A Ihara-Bass Formula for Non-Boolean Matrices and Strong Refutations of Random CSPs
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2023-07
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Conference Paper
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Abstract
We define a novel notion of “non-backtracking” matrix associated to any symmetric matrix, and we prove a “Ihara-Bass” type formula for it. We use this theory to prove new results on polynomial-time strong refutations of random constraint satisfaction problems with k variables per constraints (k-CSPs). For a random k-CSP instance constructed out of a constraint that is satisfied by a p fraction of assignments, if the instance contains n variables and nk/2/є2 constraints, we can efficiently compute a certificate that the optimum satisfies at most a p + Ok(є) fraction of constraints. Previously, this was known for even k, but for odd k one needed nk/2(log n)O(1)/є2 random constraints to achieve the same conclusion. Although the improvement is only polylogarithmic, it overcomes a significant barrier to these types of results. Strong refutation results based on current approaches construct a certificate that a certain matrix associated to the k-CSP instance is quasirandom. Such certificate can come from a Feige-Ofek type argument, from an application of Grothendieck's inequality, or from a spectral bound obtained with a trace argument. The first two approaches require a union bound that cannot work when the number of constraints is o(n⌈k/2⌉) and the third one cannot work when the number of constraints is o(nk/2√log n). We further apply our techniques to obtain a new PTAS finding assignments for k-CSP instances with nk/2/є2 constraints in the semi-random settings where the constraints are random, but the sign patterns are adversarial.
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38th Computational Complexity Conference
Volume
264
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Publisher
Dagstuhl Publishing
Event
38th Computational Complexity Conference (CCC 2023)
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Subject
CSP; k-XOR; strong refutation; sum-of-squares; tensor; graph; hypergraph; non-backtracking walk
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Funding
815464 - Unified Theory of Efficient Optimization and Estimation (EC)