The late acceptance Hill-Climbing heuristic

Abstract

This paper introduces a new and very simple search methodology called Late Acceptance Hill-Climbing (LAHC). It is a one-point iterative search algorithm, which accepts non-improving moves when a candidate cost function is better (or equal) than it was a number of iterations before. This value appears as a single algorithmic input parameter which determines the total processing time of the search procedure. The properties of this method are experimentally studied in this paper with a range of Travelling Salesman and Exam Timetabling benchmark problems. In addition, we present a fair comparison of LAHC with well-known search techniques, which employ different variants of a cooling schedule: Simulated Annealing (SA), Threshold Accepting (TA) and the Great Deluge Algorithm (GDA). Moreover, we discuss the method's success in winning an international competition to automatically solve the Magic Square problem. Our experiments have shown that the LAHC approach is simple, easy to implement and yet is an effective search procedure. For all studied problems, its average performance was distinctly better than GDA and on the same level as SA and TA. One of the major advantages of LAHC approach is the absence of a cooling schedule. This makes it significantly more robust than cooling-schedule based techniques. We present an example where the rescaling of a cost function in the Exam Timetabling Problem dramatically deteriorates the performance of three cooling-schedule based techniques, but has absolutely no influence upon the performance of LAHC.

A Step Counting Hill Climbing Algorithm applied to University Examination Timetabling

“…This suggests that the convergence time is approximately proportional to the value of the algorithmic parameter L c . This property of SCHC is similar to the LAHC one studied by Burke and Bykov (2012), where the authors mentioned its high practical importance, i.e. when the angle coefficient (ratio T /L c ) of the distribution is known, the complete search procedure can be fitted into a given available time.…”

“…This tendency was also observed in LAHC by (Burke and Bykov 2012), who proposed that some other factors, together with the size of problems, could also affect the values of the angle coefficients, such as constraints, conflict density, etc. If assuming that the amount of these factors somehow defines the hardness of a problem, then the angle coefficient might reflect, to some extent, this "bulk" hardness.…”

“…In this study we adopted software, which was used in experiments with LAHC and described in (Burke and Bykov 2012). It is developed in Delphi 2007 and run on PC Intel Core i7-3820 3.6 GHz, 32 GB RAM under OS Windows 7 64 bit.…”

“…This property is the same as for other local search methods including HC, SA, LAHC, etc. The presence of this property suggests the use of a common termination condition for such a technique; the search should be stopped exactly at the moment of convergence (the earlier or later termination reduces the effectiveness of the method (see Burke and Bykov (2012)). The identification of the convergence state can be done by a well-established procedure available in the literature described in the previous section, i.e.…”

“…The number of the backward iterations is the only LAHC parameter referred to as "history length". An extensive study of LAHC was carried in (Burke and Bykov 2012) where the salient properties of the method have been discussed. First, its total search/convergence time was proportional to the history length, which was essential for its practical use.…”