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Constraint Satisfaction Problems with Advice

Suprovat Ghoshal, Konstantin Makarychev, Yury Makarychev

TL;DR

This work investigates constraint satisfaction problems with ML oracle advice, introducing Label Advice and Variable Subset Advice models and showing near-optimal algorithms for Max Cut, Max 2-Lin, and nearly satisfiable Max 3-Lin when the average degree is sufficiently large. It pairs these algorithmic results with ETH- and PCP-based hardness, proving that without degree guarantees Max $3$-Lin remains hard to beat beyond $1/2$ and that Max $4$-Lin is hard even under high-degree regimes. The authors leverage quadratic-form optimization, Goemans–Williamson, and reductions from Gap Label Cover to translate advice into algorithmic gains and to establish tight limits, highlighting when ML advice is most effective in CSPs. Collectively, the results delineate the boundary between when oracle advice yields substantial gains and when fundamental complexity barriers persist, guiding future exploration of ML-assisted CSP algorithms and their scalability.

Abstract

We initiate the study of algorithms for constraint satisfaction problems with ML oracle advice. We introduce two models of advice and then design approximation algorithms for Max Cut, Max $2$-Lin, and Max $3$-Lin in these models. In particular, we show the following. 1. For Max-Cut and Max $2$-Lin, we design an algorithm that yields near-optimal solutions when the average degree is larger than a threshold degree, which only depends on the amount of advice and is independent of the instance size. We also give an algorithm for nearly satisfiable Max $3$-Lin instances with quantitatively similar guarantees. 2. Further, we provide impossibility results for algorithms in these models. In particular, under standard complexity assumptions, we show that Max $3$-Lin is still $1/2 + η$ hard to approximate given access to advice, when there are no assumptions on the instance degree distribution. Additionally, we also show that Max $4$-Lin is $1/2 + η$ hard to approximate even when the average degree of the instance is linear in the number of variables.

Constraint Satisfaction Problems with Advice

TL;DR

This work investigates constraint satisfaction problems with ML oracle advice, introducing Label Advice and Variable Subset Advice models and showing near-optimal algorithms for Max Cut, Max 2-Lin, and nearly satisfiable Max 3-Lin when the average degree is sufficiently large. It pairs these algorithmic results with ETH- and PCP-based hardness, proving that without degree guarantees Max -Lin remains hard to beat beyond and that Max -Lin is hard even under high-degree regimes. The authors leverage quadratic-form optimization, Goemans–Williamson, and reductions from Gap Label Cover to translate advice into algorithmic gains and to establish tight limits, highlighting when ML advice is most effective in CSPs. Collectively, the results delineate the boundary between when oracle advice yields substantial gains and when fundamental complexity barriers persist, guiding future exploration of ML-assisted CSP algorithms and their scalability.

Abstract

We initiate the study of algorithms for constraint satisfaction problems with ML oracle advice. We introduce two models of advice and then design approximation algorithms for Max Cut, Max -Lin, and Max -Lin in these models. In particular, we show the following. 1. For Max-Cut and Max -Lin, we design an algorithm that yields near-optimal solutions when the average degree is larger than a threshold degree, which only depends on the amount of advice and is independent of the instance size. We also give an algorithm for nearly satisfiable Max -Lin instances with quantitatively similar guarantees. 2. Further, we provide impossibility results for algorithms in these models. In particular, under standard complexity assumptions, we show that Max -Lin is still hard to approximate given access to advice, when there are no assumptions on the instance degree distribution. Additionally, we also show that Max -Lin is hard to approximate even when the average degree of the instance is linear in the number of variables.
Paper Structure (20 sections, 18 theorems, 8 equations, 1 figure)

This paper contains 20 sections, 18 theorems, 8 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Comparing the Models.
  3. Max Cut and Max $k$-Lin Problems.
  4. Our results
  5. Hardness Results.
  6. Discussion.
  7. Comparison with Independent Works
  8. Algorithm for Max Cut and Max 2-Lin with Advice
  9. Quadratic Forms with Advice
  10. Algorithm for Max 3-Lin with Advice
  11. Proof of Lemma \ref{['lem:soln-Psi']}
  12. Hardness of Max k-Lin with Oracle Advice
  13. ETH and Hardness of Max 3-Lin
  14. Reduction to Gap Label Cover
  15. Reduction from Gap Label Cover to Max 3-Lin
  16. ...and 5 more sections

Key Result

Theorem 1.4

There exists a polynomial-time algorithm for the Label Advice model that given an unweighted instance of Max Cut or Max 2-Lin and advice $\tilde{x}$ finds a solution of value at least $(1-O(1/\varepsilon\sqrt{\Delta})) \, OPT$ in expectation (over the random advice), where $OPT$ is the value of the

Figures (1)

  • Figure 1: Quadratic Program Maximization Algorithm

Theorems & Definitions (25)

  • Definition 1.1: Max Cut
  • Definition 1.2: Max 2-Lin
  • Definition 1.3: Max $k$-Lin
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.5
  • Theorem 1.5
  • Theorem 1.5
  • Remark 1.6
  • Theorem 3.1
  • ...and 15 more