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Statistical Regimes Across Constrainedness Regions
Carla P. Gomes, Cesar FernandezBart Selman, and Christian Bessiere
Cornell UniversityUniversitat de Lleida
LIRMM-CNRS
CP 2004Toronto
Motivation
Bring together recent results on:
• Typical Case Analysis
• Randomized Complete Search Methods
• Heavy-Tailed Phenomena
• Random CSP Models
Typical Case Analysis: Beyond NP-Completeness
Constrainedness
Com
puta
tion
al C
ost (
Mea
n)
% o
f so
lvab
le in
stan
ces
Phase TransitionPhenomenon:Discriminating “easy” vs.“hard” instances
Hogg et al 96
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 and even harder than instances from the critically constrained area.
Gent and Walsh 94Hogg and Williams 94Smith and 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 and Walsh 94Hogg and Williams 94Selman and Kirkpatrick 96Smith and Grant 97
Randomized Backtrack Search
What if we introduce a tiny element of randomness into the search heuristic – e.g., by breaking ties randomly --- and run this (still complete) randomized search procedure on the same instance over and over again?
Study of runtime distributions of a randomized backtrack search
on the same instance : Way of isolating the variance caused
solely by the algorithm Gomes et al CP 97
Time: >20003011 >20007
Easy instance – 15 % preassigned cells
Gomes, et al 97
Extreme Variance in Runtimeof Randomized Backtrack Search
Heavy-tailed distributions
0,]Pr[ 2
CsomeforxCexX
Exponential decay for standard distributions, e.g. Normal, Logonormal,
exponential:
Heavy-Tailed Power Law Decay e.g. Pareto-Levy:
0,]Pr[ xCxxX
Normal
(Frost et al 97; Gomes et al 97 ,Hoos 1999,Walsh 99,)
Heavy-tailed Dist.
Visualization of Heavy-tailed Phenomenon(Log-Log Plot of Tail o Distribution)
Visualization of Heavy-tailed Phenomenon(Log-Log Plot of Tail o Distribution)
Normal(2,1000000)
Normal(2,1)
1-F
(x)
Unso
lved f
ract
ion
Runtime (Number of backtracks) (log scale)
O,1%>200000
50%
2
Median=2
Formal Results
Abstract Search Tree Models with provably heavy-tailed behavior (Chen, Gomes, Selman 2001)
Generalization and 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:
1. Can we provide a characterization of heavy-tailed behavior: when it occurs and it does not occur?
2. Can we identify different tail regimes across different constrainedness regions?
3. Can we get further insights into the tail regime by analyzing the concrete search trees produced by the backtrack search method?
Concrete CSP ModelsComplete Randomized Backtrack Search
Outline of the Rest of the Talk
• Random Binary CSP Models• Encodings of CSP Models• Randomized 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 for the variables xi and xj.
– Solution to the constraint networkinstantiation of the variables such that all constraints are satisfied.
Random 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 forbidden tuples; t = p2 D2
Model E <N, D, p>
N – number of variables; D – size of the domains: p – proportion of forbidden pairs (out of D2N ( N-1)/ 2)
(Achlioptas et al 2000)
(Gent et al 1996)
N – from 15 to 50; (Xu and Li 2000)
Encodings
• Direct CSP Binary Encoding• Satisfiability Encoding (direct encoding)
Walsh 2000
Backtrack Search Algorithms
• Look-ahead performed::– no look-ahead (simple backtracking BT);– removal of values directly inconsistent with the last instantiation
performed (forward-checking FC);– arc consistency and propagation (maintaining arc consistency, MAC).
• Different heuristics for variable selection (the next variable to instantiate):– Random (random);– variables pre-ordered by decreasing degree in the constraint graph (deg);– smallest domain first, ties broken by decreasing degree (dom+deg)
• Different heuristics for variable value selection:– Random– Lexicographic
• For the SAT encodings we used the simplified Davis-Putnam-Logemann-Loveland procedure: Variable/Value static and random
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 performed without censorship.
Main Results
1 - Runtime distributions2 – 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 and More Sophisticated Consistency Techniques
Other Models and More Sophisticated Consistency Techniques
BT MAC
Heavy-tailed and 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 for proofdetails
Regressions for B1, B2, K
Regression for B1 and B2 Regression for k
Validation: Theoretical Model vs. Runtime Data
α= 0.26 using the model;α= 0.27 using runtime data;
Summary of Results
1 As constrainedness increases change from heavy-tailed to a non-heavy-tailed regime
Both models (B and E), CSP and SAT encodings, for the different backtrack search strategies:
Summary of Results
2 Threshold from the heavy-tailed to non-heavy-tailed regime
– Dependent on the particular search procedure;
– As the efficiency of the search method increases, the extension of the heavy-tailed region increases: the heavy-tailed threshold gets closer to the phase transition.
Summary of Results
3 Distribution of the depth of inconsistent search sub-trees
Exponentially distributed inconsistent sub-tree depth (ISTD) combined with exponential growth of the search space as the tree depth increases implies heavy-tailed runtime distributions.
As the ISTD distributions move away from the exponential distribution, the runtime distributions become non-heavy-tailed.
Research Challenges
How to exploit these results in terms of the design of more efficient search procedures?
– Randomization and restart strategies;– Search heuristics:– Look ahead and look back strategies;
Very exciting and promising research area !
Demos and papers:
www.cs.cornell.edu/gomes/http://fermat.eup.udl.es/~cesar/
www.cs.cornell.edu/selman/ http://www.lirmm.fr/~bessiere/