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Efficient Algorithms and New Characterizations for CSP Sparsification

Sanjeev Khanna, Aaron L. Putterman, Madhu Sudan

TL;DR

This work significantly extends the class of CSPs for which nearly linear-size sparsifications can be shown to exist while also extending the scope to settings with non-linear-sized sparsifications and completely classify the set of Boolean predicates P that allow non-trivial (o(nr)-size) sparsifications.

Abstract

CSP sparsification, introduced by Kogan and Krauthgamer (ITCS 2015), considers the following question: how much can an instance of a constraint satisfaction problem be sparsified (by retaining a reweighted subset of the constraints) while still roughly capturing the weight of constraints satisfied by {\em every} assignment. CSP sparsification captures as a special case several well-studied problems including graph cut-sparsification, hypergraph cut-sparsification, hypergraph XOR-sparsification, and corresponds to a general class of hypergraph sparsification problems where an arbitrary $0/1$-valued {\em splitting function} is used to define the notion of cutting a hyperedge (see, for instance, Veldt-Benson-Kleinberg SIAM Review 2022). The main question here is to understand, for a given constraint predicate $P:Σ^r \to \{0,1\}$ (where variables are assigned values in $Σ$), the smallest constant $c$ such that $\widetilde{O}(n^c)$ sized sparsifiers exist for every instance of a constraint satisfaction problem over $P$. A recent work of Khanna, Putterman and Sudan (SODA 2024) [KPS24] showed {\em existence} of near-linear size sparsifiers for new classes of CSPs. In this work (1) we significantly extend the class of CSPs for which nearly linear-size sparsifications can be shown to exist while also extending the scope to settings with non-linear-sized sparsifications; (2) we give a polynomial-time algorithm to extract such sparsifications for all the problems we study including the first efficient sparsification algorithms for the problems studied in [KPS24].

Efficient Algorithms and New Characterizations for CSP Sparsification

TL;DR

This work significantly extends the class of CSPs for which nearly linear-size sparsifications can be shown to exist while also extending the scope to settings with non-linear-sized sparsifications and completely classify the set of Boolean predicates P that allow non-trivial (o(nr)-size) sparsifications.

Abstract

CSP sparsification, introduced by Kogan and Krauthgamer (ITCS 2015), considers the following question: how much can an instance of a constraint satisfaction problem be sparsified (by retaining a reweighted subset of the constraints) while still roughly capturing the weight of constraints satisfied by {\em every} assignment. CSP sparsification captures as a special case several well-studied problems including graph cut-sparsification, hypergraph cut-sparsification, hypergraph XOR-sparsification, and corresponds to a general class of hypergraph sparsification problems where an arbitrary -valued {\em splitting function} is used to define the notion of cutting a hyperedge (see, for instance, Veldt-Benson-Kleinberg SIAM Review 2022). The main question here is to understand, for a given constraint predicate (where variables are assigned values in ), the smallest constant such that sized sparsifiers exist for every instance of a constraint satisfaction problem over . A recent work of Khanna, Putterman and Sudan (SODA 2024) [KPS24] showed {\em existence} of near-linear size sparsifiers for new classes of CSPs. In this work (1) we significantly extend the class of CSPs for which nearly linear-size sparsifications can be shown to exist while also extending the scope to settings with non-linear-sized sparsifications; (2) we give a polynomial-time algorithm to extract such sparsifications for all the problems we study including the first efficient sparsification algorithms for the problems studied in [KPS24].
Paper Structure (54 sections, 61 theorems, 125 equations, 24 algorithms)

This paper contains 54 sections, 61 theorems, 125 equations, 24 algorithms.

Table of Contents

  1. Introduction
  2. CSP sparsification
  3. Motivation
  4. Our Results
  5. Efficient Code Sparsification
  6. CSP Sparsification
  7. Technical Theorems
  8. Conclusions
  9. Addendum:
  10. Organization:
  11. A Detailed Overview of Techniques
  12. Previous work of KPS24
  13. Algorithmic challenge:
  14. Beyond Fields:
  15. Beyond linear-size sparsifications:
  16. ...and 39 more sections

Key Result

Theorem 1.1

For any code $C \subset \mathbb{F}_q^m$ of dimension $n$ and a parameter $\epsilon \in (0,1)$, there is a polynomial time (in $n,m, \log(q), \epsilon^{-1}$) randomized algorithm for computing (with high probability) a $(1 \pm \epsilon)$ code-sparsifier $C|_S$ of $C$, with $|S| = \widetilde{O}(n / \e

Theorems & Definitions (206)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 1.1: CSP Sparsification
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • ...and 196 more