Approximability of all finite CSPs with linear sketches

Chou, Chi-Ning; Golovnev, Alexander; Sudan, Madhu; Velusamy, Santhoshini

doi:10.1109/focs52979.2021.00117

Cited by 7 publications

(21 citation statements)

References 11 publications

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“…We note that a followup paper by the same authors [CGSV21] extends the result to a more general setting: Specifically they allow non-Boolean variables, allow a set of predicates rather than a single function; and allow the predicates to be applied to variables rather than literals. While their result is more general all results in this paper work in the more restricted setting of [CGSV22] and so we will describe our results in their language (which can be somewhat simpler for problems that are expressible in their setting).…”

Section: Motivation and Related Workmentioning

confidence: 79%

“…There has been an increasing interest in studying the approximability of CSPs in the streaming setting [KK15, KKS15, KKSV17, GVV17, GT19, KK19, CGV20, CGSV21, CGSV22, SSV21, BHP + 22, CGS + 22]. In particular, recently Chou, Golovnev, Sudan, and Velusamy [CGSV21,CGSV22] gave a dichotomy result for sketching approximability of all finite CSPs. Specifically, they proved the following theorem.…”

Section: Motivation and Related Workmentioning

confidence: 99%

“…Theorem 1.1 ( [CGSV22]). For every k, every predicate f : {−1, 1} k → {0, 1} and every 0 ≤ β < γ ≤ 1 one of the following holds: (1) (γ, β)-Max-CSP(f ) is solvable by an O(log(n))-space sketching algorithm, or (2) for every ε > 0, (γ − ε, β + ε)-Max-CSP(f ) is not solvable by any o( √ n)-space sketching algorithm.…”

Section: Motivation and Related Workmentioning

confidence: 99%

“…While the results of [CGSV22] imply a dichotomy, to explicitly get the optimal sketching approximation ratio for a given predicate f , they need to solve an optimization problem which in general needs computer-aided analysis. In order to get more explicit results one needs to restrict the families of functions considered, and even then it is unclear if there can be a closed-form expression.…”

Section: Motivation and Related Workmentioning

confidence: 99%

“…At first glance, it seems that if f is not sketching approximation resistant then it's not necessarily sketching approximable. Nonetheless, [CGSV22] proved that every f is either approximable or approximation resistant. 3…”

Section: Sketching Approximability and Approximation Resistancementioning

confidence: 99%

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Sketching Approximability of (Weak) Monarchy Predicates

Chou¹,

Golovnev²,

Shahrasbi³

et al. 2022

Preprint

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We analyze the sketching approximability of constraint satisfaction problems on Boolean domains, where the constraints are balanced linear threshold functions applied to literals. In particular, we explore the approximability of monarchy-like functions where the value of the function is determined by a weighted combination of the vote of the first variable (the president) and the sum of the votes of all remaining variables. The pure version of this function is when the president can only be overruled by when all remaining variables agree. For every k ≥ 5, we show that CSPs where the underlying predicate is a pure monarchy function on k variables have no non-trivial sketching approximation algorithm in o( √ n) space. We also show infinitely many weaker monarchy functions for which CSPs using such constraints are non-trivially approximable by O(log(n)) space sketching algorithms. Moreover, we give the first example of sketching approximable asymmetric Boolean CSPs. Our results work within the framework of Chou, Golovnev, Sudan, and Velusamy (FOCS 2021) that characterizes the sketching approximability of all CSPs. Their framework can be applied naturally to get a computer-aided analysis of the approximability of any specific constraint satisfaction problem. The novelty of our work is in using their work to get an analysis that applies to infinitely many problems simultaneously.

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Section: Motivation and Related Workmentioning

confidence: 79%

Section: Motivation and Related Workmentioning

confidence: 99%

Section: Motivation and Related Workmentioning

confidence: 99%

Section: Motivation and Related Workmentioning

confidence: 99%

Section: Sketching Approximability and Approximation Resistancementioning

confidence: 99%

See 3 more Smart Citations

Sketching Approximability of (Weak) Monarchy Predicates

Chou¹,

Golovnev²,

Shahrasbi³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

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Streaming approximation resistance of every ordering CSP

Singer,

Sudan,

Velusamy

2024

comput. complex.

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An ordering constraint satisfaction problem (OCSP) is defined by a family $$\mathcal F$$ F of predicates mapping permutations on $$\{1,\ldots,k\}$$ { 1 , … , k } to $$\{0,1\}$$ { 0 , 1 } . An instance of ($$\mathcal F$$ F ) on n variables consists of a list of constraints, each consisting of a predicate from $$\mathcal F$$ 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’ () 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 $$\mathcal F$$ F , ($$\mathcal F$$ 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 , our result shows that for every $$\epsilon>0$$ ϵ > 0 , is not $$(1/2+\epsilon)$$ ( 1 / 2 + ϵ ) -approximable in o(n) space. The previous best inapproximability result, due to Guruswami & Tao (2019), only ruled out 3/4-approximations in $$o(\sqrt n)$$ o ( 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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