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. Natural ordering constraint satisfaction problems include "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, algorithms using o(\sqrt{n . 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 propertythat 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 CSPsover large finite alphabets to OCSPs , we give optimal inapproximability results for all OCSPs.
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. OCSPs 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. 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 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 over arbitrary (finite) alphabets. We design a family of appropriate CSPs (one for every q) from any given OCSP, and apply their work to this family of CSPs. We show that the hard instances from this earlier work have a particular "small-set expansion" property . 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.
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Natural ordering constraint satisfaction problems include
OCSPs capture natural problems including
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algorithms using o(\sqrt{n
OCSP(\Pi) is approximation-resistant to o(n)-space streaming algorithms. This space bound is tight up to polylogarithmic factors
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in o(n) space
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4 approximation
4-approximation in o(\sqrt n) space
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a recent work
recent works
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Velingker,
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, linear-space
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some
a broad class of (non-ordering)
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design a family of appropriate CSPs (one for every q) from any given OCSP, and apply their work to this family of CSPs. We
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the propertythat in every partition of the hypergraph formed by the constraints into small blocks, most of the hyperedges are incident on vertices from distinct blocks
a particular "small-set expansion" property
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a natural reduction from CSPsover large finite alphabets to OCSPs
the hardness results of the resulting families of CSPs