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Towards Geometry-Preserving Reductions Between Constraint Satisfaction Problems (and other problems in NP)

Gabriel Istrate

TL;DR

This work defines two kinds of geometry-preserving reductions between constraint satisfaction problems and other NP-search problems that are motivated by phase transitions in combinatorial optimization problems.

Abstract

Motivated by phase transitions in combinatorial optimization problems, we define two kinds of geometry-preserving reductions between constraint satisfaction problems and other NP-search problems. We give a couple of examples and counterexamples for these reductions.

Towards Geometry-Preserving Reductions Between Constraint Satisfaction Problems (and other problems in NP)

TL;DR

This work defines two kinds of geometry-preserving reductions between constraint satisfaction problems and other NP-search problems that are motivated by phase transitions in combinatorial optimization problems.

Abstract

Motivated by phase transitions in combinatorial optimization problems, we define two kinds of geometry-preserving reductions between constraint satisfaction problems and other NP-search problems. We give a couple of examples and counterexamples for these reductions.

Paper Structure

This paper contains 8 sections, 6 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Geometry-preserving reductions via local constructions
  4. An(other) example of an overlap-preserving reduction
  5. An(other) example of a cover-preserving reduction
  6. Reductions that are not geometry-preserving
  7. Geometry-preserving reductions: an Algebraic Perspective
  8. Conclusions

Key Result

Theorem 1

The natural reduction from $k$-COL to $k$-SAT is cover-preserving and overlap-preserving.

Theorems & Definitions (30)

  • Definition 1
  • Definition 2
  • Definition 3
  • Example 1
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9
  • ...and 20 more