Typical Performance of Approximation Algorithms for NP-hard Problems

TL;DR

This paper analyzes the typical performance of three approximation algorithms for randomized minimum vertex cover problems across various random graph ensembles, revealing threshold behaviors and classification conditions.

Contribution

It introduces a comprehensive analysis of three algorithms' thresholds in random graph ensembles, using statistical mechanics and generating functions.

Findings

Algorithms have a threshold in performance related to average degree.

Below the threshold, algorithms find optimal solutions with high probability.

Three distinct cases of performance thresholds are identified and classified.

Abstract

Typical performance of approximation algorithms is studied for randomized minimum vertex cover problems. A wide class of random graph ensembles characterized by an arbitrary degree distribution is discussed with some theoretical frameworks. Here three approximation algorithms are examined; the linear-programming relaxation, the loopy-belief propagation, and the leaf-removal algorithm. The former two algorithms are analyzed using the statistical-mechanical technique while the average-case analysis of the last one is studied by the generating function method. These algorithms have a threshold in the typical performance with increasing the average degree of the random graph, below which they find true optimal solutions with high probability. Our study reveals that there exist only three cases determined by the order of the typical-performance thresholds. We provide some conditions for classifying the graph ensembles and demonstrate explicitly examples for the difference in the threshold.