Contents

## Statistical Regimes Across Constrainedness Regions Carla P. Gomes, Cesar Fern in addition to

Blaine, Sydney, Executive Producer has reference to this Academic Journal, PHwiki organized this Journal Statistical Regimes Across Constrainedness Regions Carla P. Gomes, Cesar Fern in addition to ez Bart Selman, in addition to Christian Bessiere Cornell University Universitat de Lleida LIRMM-CNRS CP 2004 Toronto Motivation Bring together recent results on: Typical Case Analysis R in addition to omized Complete Search Methods Heavy-Tailed Phenomena R in addition to om CSP Models Typical Case Analysis: Beyond NP-Completeness Constrainedness Computational Cost (Mean) % of solvable instances Phase Transition Phenomenon: Discriminating easy vs. hard instances Hogg et al 96

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Exceptional Hard Instances Seem to defy the easy-hard pattern: such instances occur in the under-constrained area; they are considerably harder than other similar instances in addition to even harder than instances from the critically constrained area. Gent in addition to Walsh 94 Hogg in addition to Williams 94 Smith in addition to Grant 97 Are Exceptionally Hard Instances Truly Hard Different algorithms encounter different exceptionally hard instances. “Hardness” of exceptionally hard instances not necessarily hardness of the instances, but rather a the combination of the instance with the details of the search method; Gent in addition to Walsh 94 Hogg in addition to Williams 94 Selman in addition to Kirkpatrick 96 Smith in addition to Grant 97 R in addition to omized Backtrack Search What if we introduce a tiny element of r in addition to omness into the search heuristic e.g., by breaking ties r in addition to omly — in addition to run this (still complete) r in addition to omized search procedure on the same instance over in addition to over again Study of runtime distributions of a r in addition to omized backtrack search on the same instance : Way of isolating the variance caused solely by the algorithm Gomes et al CP 97

Extreme Variance in Runtime of R in addition to omized Backtrack Search Easy instance 15 % preassigned cells Gomes, et al 97 Heavy-tailed distributions Exponential decay as long as st in addition to ard distributions, e.g. Normal, Logonormal, exponential: Heavy-Tailed Power Law Decay e.g. Pareto-Levy: Normal (Frost et al 97; Gomes et al 97 ,Hoos 1999,Walsh 99,) Visualization of Heavy-tailed Phenomenon (Log-Log Plot of Tail o Distribution) Heavy-tailed Dist. Normal (2,1000000) Normal (2,1) 1-F(x) Unsolved fraction Runtime (Number of backtracks) (log scale)

Formal Results Abstract Search Tree Models with provably heavy-tailed behavior (Chen, Gomes, Selman 2001) Generalization in addition to Assignment of Semantics to the Abstract Search Tree Models (Williams, Gomes, Selman 2003) Provably Polytime Restart Strategies (Williams, Gomes, Selman 2003) What about concrete CSP models (so far no good characterization of runtime distributions of concrete CSP models) Research Questions: Can we provide a characterization of heavy-tailed behavior: when it occurs in addition to it does not occur Can we identify different tail regimes across different constrainedness regions Can we get further insights into the tail regime by analyzing the concrete search trees produced by the backtrack search method Concrete CSP Models Complete R in addition to omized Backtrack Search

Outline of the Rest of the Talk R in addition to om Binary CSP Models Encodings of CSP Models R in addition to omized Backtrack Search Algorithms Search Trees Statistical Tail Regimes Across Cosntrainedness Regions Empirical Results Theoretical Model Conclusions Binary Constraint Networks A finite binary constraint network P = (X, D,C) a set of n variables X = {x1, x2, , xn} For each variable, set of finite domains D = { D(x1), D(x2), , D(xn)} A set C of binary constraints between pairs of variables; a constraint Cij, on the ordered set of variables (xi, xj) is a subset of the Cartesian product D(xi) x D(xj) that specifies the allowed combinations of values as long as the variables xi in addition to xj. Solution to the constraint network instantiation of the variables such that all constraints are satisfied. R in addition to om Binary CSP Models Model B < N, D, c, t > N number of variables; D size of the domains; c number of constrained pairs of variables; p1 proportion of binary constraints included in network ; c = p1 N ( N-1)/ 2; t tightness of constraints; p2 – proportion of as long as bidden tuples; t = p2 D2 Model E

Encodings Direct CSP Binary Encoding Satisfiability Encoding (direct encoding) Walsh 2000 Backtrack Search Algorithms Look-ahead per as long as med:: no look-ahead (simple backtracking BT); removal of values directly inconsistent with the last instantiation per as long as med ( as long as ward-checking FC); arc consistency in addition to propagation (maintaining arc consistency, MAC). Different heuristics as long as variable selection (the next variable to instantiate): R in addition to om (r in addition to om); variables pre-ordered by decreasing degree in the constraint graph (deg); smallest domain first, ties broken by decreasing degree (dom+deg) Different heuristics as long as variable value selection: R in addition to om Lexicographic For the SAT encodings we used the simplified Davis-Putnam-Logemann-Lovel in addition to procedure: Variable/Value static in addition to r in addition to om Inconsistent Subtrees Bessiere at al 2004

Distributions Runtime distributions of the backtrack search algorithms; Distribution of the depth of the inconsistency trees found during the search; All runs were per as long as med without censorship. Main Results 1 – Runtime distributions 2 Inconsistent Sub-tree Depth Distributions Dramatically different statistical regimes across the constrainedness regions of CSP models; Runtime distributions

Distribution of Depth of Inconsistent Subtrees Applet Applet Depth of Inconsistent Search Tree vs. Runtime Distributions

Other Models in addition to More Sophisticated Consistency Techniques BT MAC Heavy-tailed in addition to non-heavy-tailed regions. As the sophistication of the algorithm increases the heavy-tailed region extends to the right, getting closer to the phase transition Model B SAT encoding: DPLL Theoretical Model

Depth of Inconsistent Search Tree vs. Runtime Distributions Theoretical Model X search cost (runtime); ISTD depth of an inconsistent sub-tree; Pistd [IST = N] probability of finding an inconsistent sub-tree of depth N during search; P[X>x N] probability of the search cost being larger x, given an inconsistent tree of depth N Depth of Inconsistent Search Tree vs. Runtime Distributions: Theoretical Model See paper as long as proof details Regressions as long as B1, B2, K Regression as long as B1 in addition to B2 Regression as long as k

Motivation Great strides in designing more efficient complete backtrack search methods as long as solving constraint satisfaction problems: strong search heuristics; Look-ahead in addition to look-back techniques; R in addition to omization in addition to restarts. Motivation The study of problem structure — insights in terms of the interplay between structure, search algorithms, in addition to more generally, typical case complexity: Phase transition phenomena Exceptionally hard instances R in addition to omized Backtrack Search Heavy-tailed phenomena in combinatorial search

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