On Asymptotic Complexity of Linear Ordering the Paley Tournament Graphs: First Prototypes with Self-Avoiding Walks Franc Brglez and Yang Ho and Johnny Nguyen Computer Science, NC State University Raleigh, NC 27695, USA Email: brglez@ncsu.edu Abstract—The linear ordering problem (LOP) arises in a number of divers domains. The notoriously hard class of LOP in- stances, 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 L (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). I. I NTRODUCTION The linear ordering problem (LOP) arises in a number of divers domains, ranging from applications in triangulation of input-output matrices in economics, chronological ordering of artifacts in archeology, job-shop scheduling, scheduling of tournaments, query processing of wireless data broadcast, ordering of mutations as well as amino acid replacements in polypeptide, learning by ordering for better language trans- lation, mathematical psychology, among others. For a com- prehensive overview of the linear ordering problem, and the solutions provided by exact and heuristic methods, see [1]. While there is a prolific number of heuristic solvers, only a few exact methods and LOP solvers are known to exist. The relatively short list of solvers that implement exact algorithms includes [2], [3], and [4]. The importance of finding an exact solution for given LOP cannot be overstated: clearly they are important in a number of practical applications where runtime constraints are not critical and we can afford to wait for several days to get an exact solution. However, they also are important as the initial source of best-known-solutions (BKVs). These solutions provide a foundation for an experimental statistical model that reliably predicts the asymptotic complexity of a heuristic linear ordering solver – when running a well-defined class of LOP instances of increasing size L. The notoriously hard set of LOP instances, even for sizes L < 50, is represented by Paley tournament graphs [5]. These and many other instances are introduced in the survey paper [6] and the appendix [7]. To access all instances and the Excel file lolib method exp.xls with BKVs, see [8]. Names of files representing matrices of Paley graph tournaments contain the string pal which we also use in Eq. 1 below. The asymptotic experiments with the LOP solver introduced in this paper return two new BKVs for L = 31 and L = 43. The improved BKVs are significant, see the paragraph following Eq. 1 for details. BKV(L, pal)= {pal(11, 35), pal(13, 55), pal(19, 107), (1) pal(23, 161), pal(27, 252), pal(31, 300) * , pal(43, 597) * , pal(55, 1084) * } In the Excel file lolib method exp.xls under [8] we find best- known-values for pal instances, but no runtimes. The solvers listed are {TS, MA, VNS, SA, SS, GRASP}. The condensed summary is below: • pal(31, 285) as the best value by some but not all solvers • pal(43, 543) as the best value by some but not all solvers • pal(55, 1045) as the best value by some but not all solvers The same Excel file also lists pal(31, 300) * , pal(43, 597) * , pal(55, 1084) * as the Upper Bound Norm. In view of results with our solver, we suggest to use the values in Eq. 1 as the set of target BKVs when designing asymptotic experiments with pal instances for any LOP solvers in the future. The number of heuristic methods and solvers is indeed prolific, one more reason to categorize them [1]. For a brief chronological perspective, we list some of the representative approaches: [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19]. The only paper that includes experiments with pal instances is [17]. None of these experiments reach the BKVs posted in the appendix [7]. What they report is a measure of distances from these BKVs. Uncensored Experiments. Our experiments with combina- torial problems in three different domain have demonstrated intrinsic merits of statistical models that can predict the asymptotic runtime complexity of any solver that returns BKVs associated a well-defined instance set in the problem domain of choice: see Figure 7 in [20], a number of plots in [21], and Figure 5 in [22]. We expect no less for the LOP solver outlined in this paper, given the instance set defined in Eq. 1.