A New Look at the Easy-Hard-Easy Pattern of Combinatorial Search Difficulty

The easy-hard-easy pattern in the difficulty of combinatorial search problems as constraints are added has been explained as due to a competition between the decrease in number of solutions and increased pruning. We test the generality of this explanation by examining one of its predictions: if the number of solutions is held fixed by the choice of problems, then increased pruning should lead to a monotonic decrease in search cost. Instead, we find the easy-hard-easy pattern in median search cost even when the number of solutions is held constant, for some search methods. This generalizes previous observations of this pattern and shows that the existing theory does not explain the full range of the peak in search cost. In these cases the pattern appears to be due to changes in the size of the minimal unsolvable subproblems, rather than changing numbers of solutions.Comment: See http://www.jair.org/ for any accompanying file

Metareasoning about propagators for constraint satisfaction

Given the breadth of constraint satisfaction problems (CSPs) and the wide variety of CSP solvers, it is often very difficult to determine a priori which solving method is best suited to a problem. This work explores the use of machine learning to predict which solving method will be most effective for a given problem. We use four different problem sets to determine the CSP attributes that can be used to determine which solving method should be applied. After choosing an appropriate set of attributes, we determine how well j48 decision trees can predict which solving method to apply. Furthermore, we take a cost sensitive approach such that problem instances where there is a great difference in runtime between algorithms are emphasized. We also attempt to use information gained on one class of problems to inform decisions about a second class of problems. Finally, we show that the additional costs of deciding which method to apply are outweighed by the time savings compared to applying the same solving method to all problem instances

Backbone Fragility and the Local Search Cost Peak

The local search algorithm WSat is one of the most successful algorithms for solving the satisfiability (SAT) problem. It is notably effective at solving hard Random 3-SAT instances near the so-called `satisfiability threshold', but still shows a peak in search cost near the threshold and large variations in cost over different instances. We make a number of significant contributions to the analysis of WSat on high-cost random instances, using the recently-introduced concept of the backbone of a SAT instance. The backbone is the set of literals which are entailed by an instance. We find that the number of solutions predicts the cost well for small-backbone instances but is much less relevant for the large-backbone instances which appear near the threshold and dominate in the overconstrained region. We show a very strong correlation between search cost and the Hamming distance to the nearest solution early in WSat's search. This pattern leads us to introduce a measure of the backbone fragility of an instance, which indicates how persistent the backbone is as clauses are removed. We propose that high-cost random instances for local search are those with very large backbones which are also backbone-fragile. We suggest that the decay in cost beyond the satisfiability threshold is due to increasing backbone robustness (the opposite of backbone fragility). Our hypothesis makes three correct predictions. First, that the backbone robustness of an instance is negatively correlated with the local search cost when other factors are controlled for. Second, that backbone-minimal instances (which are 3-SAT instances altered so as to be more backbone-fragile) are unusually hard for WSat. Third, that the clauses most often unsatisfied during search are those whose deletion has the most effect on the backbone. In understanding the pathologies of local search methods, we hope to contribute to the development of new and better techniques

Population migration: A meta-heuristics for stochastic approaches to constraint satisfaction problems

A meta-heuristics for escaping from local optima to solve constraint satisfaction problems is proposed, which enables self-adaptive dynamic control of the temperature to adjust the locality of stochastic search. In our method, several groups with different temperatures are prepared. To each group the same number of candidate solutions are initially allotted. Then, the main process is repeated until the procedure comes to a certain convergence. The main process is composed of two phases: stochastic searching and population tuning. As for the latter phase, after evaluating the adaptation value of every group, migration of some number of candidate solutions in groups with lower values to groups with higher values are induced. Population migration is a kind ofparallel version of simulated annealing, where several temperatures are spatially distributed. Some experiments are performed to verify the efficiency of the method applied to constraint satisfaction problems. It is also demonstrated that population migration is exceptionally effective in the critical region where phase transitions occur

Some Enhancement Methods For Backtracking-Search In Solving Multiple Permutation Problems

In this dissertation, we present some enhancement methods for backtracking-search in solving multiple permutation problems. Some well-known NP-complete multiple permutation problems are Quasigroup Completion Problem and Sudoku. Multiple permutation problems have been getting a lot of attention in the literature in the recent years due to having a highly structured nature and being a challenging combinatorial search problem. Furthermore, it has been shown that many real-world problems in scheduling and experimental design take the form of multiple permutation problems. Therefore, it has been suggested that they can be used as a benchmark problem to test various enhancement methods for solving constraint satisfaction problems. Then it is hoped that the insight gained from studying them can be applied to other hard structured as well as unstructured problems. Our supplementary and novel enhancement methods for backtracking-search in solving these multiple permutation problems can be summarized as follows: We came up with a novel way to encode multiple permutation problems and then we designed and developed an arc-consistency algorithm that is tailored towards this modeling. We implemented five versions of this arc-consistency algorithm where the last version eliminates almost all of the possible propagation redundancy. Then we introduced the novel notion of interlinking dynamic variable ordering with dynamic value ordering, where the dynamic value ordering is also used as a second tie-breaker for the dynamic variable ordering. We also proposed the concept of integrating dynamic variable ordering and dynamic value ordering into an arc-consistency algorithm by using greedy counting assertions. We developed the concept of enforcing local-consistency between variables from different redundant models of the problem. Finally, we introduced an embarrassingly parallel task distribution process at the beginning of the search. We theoretically proved that the limited form of the Hall\u27s theorem is enforced by our modeling of the multiple permutation problems. We showed with our empirical results that the ``fail-first principle is confirmed in terms of minimizing the total number of explored nodes, but is refuted in terms of minimizing the depth of the search tree when finding a single solution, which correlates with previously published results. We further showed that the performance (total number instances solved at the phase transition point within a given time limit) of a given search heuristic is closely related to the underlying pruning algorithm that is being employed to maintain some level of local-consistency during backtracking-search. We also extended the previously established hypothesis, which stated that the second peak of hardness for NP-complete problems is algorithm dependent, to second peak of hardness for NP-complete problems is also search-heuristic dependent. Then we showed with our empirical results that several of our enhancement methods on backtracking-search perform better than the constraint solvers MAC-LAH and Minion as well as the SAT solvers Satz and MiniSat for previously tested instances of multiple permutation problems on these solvers