DIRECT ALGORITHMS FOR FINDING MINIMAL UNSATISFIABLE SUBSETS IN OVER-CONSTRAINED CSPs

Abstract

In many situations, an explanation of the reasons behind inconsistency in an overconstrained CSP is required. This explanation can be given in terms of minimal unsatisfiable subsets (MUSes) of constraints. This paper presents algorithms for finding minimal unsatisfiable subsets (MUSes) of constraints in overconstrained CSPs with finite domains and binary constraints. The approach followed is to generate subsets in the subset space, test them for consistency and record the inconsistent subsets found. We present three algorithms as variations of this basic approach. Each algorithm generates subsets in the subset space in a different order and curtails search by employing various search pruning mechanisms. The proposed algorithms are anytime algorithms: a time limit can be set on an algorithm's search and the algorithm can be made to find a subset of MUSes. Experimental evaluation of the proposed algorithms demonstrates that they perform two to three orders of magnitude better than the existing indirect algorithms. Furthermore, the algorithms are able to find MUSes in large CSP benchmarks.

A hybrid algorithm for finding minimal unsatisfiable subsets in over-constrained CSPs

“…6 These methods search the subset space of the set of constraints of an overconstrained CSP and find its complete set of MUSes. Incomplete methods reported find a single unsatisfiable subset of constraints or core that is often not guaranteed to be minimal.…”

“…These algorithms are related to the direct algorithms presented in Ref. 6. Each stem yields a set of USes, not all of which may be minimally unsatisfiable; but as search progresses and other stem's superset spaces are searched, USes that are not minimally unsatisfiable are discarded, as they are found to be subsumed by new USes.…”

“…Algorithm searchSuperSets generates supersets of a stem S by generating the subsets of constraints in C − S, where C is the set of CSP's constraints, and appending the subsets to stem S. We present two implementations of searchSuperSets, based on two different subset generating strategies. 6 The first algorithm views the subset space as a tree with subset sizes increasing level to level (Figure 6), and second algorithm, as a tree with subset sizes decreasing level to level (Figure 7). We refer to the former as small-first subset space tree and the latter as large-first subset space tree.…”

“…// S -set of constraints in the stem; // C -set of constraints in the CSP; // prevStems -set of previously searched Stems; 2 MU Ses ← φ; 3 parentSubset ← φ; 4 pushStack(parentSubset); 5 while ¬emptyStack() do 6 parentSubset ← popStack(); The algorithm, searchSmallSuperSets, generates supersets of S by generating the subsets of C − S in depth-first, right-to-left order and appending them to S. Example 4 illustrates this process. A generated superset X, is tested for satisfiability only if it is not subsumed by previously found unsatisfiable supersets of the current stem or by a previously searched stem; otherwise search is pruned at X and X's descendants are not generated and tested.…”

“…4,5 Recently, complete methods for finding MUSes in CSPs with binary constraints and finite domains have been reported. 6 Incomplete methods reported find a subset of the MUSes of an overconstrained CSP, 7 and some of them just find small inconsistent 'cores,' without guaranteeing their minimality.…”