Abstract
An ordering constraint satisfaction problem (OCSP) is given by a positive integer k and a constraint predicate Π mapping permutations on {1,…,k} to {0,1}. Given an instance of OCSP(Π) 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 Π. 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 (singlepass) streaming setting, where an instance is presented as a stream of constraints. We show that for every Π, OCSP(Π) is approximationresistant 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 ε > 0, MAS is not 1/2+εapproximable in o(n) space. The previous best inapproximability result only ruled out a 3/4approximation in o(√ n) space.
Our results build on recent works of Chou, Golovnev, Sudan, Velingker, and Velusamy who show tight, linearspace inapproximability results for a broad class of (nonordering) 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 "smallset 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.
BibTeX  Entry
@InProceedings{singer_et_al:LIPIcs.APPROX/RANDOM.2021.17,
author = {Singer, Noah and Sudan, Madhu and Velusamy, Santhoshini},
title = {{Streaming Approximation Resistance of Every Ordering CSP}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
pages = {17:117:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772075},
ISSN = {18688969},
year = {2021},
volume = {207},
editor = {Wootters, Mary and Sanit\`{a}, Laura},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/14710},
URN = {urn:nbn:de:0030drops147106},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2021.17},
annote = {Keywords: Streaming approximations, approximation resistance, constraint satisfaction problems, ordering constraint satisfaction problems}
}
Keywords: 

Streaming approximations, approximation resistance, constraint satisfaction problems, ordering constraint satisfaction problems 
Collection: 

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021) 
Issue Date: 

2021 
Date of publication: 

15.09.2021 