Abscon 112 Toward more Robustness

This paper describes the three main improvements made to the solver Abscon 109 [9]. The new version, Abscon 112, is able to automatically break some variable symmetries, infer allDifferent constraints from cliques of variables that are pair-wise irreflexive, and use an optimized version of the STR (Simple Tabular Reduction) technique initially introduced by J. Ullmann for table constraints. 1 From Local to Global Variable Symmetries In [10], we have proposed to automatically detect variable symmetries of CSP instances by computing for each constraint scope a partition exhibiting locally symmetrical variables. From this local information that can be obtained in polynomial time, we can build a so-called lsv-graph whose automorphisms correspond to (global) variable symmetries. Interestingly enough, our approach allows us to disregard the representation (extension, intension, global) of constraints. Besides, the size of the lsv-graph is linear wrt the number of constraints (and their arity). To break symmetries from the generators returned by a graph automorphism algorithm, a classical approach is to post lexicographic ordering constraints defined on two vectors of variables. We have proposed a new variant of an algorithm enforcing GAC (generalized arc consistency) on such constraints which is able to deal with shared variables. This algorithm is quite simple to implement and well-adapted to general-purpose constraint solvers. Our experimental results show the robustness of the overall approach with different search heuristics: on a large number of series, more instances can be solved while the cpu time required for symmetry identification is observed as negligible. These results confirm that automatically breaking symmetries constitutes a significant breakthrough for black-box CSP solvers. In order to show the practical interest of this approach, we have then conducted an extensive experimentation on a cluster of Xeon 3.0GHz with 1GiB of RAM under Linux. Here, performance is measured in terms of cpu time (in seconds) and number of visited nodes. We have integrated to the classical MAC algorithm, that is to say the algorithm that maintains (generalized) arc consistency at each node of the search tree, several variants of the symmetry breaking MAC MACLe MACLex MAC ∗ Le MAC ∗ Lex scen11-f12 cpu 1.88 2.0 2.05 1.71 1.82 nodes 390 614 721 4, 140 4, 140 scen11-f11 cpu 1.77 1.97 1.95 1.82 1.83 nodes 390 614 721 4, 216 4, 216 scen11-f10 cpu 1.77 1.82 2.08 1.81 1.9 nodes 468 327 722 109 77 scen11-f9 cpu 2.15 1.96 2.23 1.84 1.97 nodes 1, 064 576 922 109 90 scen11-f8 cpu 2.1 2.09 2.28 2.02 2.0 nodes 1, 354 558 997 112 115 scen11-f7 cpu 4.83 2.28 2.37 1.91 2.05 nodes 8, 369 955 1, 247 121 135 scen11-f6 cpu 8.29 2.14 2.37 2.1 2.08 nodes 17, 839 571 1, 333 172 157 scen11-f5 cpu 32.0 2.2 3.13 2.19 2.13 nodes 85, 104 988 3, 465 253 226 scen11-f4 cpu 112 2.66 3.88 2.36 2.53 nodes 345K 1, 983 5, 007 593 903 scen11-f3 cpu 403 3.41 7.98 2.55 2.45 nodes 1, 300K 3, 926 17, 259 946 696 scen11-f2 cpu time-out 4.32 16.4 2.95 2.92 nodes − 6, 014 40, 615 1, 700 1, 591 scen11-f1 cpu time-out 7.56 19.7 3.49 3.4 nodes − 14, 997 47, 318 3, 199 2, 609 Table 1. Cost of running MAC and its symmetry breaking variants on hard RLFAP instances (38 generators). The variable ordering heuristic is dom/wdeg. approach described in [10]. For this experimentation, no restarts and no nogood recording were activated. To identify variable symmetries, we have used Saucy. For each generator of the symmetry group returned by Saucy, we have considered four distinct symmetry breaking procedures. For the first one, denoted by MACLe, a binary constraint of difference Le (constraint of the form x ≤ y) that involves the two first variables of the first cycle of the generator is posted. For the second one, denoted by MACLex, a lexicographic ordering constraint Lex (involving all variables of all cycles of the generator) is posted. Clearly, a Lex constraint is stronger than the corresponding Le constraint: its filtering capability is greater. Notice that when the two first variables of the first cycle of the generator are included in the scope of a (non-global) constraint c of the network, one can merge c with a binary constraint Le. In practice, if c is defined in intension, its associated predicate is modified whereas if c is defined in extension, the set of tuples disallowing the constraint Le are removed from the table associated with c. When such a merging method is applied, one obtains two additional procedures, denoted by MAC∗ Le and MAC ∗ Lex. Here, we only provide some results (see Table 1) obtained for the hardest instances (which involve 680 variables and a greatest domain size of about 50 values) built from the real-world Radio Link Frequency Assignment Problem (RLFAP). Clearly, the symmetry breaking methods allow us to be far more efficient than the classical MAC algorithm. In practice, MAC∗ Lex has been observed as the best method and has been used for the CSP competition. 2 Exploiting Cliques Some instances contain hidden structures such as backbones, (strong) backdoors and unsatisfiable cores. Cliques also belong to this category. A clique is a graph such that there exists an edge between any two vertices. Interestingly, sometimes, we observe that for any pair (x, y) of variables of a sub-network P ′ of P whose constraint graph is a clique, the relation associated with the constraint involving x and y is irreflexive. Otherwise stated, we know that ∀{x, y} ⊂ vars(P ′), x 6= y. We can then infer an additional global constraint allDifferent that can be useful to better prune the search space. However, in some constraint solvers, the filtering procedure (propagator) attached to allDifferent achieves a local consistency weaker than generalized arc consistency. But, even in this case, inferring allDifferent global constraints can be quite effective provided that the following (trivial) proposition is exploited. Proposition 1. Let c : allDifferent(x1, . . . , xr) be a constraint. If we have | ∪i=1 dom(xi)| < r, then c is disentailed (i.e. the set of supports of c is empty). This approach is quite simple, and to the best of our knowledge, employed by some other solvers engaged in the 2008 competition. It suffices to detect cliques in a greedy manner, determine if irreflexivity is guaranteed between each pair of variables, and post a constraint allDifferent that at least exploits Proposition 1. Interestingly, it is not so rare to find cliques in non-random problems. As an illustration, the instance blackHole-4-4-e-0 (see its constraint graph in Figure 1) contains a 16-clique that enables us to infer a global constraint allDifferent. As one can show that this additional constraint is disentailed by using Proposition 1, the instance is directly proved to be unsatisfiable. 3 Simple Tabular Reduction Table constraints play an important role within constraint programming. Recently, many schemes or algorithms have been proposed to propagate table constraints or/and to compress their representation. In [7], we have shown that simple tabular reduction (STR), a technique proposed by J. Ullmann [11] to dynamically maintain the tables of supports, is very often the most efficient practical approach to enforce generalized arc consistency within MAC. We have also described an optimization of STR which allows limiting the number of operations related to validity checking or search of supports. Interestingly enough,