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On state space structure and average case complexity in random K-SAT problems
KTH, School of Computer Science and Communication (CSC), Computational Biology, CB.
2008 (English)Licentiate thesis, comprehensive summary (Other scientific)
Abstract [en]

This thesis gives an introduction to a currently active area in the cross-section between theoretical computer science and theoretical physics. In the last ten years it has been suggested that critical behaviour, usually seen in models from condensed matter physics, may be responsible for the intractability of NP complete computation problems. This would suggest a very deep connection between the two fields on the most fundamental level. How deep this connection really is is subject to ongoing research as well as the topic of this thesis. Some of the conjectrues from the physics community regarding computational hardness in certain problem classes has turned out to be wrong or misinterpreted but the gained interest in both fields has promising potiential in moving the research frontier forward.

The material presented in this thesis is the result of nearly two years work in trying to clearify how the results from physics can be interpreted in the language of actuall computation problems.

Abstract [sv]

Denna avhandling ger en introduktion till ett mycket aktivt forskningsområde i gränslandet mellan teortisk datalogi och teoretisk fysik. Under de senaste tio åren har det framkommit forskningsresultat som pekar på att kritiska fenomen, vanligen hemmahörande i modeller från teoretisk materialfysik, kan vara nyckeln till att förstå varför NP fullständiga problem är så svåra att lösa. Detta skulle innebära en mycket djup och fundamental koppling mellan de bägge områdena. Hur djup denna koppling verkligen är är temat i mycket av pågående forskning såväl som ämnet för denna avhandling. Vissa förutsägelser från den teoretiska fysiken har visat sig felaktiga eller feltolkade men det ökade intresset för dylika frågor inom bägge forskningområden ger hopp om att tillsammans kunna flytta from forskningsfronten.

Place, publisher, year, edition, pages
Stockholm: KTH , 2008. , viii, 31 p.
Series
Trita-CSC-A, ISSN 1653-5723
National Category
Computer Science
Identifiers
URN: urn:nbn:se:kth:diva-4765ISBN: 978-91-7178-971-6 (print)OAI: oai:DiVA.org:kth-4765DiVA: diva2:13835
Presentation
2008-05-14, FR42, NORDITA, Roslagstullsbacken 23, 13:00
Opponent
Supervisors
Note
QC 20101103Available from: 2008-05-19 Created: 2008-05-19 Last updated: 2010-11-03Bibliographically approved
List of papers
1. Behavior of heuristics on large and hard satisfiability problems
Open this publication in new window or tab >>Behavior of heuristics on large and hard satisfiability problems
2006 (English)In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics: Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, ISSN 1063-651X, E-ISSN 1095-3787, Vol. 74, no 3, 037702- p.Article in journal (Refereed) Published
Abstract [en]

 We study the behavior of a heuristic for solving random satisfiability problems by stochastic local search near the satisfiability threshold. The heuristic for average satisfiability (ASAT), is similar to the Focused Metropolis Search heuristic, and shares the property of being focused, i.e., only variables in unsatisfied clauses are updated in each step. It is significantly simpler than the benchmark WALKSAT heuristic. We show that ASAT solves instances as large as N=10(6) in linear time, on average, up to a ratio of 4.21 clauses per variable in random three-satisfiability. For K higher than 3, ASAT appears to solve instances of K-satisfiability up to the Montanari-Ricci-Tersenghi-Parisi full replica symmetry breaking (FSRB) threshold denoted alpha(s)(K) in linear time.

Keyword
Atomic physics, Benchmarking, Problem solving, Quantum theory, Random processes
National Category
Physical Sciences
Identifiers
urn:nbn:se:kth:diva-8491 (URN)10.1103/PhysRevE.74.037702 (DOI)000240870300105 ()2-s2.0-33748771720 (Scopus ID)
Note
QC 20100901Available from: 2008-05-19 Created: 2008-05-19 Last updated: 2017-12-14Bibliographically approved
2. Clustering of solutions in hard satisfiability problems
Open this publication in new window or tab >>Clustering of solutions in hard satisfiability problems
2007 (English)In: Journal of Statistical Mechanics: Theory and Experiment, ISSN 1742-5468, no 10, P10012- p.Article in journal (Refereed) Published
Abstract [en]

We study numerically the solution space structure of random 3-SAT problems close to the SAT/UNSAT transition. This is done by considering chains of satisfiability problems, where clauses are added sequentially to a problem instance. Using the overlap measure of similarity between different solutions found on the same problem instance, we examine geometrical changes as a function of α. In each chain, the overlap distribution is first smooth, but then develops a tiered structure, indicating that the solutions are found in well separated clusters. On chains of not too large instances, all remaining solutions are eventually observed to be found in only one small cluster before vanishing. This condensation transition point is estimated by finite size scaling to be αc = 4.26 with an apparent critical exponent of about 1.7. The average overlap value is also observed to increase with α up to the transition, indicating a reduction in solutions space size, in accordance with theoretical predictions. The solutions are generated by a local heuristic, ASAT, and compared to those found by the Survey Propagation algorithm up to αc.

Keyword
Energy landscapes (experiment), Finite-size scaling, Network dynamics, Networks, Random graphs
National Category
Bioinformatics and Systems Biology
Identifiers
urn:nbn:se:kth:diva-8492 (URN)10.1088/1742-5468/2007/10/P10012 (DOI)000250725400006 ()2-s2.0-35948985998 (Scopus ID)
Note
QC 20100903Available from: 2008-05-19 Created: 2008-05-19 Last updated: 2010-09-03Bibliographically approved
3. Circumspect descent prevails in solving random constraint satisfaction problems
Open this publication in new window or tab >>Circumspect descent prevails in solving random constraint satisfaction problems
Show others...
2008 (English)In: Proceedings of the National Academy of Sciences of the United States of America, ISSN 0027-8424, E-ISSN 1091-6490, Vol. 105, no 40, 15253-15257 p.Article in journal (Refereed) Published
Abstract [en]

We study the performance of stochastic local search algorithms for random instances of the K-satisfiability (K-SAT) problem. We present a stochastic local search algorithm, ChainSAT, which moves in the energy landscape of a problem instance by never going upwards in energy. ChainSAT is a focused algorithm in the sense that it focuses on variables occurring in unsatisfied clauses. We show by extensive numerical investigations that ChainSAT and other focused algorithms solve large K-SAT instances almost surely in linear time, up to high clause-to-variable ratios a; for example, for K = 4 we observe linear-time performance well beyond the recently postulated clustering and condensation transitions in the solution space. The performance of ChainSAT is a surprise given that by design the algorithm gets trapped into the first local energy minimum it encounters, yet no such minima are encountered. We also study the geometry of the solution space as accessed by stochastic local search algorithms.

Keyword
geometry of solutions local search, performance, random K-SAT
National Category
Other Biological Topics
Identifiers
urn:nbn:se:kth:diva-8493 (URN)10.1073/pnas.0712263105 (DOI)000260360500010 ()2-s2.0-55749098814 (Scopus ID)
Note
QC 20100903. Uppdaterad från Submitted till Published (20100903)Available from: 2008-05-19 Created: 2008-05-19 Last updated: 2017-12-14Bibliographically approved
4. Temperature dependent sampling in hard satisfiability problems
Open this publication in new window or tab >>Temperature dependent sampling in hard satisfiability problems
2008 (English)Article in journal (Other academic) Submitted
Abstract [en]

The solution sets of constraint satisfaction problems (CSPs)have been conjectured to to split up into clusters (connectedcomponents) when they are close to critically constrained,and this has been assumed to be related to computationalhardness. Simple stochastic local search (SLS) heuristics hashowever shown to be very efficient for many of these problems,and it has been unclear if this is related to clustering.In this work an SLS is used to sample the space of solutionsand the results are compared to the actual solution space generatedwith a complete solver. We show that different heuristicsfind different types of clusters. An increased greedinessresults in a sampling more uniform over the set of clusters,rather than over the set of solutions and the fastest solver visitsthe smaller cluster much more frequently than its solutionsdensity would imply. We also show that the level of randomness(temperature) is related in a non-trivial way to the abilityto escape from local minimas. This approach confirms thatclusters are important in understanding computational hardnessand may begin to explain the efficiency of SLSs in fragmentedsolution spaces.

National Category
Computer Science
Identifiers
urn:nbn:se:kth:diva-8494 (URN)
Note
QS 20120314Available from: 2008-05-19 Created: 2008-05-19 Last updated: 2012-03-14Bibliographically approved
5. Exhaustive enumeration unveils clustering and freezing in the random 3-satisfiability problem
Open this publication in new window or tab >>Exhaustive enumeration unveils clustering and freezing in the random 3-satisfiability problem
2008 (English)In: Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, ISSN 1539-3755, Vol. 78, no 4, 040101- p.Article in journal (Refereed) Published
Abstract [en]

We study geometrical properties of the complete set of solutions of the random 3-satisfiability problem. We show that even for moderate system sizes the number of clusters corresponds surprisingly well with the theoretic asymptotic prediction. We locate the freezing transition in the space of solutions, which has been conjectured to be relevant in explaining the onset of computational hardness in random constraint satisfaction problems.

Keyword
Boolean functions, Freezing, Clustering, Complete sets, Computational hardnesses, Exhaustive enumerations, Freezing transitions, Geometrical properties, Number of clusters, Random constraint satisfaction problems, Satisfiability problems, System sizes
National Category
Computer Science
Identifiers
urn:nbn:se:kth:diva-25885 (URN)10.1103/PhysRevE.78.040101 (DOI)000260573800002 ()
Note
QC 20101103. Uppdaterad från submitted till published (20101103). Tidigare titel: Exhaustive enumeration unveils clustering and freezing in random 3-SATAvailable from: 2010-11-03 Created: 2010-11-03 Last updated: 2010-11-03Bibliographically approved

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