On Asymptotic Complexity of Linear Ordering the Paley Tournament Graphs: First Prototypes with Self-Avoiding Walks

The linear ordering problem (LOP) arises in a number of divers domains. The notoriously hard class of LOP instances, even for sizes L < 50, is represented by Paley tournament graphs. Instances of these graphs, along with best-known-values (BKVs), some of them proven optimum, are now readily available on the Internet. There is no published record of asymptotic complexity to solve instances of Paley tournament graphs, also known as the pal instances. In this paper we propose a new stochastic solver based on a variation of a self-avoiding walk. The solver we propose has the platform-independent asymptotic walkLength complexity of 0.09336 ⇤ 1.7727 (in number of steps). We demonstrate close correlation of walkLength with the asymptotic runtime complexity on different platforms as well under two rapid prototype implementations: one in Tcl, one in Python. Most importantly, the paper presents two significant improvements in BKVs: pal(31, 300) improves on pal(31, 285) and pal(43, 597) improves on pal(43, 543).