Conference Paper

The Impact of Balancing on Problem Hardness in a Highly Structured Domain.

Conference: Proceedings, The Twenty-First National Conference on Artificial Intelligence and the Eighteenth Innovative Applications of Artificial Intelligence Conference, July 16-20, 2006, Boston, Massachusetts, USA
Source: DBLP

ABSTRACT Random problem distributions have played a key role in the study and design of algorithms for constraint sat- isfaction and Boolean satisfiability, as well as in our understanding of problem hardness, beyond standard worst-case complexity. We consider random problem distributions from a highly structured problem domain that generalizes the Quasigroup Completion problem (QCP) and Quasigroup with Holes (QWH), a widely used domain that captures the structure underlying a range of real-world applications. Our problem domain is also a generalization of the well-known Sudoku puz- zle: we consider Sudoku instances of arbitrary order, with the additional generalization that the block re- gions can have rectangular shape, in addition to the standard square shape. We evaluate the computational hardness of Generalized Sudoku instances, for different parameter settings. Our experimental hardness results show that we can generate instances that are consider- ably harder than QCP/QWH instances of the same size. More interestingly, we show the impact of different bal- ancing strategies on problem hardness. We also provide insights into backbone variables in Generalized Sudoku instances and how they correlate to problem hardness.

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    • "An example of easy satisfiable formulas are commercial sudokus encoded as SAT formulas. The complexity of this problem is studied in (Lynce and Ouaknine 2006; Ansótegui et al. 2006). In particular, they give the percentage of problems (from a database of sudoku problems) that can be solved with unit propagation and other forms of restricted inference rules. "
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    ABSTRACT: The search of a precise measure of what hardness of SAT instances means for state-of-the-art solvers is a relevant re- search question. Among others, the space complexity of tree- like resolution (also called hardness), the minimal size of strong backdoors and of cycle-cutsets, and the treewidth can be used for this purpose. We propose the use of the tree-like space complexity as a solid candidate to be the best measure for solvers based on DPLL. To support this thesis we provide a comparison with the other mentioned measures. We also conduct an experi- mental investigation to show how the proposed measure char- acterizes the hardness of random and industrial instances.
    Proceedings of the Twenty-Third AAAI Conference on Artificial Intelligence, AAAI 2008, Chicago, Illinois, USA, July 13-17, 2008; 01/2008
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    • "Namely, their most difficult instances exhibit " heavy-tailed " behavior making them extremely sensitive to variable and value ordering heuristics (Hulubei & O'Sullivan, 2004). This may be because QWH and its generalizations often exhibit small " backdoors " : identifying a dense inner problem and solving it first can be the key to tractability (Williams et al., 2003; Ansótegui et al., 2006). Below we formally define this problem. "
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    ABSTRACT: We present a new probabilistic framework for finding likely variable assignments in difficult constraint satisfaction problems. Finding such assignments is key to efficient search, but prac-tical efforts have largely been limited to random guessing and heuristically designed weighting systems. In contrast, we derive a new version of Belief Propagation (BP) using the method of Ex-pectation Maximization (EM). This allows us to differentiate between variables that are strongly biased toward particular values and those that are largely extraneous. Using EM also eliminates the threat of nonconvergence associated with regu-lar BP. Theoretically, the derivation exhibits ap-pealing primal/dual semantics. Empirically, it produces an "EMBP"-based heuristic that out-performs existing techniques for guiding variable and value ordering during backtracking search.
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    • ", d} that instantiates the constraint. (So for instance, r 3 [1] = 5 indicates that the first column of the third row contains the entry 5.) Finally, the Θ parameterizing EM's Q() distribution over Z is the vector of variable biases {θ a,b (v)} that we wish to optimize. That is, for each cell indexed by row a, column b, and each value v that can be placed in that cell, we are asking EM for the bias θ a,b (v) representing the probability that this cell takes the value v in a claimed solution. "
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    ABSTRACT: We present a new probabilistic framework for finding likely variable assignments in difficult constraint satisfaction prob- lems. Finding such assignments is key to efficient search, but practical efforts have largely been limited to random guess- ing and heuristically designed weighting systems. In contrast, we derive a new version of Belief Propagation (BP) using the method of Expectation Maximization (EM). This allows us to differentiate between variables that are strongly biased to- ward particular values and those that are largely extraneous. Using EM also eliminates the threat of non-convergence asso- ciated with regular BP. Theoretically, the derivation exhibits appealing primal/dual semantics. Empirically, it produces an "EMBP"-based heuristic for solving constraint satisfac- tion problems, as illustrated with respect to the Quasigroup with Holes domain. EMBP outperforms existing techniques for guiding variable and value ordering during backtracking search on this problem.
    Proceedings of the Twenty-Second AAAI Conference on Artificial Intelligence, July 22-26, 2007, Vancouver, British Columbia, Canada; 01/2007
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