Abstract
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 & Tao (2019), only ruled out 3/4-approximations in \(o(\sqrt n)\) space. Our results build on recent works of Chou et al. (2022b, 2024) 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.
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Acknowledgements
A previous version of this paper appeared in APPROX 2021 (Singer et al. 2021a). The results starting from that version improve on an earlier version of the paper (Singer et al. 2021b) 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 et al. (2022b), whereas our previous version used the strongest lower bounds for CSPs that were available at the time from an earlier work of Chou et al. (2024).7
We would like to thank the anonymous referees at Computational Complexity and APPROX for their helpful comments.
N.G.S. was supported by an NSF Graduate Research Fellowship (Award DGE2140739). Work was done in part when swa hor was an undergraduate student at Harvard University.
M.S. was supported in part by a Simons Investigator Award and NSF Award CCF 1715187.
S.V. was supported in part by a Google Ph.D. Fellowship, a Simons Investigator Award to Madhu Sudan, and NSF Award CCF 2152413. Work was done in part when the author was an graduate student at Harvard University.
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Singer, N.G., Sudan, M. & Velusamy, S. Streaming approximation resistance of every ordering CSP. comput. complex. 33, 6 (2024). https://doi.org/10.1007/s00037-024-00252-5
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DOI: https://doi.org/10.1007/s00037-024-00252-5