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A constraint satisfaction problem (CSP),\(\textsf {Max-CSP}(\mathcal {F})\), is specified by a finite set of constraints\(\mathcal {F}\subseteq \lbrace [q]^k \rightarrow \lbrace 0,1\rbrace \rbrace\)for positive integersqandk. An instance of the problem onnvariables is given bymapplications of constraints from\(\mathcal {F}\)to subsequences of thenvariables, and the goal is to find an assignment to the variables that satisfies the maximum number of constraints. In the (γ ,β)-approximation version of the problem for parameters 0 ≤ β ≤ γ ≤ 1, the goal is to distinguish instances where at least γ fraction of the constraints can be satisfied from instances where at most β fraction of the constraints can be satisfied. In this work, we consider the approximability of this problem in the context of sketching algorithms and give a dichotomy result. Specifically, for every family\(\mathcal {F}\)and every β < γ, we show that either a linear sketching algorithm solves the problem in polylogarithmic space or the problem is not solvable by any sketching algorithm in\(o(\sqrt {n})\)space. In particular, we give non-trivial approximation algorithms using polylogarithmic space for infinitely many constraint satisfaction problems. We also extend previously known lower bounds for general streaming algorithms to a wide variety of problems, and in particular the case ofq=k=2, where we get a dichotomy, and the case when the satisfying assignments of the constraints of\(\mathcal {F}\)support a distribution on\([q]^k\)with uniform marginals. Prior to this work, other than sporadic examples, the only systematic classes of CSPs that were analyzed considered the setting of Boolean variablesq= 2, binary constraintsk=2, and singleton families\(|\mathcal {F}|=1\)and only considered the setting where constraints are placed on literals rather than variables. Our positive results show wide applicability of bias-based algorithms used previously by [47] and [41], which we extend to include richer norm estimation algorithms, by giving a systematic way to discover biases. Our negative results combine the Fourier analytic methods of [56], which we extend to a wider class of CSPs, with a rich collection of reductions among communication complexity problems that lie at the heart of the negative results. In particular, previous works used Fourier analysis over the Boolean cube to initiate their results and the results seemed particularly tailored to functions on Boolean literals (i.e., with negations). Our techniques surprisingly allow us to get to generalq-ary CSPs without negations by appealing to the same Fourier analytic starting point over Boolean hypercubes.
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Sketching Approximability of (Weak) Monarchy Predicates
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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- Award ID(s):
- 2152413
- PAR ID:
- 10400120
- Editor(s):
- Chakrabarti, Amit; Swamy, Chaitanya
- Date Published:
- Journal Name:
- Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, {APPROX/RANDOM} 2022
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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