An ordering constraint satisfaction problem (OCSP) is given by a positive integer $k$ and a constraint predicate $\Pi$ mapping permutations on $\{1,\ldots,k\}$ to $\{0,1\}$. Given an instance of OCSP$(\Pi)$ on $n$ variables and $m$ constraints, the goal is to find an ordering of the $n$ variables that maximizes the number of constraints that are satisfied, where a constraint specifies a sequence of $k$ distinct variables and the constraint is satisfied by an ordering on the $n$ variables if the ordering induced on the $k$ variables in the constraint satisfies $\Pi$. Ordering constraint satisfaction problems capture natural problems including ``Maximum acyclic subgraph (MAS)'' and ``Betweenness''.

In this work we consider the task of approximating the maximum number of satisfiable constraints in the (single-pass) streaming setting, where an instance is presented as a stream of constraints. We show that for every $\Pi$, OCSP$(\Pi)$ is approximation-resistant to $o(n)$-space streaming algorithms, i.e., algorithms using $o(n)$ space cannot distinguish streams where almost every constraint is satisfiable from streams where no ordering beats the random ordering by a noticeable amount. This space bound is tight up to polylogarithmic factors. In the case of MAS our result shows that for every $\epsilon>0$, MAS is not $1/2+\eps$-approximable in $o(n)$ space. The previous best inapproximability result only ruled out a $3/4$-approximation in $o(\sqrt n)$ space.

Our results build on recent works of Chou, Golovnev, Sudan, Velingker, and Velusamy who show tight, linear-space inapproximability results for a broad class of (non-ordering) constraint satisfaction problems (CSPs) over arbitrary (finite) alphabets.
Our results are obtained by building a family of appropriate CSPs (one for every $q$) from any given OCSP, and applying their work to this family of CSPs.
To convert the resulting hardness results for CSPs back to our OCSP,
we show that the hard instances from this earlier work have the following ``small-set expansion'' property: If the CSP instance is viewed as a hypergraph in the natural way, then for every partition of the hypergraph into small blocks most of the hyperedges are incident on vertices from distinct blocks. By exploiting this combinatorial property, in combination with the hardness results of the resulting families of CSPs, we give optimal inapproximability results for all OCSPs.

Edited abstract

An ordering constraint satisfaction problem (OCSP) is given by a positive integer $k$ and a constraint predicate $\Pi$ mapping permutations on $\{1,\ldots,k\}$ to $\{0,1\}$. Given an instance of OCSP$(\Pi)$ on $n$ variables and $m$ constraints, the goal is to find an ordering of the $n$ variables that maximizes the number of constraints that are satisfied, where a constraint specifies a sequence of $k$ distinct variables and the constraint is satisfied by an ordering on the $n$ variables if the ordering induced on the $k$ variables in the constraint satisfies $\Pi$. Ordering constraint satisfaction problems capture natural problems including ''Maximum acyclic subgraph (MAS)'' and ''Betweenness''.

In this work we consider the task of approximating the maximum number of satisfiable constraints in the (single-pass) streaming setting, where an instance is presented as a stream of constraints. We show that for every $\Pi$, OCSP$(\Pi)$ is approximation-resistant to $o(\sqrt{n})$-space streaming algorithms, i.e., algorithms using $o(\sqrt{n})$ space cannot distinguish streams where almost every constraint is satisfiable from streams where no ordering beats the random ordering by a noticeable amount. In the case of MAS our result shows that for every $\epsilon>0$, MAS is not $1/2+\epsilon$-approximable. The previous best inapproximability result only ruled out a $3/4$ approximation.

Our results build on a recent work of Chou, Golovnev, Sudan, and Velusamy who show tight inapproximability results for some constraint satisfaction problems over arbitrary (finite) alphabets. We show that the hard instances from this earlier work have the following ''small-set expansion'' property: in every partition of the hypergraph formed by the constraints into small blocks, most of the hyperedges are incident on vertices from distinct blocks. By exploiting this combinatorial property, in combination with a natural reduction from CSPs over large finite alphabets to OCSPs, we give optimal inapproximability results for all OCSPs.

The current version of this paper improves on a previous version of this paper that gave only $\Omega(\sqrt{n})$ space lower bounds for all OCSPs. Our improvement to $\Omega(n)$ space lower bounds comes by invoking the more recent results of Chou, Golovnev, Sudan, Velingker, and Velusamy, whereas our previous version used the strongest lower bounds for CSPs that were available at the time from an earlier work of Chou, Golovnev, Sudan, and Velusamy.

An ordering constraint satisfaction problem (OCSP) is given by a positive integer $k$ and a constraint predicate $\Pi$ mapping permutations on $\{1,\ldots,k\}$ to $\{0,1\}$. Given an instance of OCSP$(\Pi)$ on $n$ variables and $m$ constraints, the goal is to find an ordering of the $n$ variables that maximizes the number of constraints that are satisfied, where a constraint specifies a sequence of $k$ distinct variables and the constraint is satisfied by an ordering on the $n$ variables if the ordering induced on the $k$ variables in the constraint satisfies $\Pi$. Ordering constraint satisfaction problems capture natural problems including ''Maximum acyclic subgraph (MAS)'' and ''Betweenness''.

In this work we consider the task of approximating the maximum number of satisfiable constraints in the (single-pass) streaming setting, where an instance is presented as a stream of constraints. We show that for every $\Pi$, OCSP$(\Pi)$ is approximation-resistant to $o(\sqrt{n})$-space streaming algorithms, i.e., algorithms using $o(\sqrt{n})$ space cannot distinguish streams where almost every constraint is satisfiable from streams where no ordering beats the random ordering by a noticeable amount. In the case of MAS our result shows that for every $\epsilon>0$, MAS is not $1/2+\epsilon$-approximable. The previous best inapproximability result only ruled out a $3/4$ approximation.

Our results build on a recent work of Chou, Golovnev, Sudan, and Velusamy who show tight inapproximability results for some constraint satisfaction problems over arbitrary (finite) alphabets. We show that the hard instances from this earlier work have the following ''small-set expansion'' property: in every partition of the hypergraph formed by the constraints into small blocks, most of the hyperedges are incident on vertices from distinct blocks. By exploiting this combinatorial property, in combination with a natural reduction from CSPs over large finite alphabets to OCSPs, we give optimal inapproximability results for all OCSPs.