[Paper Review] Inapproximability of Combinatorial Optimization Problems

Luca Trevisan|ArXiv.org|Sep 24, 2004

Optimization and Packing ProblemsEngineering17 citations

TL;DR

This survey establishes fundamental inapproximability results for key combinatorial optimization problems using probabilistically checkable proofs (PCP) and reductions, demonstrating that certain problems like Vertex Cover, Independent Set, and Max SAT cannot be approximated within specific constant factors unless P = NP. It provides a comprehensive overview of the theoretical foundations and open problems in approximation hardness, highlighting the role of PCP theorems and the Unique Games Conjecture in advancing the field.

ABSTRACT

We survey results on the hardness of approximating combinatorial optimization problems.

Motivation & Objective

• To systematically survey the state of inapproximability results for NP-hard combinatorial optimization problems.
• To explain how probabilistically checkable proofs (PCP) and reductions are used to establish lower bounds on approximation ratios.
• To identify and discuss key open problems in the field, including the Unique Games Conjecture and inapproximability of Metric TSP and Sparsest Cut.
• To clarify the distinction between inapproximability results and integrality gaps in optimization.
• To provide a technical foundation for researchers exploring the limits of approximation algorithms.

Proposed method

• Uses PCP theorems to transform NP-hardness proofs into inapproximability results for optimization problems.
• Employs reductions from 3SAT and other NP-complete problems to construct instances with controlled approximation ratios.
• Applies optimized PCP constructions tailored for specific problems like Max SAT, Max CUT, and Independent Set.
• Analyzes the soundness and query complexity of PCP verifiers to derive tight inapproximability factors.
• Leverages the Unique Games Conjecture as a central hypothesis to derive optimal inapproximability results.
• Examines the role of randomness, alphabet size, and witness length in PCP constructions to strengthen hardness results.

Experimental results

Research questions

• RQ1 What is the best possible approximation ratio achievable for Vertex Cover, assuming P ≠ NP?
• RQ2 Can the inapproximability of Max SAT be improved beyond known PCP-based bounds?
• RQ3 Is there a constant-factor inapproximability result for the Metric TSP, and what techniques are needed to prove it?
• RQ4 Can the Unique Games Conjecture be proven or disproven, and what would its resolution imply for approximation hardness?
• RQ5 Are there inapproximability results for graph partitioning problems like Sparsest Cut, given the lack of known lower bounds?

Key findings

• The paper establishes that Vertex Cover has no (2−ε)-approximation unless P = NP, based on the Unique Games Conjecture.
• For Max SAT, a 2-approximation is possible, but no better than 8/7-approximation is achievable for Max E3SAT unless P = NP.
• Independent Set and Coloring problems do not admit any polynomial-time approximation scheme with performance ratio better than n^{1−ε} for ε > 0, unless P = NP.
• The Unique Games Conjecture implies optimal inapproximability results for Max CUT and Vertex Cover in hypergraphs.
• The best known approximation for Sparsest Cut is O(√log n), but no inapproximability result is known, leaving a major open gap.
• The inapproximability of Metric TSP is currently bounded below by 220/219, but no 1.1-inapproximability result has been proven, indicating a need for new techniques.