Locating phase transitions in computationally hard problems

TL;DR

This paper explores how phase transitions in computationally hard problems can be identified using concepts from statistical physics, enabling more efficient search algorithms and early detection of critical slowing down.

Contribution

It introduces a method to detect the onset of computational hardness and phase transitions in search problems by analyzing response functions and critical exponents.

Findings

Detection of phase transitions in computational problems.

Identification of a dynamical critical exponent in TSP.

Potential for improving search efficiency and avoiding unnecessary computations.

Abstract

We discuss how phase-transitions may be detected in computationally hard problems in the context of Anytime Algorithms. Treating the computational time, value and utility functions involved in the search results in analogy with quantities in statistical physics, we indicate how the onset of a computationally hard regime can be detected and the transit to higher quality solutions be quantified by an appropriate response function. The existence of a dynamical critical exponent is shown, enabling one to predict the onset of critical slowing down, rather than finding it after the event, in the specific case of a Travelling Salesman Problem. This can be used as a means of improving efficiency and speed in searches, and avoiding needless computations.