The Connectivity of Boolean The Connectivity of Boolean Satisfiability: Satisfiability: Structural and Computational Structural and Computational Dichotomies Dichotomies Elitza Maneva (UC Berkeley) Elitza Maneva (UC Berkeley) Joint work with Joint work with Parikshit Gopalan, Phokion Kolaitis and Christos Parikshit Gopalan, Phokion Kolaitis and Christos Papadimitriou Papadimitriou
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The Connectivity of Boolean Satisfiability: Structural and Computational Dichotomies Elitza Maneva (UC Berkeley) Joint work with Parikshit Gopalan, Phokion.
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The Connectivity of Boolean Satisfiability: The Connectivity of Boolean Satisfiability: Structural and Computational DichotomiesStructural and Computational Dichotomies
Parikshit Gopalan, Phokion Kolaitis and Christos PapadimitriouParikshit Gopalan, Phokion Kolaitis and Christos Papadimitriou
Features of our dichotomyFeatures of our dichotomy
• Refers to the structure of the entire space of solutions
• The dichotomy cuts across Boolean clones
• Motivated by recent heuristics for random input CSP.
Space of solutionsSpace of solutions1111111111
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n-dimensional hypercube
Space of solutionsSpace of solutions1111111111
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Space of solutionsSpace of solutions1111111111
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Space of solutionsSpace of solutions
Connectivity of graph of solutions?Connectivity of graph of solutions?
1111111111
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1111111111
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Our dichotomyOur dichotomy
• Computational problems– CONN: Is the solution graph connected?– st-CONN: Are two solutions connected?
• Structural property– Possible diameter of components
PSPACE-complete
PSPACE-complete
exponential
NP-complete
CONN
st-CONN
diameter
SAT
Tight Tight CSPCSP Non-tight Non-tight CSP CSP
in co-NP
in P
linear
P and NP-complete
Motivation for our studyMotivation for our study
Heuristics for random CSP are influenced by the structure of the solution space
Random 3-SAT with parameter :
n variables, n clauses are chosen at random
4.154.15 4.274.2700
Easy Hard Unsat
Motivation for our studyMotivation for our study
Heuristics for random CSP are influenced by the structure of the solution space
Survey propagation algorithm [Mezard, Parisi, Zecchina ‘02]• designed to work for clustered random problems• very successful for such random instances• based on statistical physics analysis
Clustering in random CSPClustering in random CSPWhat is known?What is known?
2-SAT: a single cluster up to the satisfiability threshold 2-SAT: a single cluster up to the satisfiability threshold
3-SAT to 7-SAT: not known, but conjectured to have 3-SAT to 7-SAT: not known, but conjectured to have clusters before the satisfiability thresholdclusters before the satisfiability threshold
8-SAT and above: exponential number of clusters8-SAT and above: exponential number of clusters[Achlioptas, Ricci-Tersenghi `06][Achlioptas, Ricci-Tersenghi `06][Mezard, Mora, Zecchina `05] [Mezard, Mora, Zecchina `05]
• Trichotomy for CONN? Trichotomy for CONN? – P for component-wise bijunctiveP for component-wise bijunctive– coNP-complete for non-Schaefer tight relationscoNP-complete for non-Schaefer tight relations– open for Horn/dual-Hornopen for Horn/dual-Horn
• Which Boolean CSPs have a clustered phase?Which Boolean CSPs have a clustered phase?