From academically interesting to practically relevant. Heavy Tailed behavior seems to be pervasive in combinatorial search in addition to has been observed in several other domains: Graph Coloring, Planning, Scheduling, Verification, Circuit synthesis, Decoding, etc. www.cs.cornell.edu/gomes

From academically interesting to practically relevant. Heavy Tailed behavior seems to be pervasive in combinatorial search in addition to has been observed in several other domains: Graph Coloring, Planning, Scheduling, Verification, Circuit synthesis, Decoding, etc. www.cs.cornell.edu/gomes www.phwiki.com

From academically interesting to practically relevant. Heavy Tailed behavior seems to be pervasive in combinatorial search in addition to has been observed in several other domains: Graph Coloring, Planning, Scheduling, Verification, Circuit synthesis, Decoding, etc. www.cs.cornell.edu/gomes

Dawson, Michelle, Morning Show On-Air Personality has reference to this Academic Journal, PHwiki organized this Journal Beyond Satisfiability: Model Counting, Quantification, in addition to R in addition to omization Carla P. Gomes Cornell University Connections II Caltech 2006 Satisfiability in addition to Beyond SAT MAXSAT, SMT SAT QBF Motivation: Significant progress in SAT From 100 variables, 200 constraints (early 90’s) to 1,000,000 vars. in addition to 5,000,000 clauses in 15 years. Applications: Hardware in addition to Software Verification, Planning, Scheduling, Optimal Control, Protocol Design, Routing, Multi-agent systems, E-Commerce (E-auctions in addition to electronic trading agents), etc.

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Motivation: Defying NP-Completeness While real-world instances with over 1,000,000 variables are often solved in a few minutes, r in addition to om SAT instances with only a few hundred variables often cannot be solved! Current state of the art complete or exact solvers (SAT/CSP/MIP) can h in addition to le very large problem instances of real-world combinatorial: We are dealing with as long as midable search spaces of exponential size — to prove unsatisifability or optimality we have to implicitly search the entire search, the problems we are able to solve are much larger than one would predict given that such problems are in general NP complete or harder often Worst Case Complexity Gap Between Theory in addition to Practice R in addition to om Instances Short Proofs (in practice) Exponential Proofs (worst case) Worst Case Complexity Underst in addition to ing the Gap Between Theory in addition to Practice

Outline SAT R in addition to om Problems Structured Problems Connections between Heavy-tailed Distributions, Backdoors, in addition to Restart Strategies in Complete Search Methods as long as Combinatorial Problems: SAT Streamlining Constraint Reasoning, R in addition to omization, in addition to Model Counting; QBF – Quantification Conclusions SAT Propositional Satisfiability problem: (SAT) Satifiability (SAT): Given a as long as mula in propositional calculus, does it have a model, i.e., is there an assignment to its variables making it true ( a b c ) AND ( b c) AND ( a c) possible assignments SAT: prototypical hard combinatorial search in addition to reasoning problem. Problem is NP-Complete. (Cook 1971)

SAT R in addition to om Instances R in addition to om 3-SAT as of 2005 Linear time algs. Mitchell, Selman, in addition to Levesque ’92 R in addition to om 3-SAT as of 2004 Linear time algs. Upper bounds by combinatorial arguments (’92 – ’05)

Exact Location of Threshold Surprisingly challenging problem Current rigorously proved results: 3SAT threshold lies between 3.42 in addition to 4.506. Motwani et al. 1994; Broder et al. 1992; Frieze in addition to Suen 1996; Dubois 1990, 1997; Kirousis et al. 1995; Friedgut 1997; Archlioptas et al. 1999; Beame, Karp, Pitassi, in addition to Saks 1998; Impagliazzo in addition to Paturi 1999; Bollobas, Borgs, Chayes, Han Kim, in addition to Wilson1999; Achlioptas, Beame in addition to Molloy 2001; Frieze 2001; Zecchina et al. 2002; Kirousis et al. 2004; Gomes in addition to Selman, Nature ’05; Achlioptas et al. Nature ’05; in addition to ongoing Empirical: 4.25 — Mitchell, Selman, in addition to Levesque ’92, Craw as long as d ’93. Tremendous interaction with other communities OR, Physics, Mathematics SAT Structured Problems Surprising “power” of SAT as long as solving certain real world combinatorial problems (clearly outper as long as ming Integer Programming).

From academically interesting to practically relevant. We now have regular SAT solver competitions. Germany ’89, Dimacs ’93, China ’96, SAT-02, SAT-03, , SAT-06 Sat 06 – Seattle Aug. 12-15, 2006 SAT Competitions: Classical SAT solvers (CNF) Pseudo Boolean Solvers QBF MAXSAT SMT – SAT Mod Theory SAT Competition 2006: Industrial Instances

SAT Competition 2006: Industrial Track SAT Competition 2006: Benchmark Instances Classical SAT Solvers Instances up to 1,000,000 variables in addition to 15,000,000 clauses – at least one solver could prove SAT/UNSAT Timelimit per instance: 15 minutes Progress SAT Solvers Source: Marques Silva 2002

An abstraction of a structured combinatorial problems: Encodings in addition to hardness profiles Latin Square Completion Better characterization beyond worst case Latin Square (Order 4) NP-Complete Gomes in addition to Selman 97 Routing in Fiber Optic Networks Scheduling in addition to timetabling Design of Scientific Experiments Many more applications Sudoku Underlying Latin Square structure characterizes many real world applications

Dawson, Michelle WUFN-FM Morning Show On-Air Personality www.phwiki.com

Encodings Constraint Satisfaction Integer Programming SAT All the encodings exhibit similar qualitative behavior wrt to hardness profile 2.Scaling varies with encoding; Integer Programming (Assignment Formulation) Row/color line Column/color line Row/column line Max number of colored cells Scaling: up to order 20 Variables New Phase Transition Phenomenon: Integrality of LP Note: st in addition to ard phase transition curves are w.r.t existence of solution) holes/n^1.55 No of backtracks Max value of LP Relaxation Gomes in addition to Leahu 04

Integer Programming: Packing Formulation one pattern per color at most one pattern covering each cell Max number of colored cells (1-1/e) Approximation Algorithm Gomes in addition to Shmoys 03 Constraint Satisfaction Problem (CSP) Variables Constraints – row column Scaling: up to order 33 Hybrid CSP + LP CSP propagation (1-1/e) – Approximation Algorithm (based on the packing as long as mulation) Scaling: up to order 36 Gomes in addition to Shmoys 04

Experimental Results, contd. 7-9 quantifier levels Duaffle (even without learning) on the dual encoding dramatically outper as long as ms all leading CNF-based QBF solvers on these challenging instances Summary SAT progress Path from 100 var instances (early 90’s) to 1,000,000+ var instances (current). Still moving as long as ward Capturing in addition to exploiting structure is key when dealing with Large Real-world instances: connections between heavy-tails, backdoor sets, r in addition to omization, in addition to restarts. streamlining in addition to xor streamlining Beyond satisfaction / New applications: Model counting in addition to Quantification. The End www.cs.cornell.edu/gomes

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