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Rigorous location of phase transitions in hard optimization problems

Author

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  • Dimitris Achlioptas

    (Microsoft Research, One Microsoft Way)

  • Assaf Naor

    (Microsoft Research, One Microsoft Way)

  • Yuval Peres

    (University of California)

Abstract

It is widely believed that for many optimization problems, no algorithm is substantially more efficient than exhaustive search. This means that finding optimal solutions for many practical problems is completely beyond any current or projected computational capacity. To understand the origin of this extreme ‘hardness’, computer scientists, mathematicians and physicists have been investigating for two decades a connection between computational complexity and phase transitions in random instances of constraint satisfaction problems. Here we present a mathematically rigorous method for locating such phase transitions. Our method works by analysing the distribution of distances between pairs of solutions as constraints are added. By identifying critical behaviour in the evolution of this distribution, we can pinpoint the threshold location for a number of problems, including the two most-studied ones: random k-SAT and random graph colouring. Our results prove that the heuristic predictions of statistical physics in this context are essentially correct. Moreover, we establish that random instances of constraint satisfaction problems have solutions well beyond the reach of any analysed algorithm.

Suggested Citation

  • Dimitris Achlioptas & Assaf Naor & Yuval Peres, 2005. "Rigorous location of phase transitions in hard optimization problems," Nature, Nature, vol. 435(7043), pages 759-764, June.
    • Handle: RePEc:nat:nature:v:435:y:2005:i:7043:d:10.1038_nature03602
    DOI: 10.1038/nature03602
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    Citations

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    Cited by:

    1. Olawale Titiloye & Alan Crispin, 2012. "Parameter Tuning Patterns for Random Graph Coloring with Quantum Annealing," PLOS ONE, Public Library of Science, vol. 7(11), pages 1-9, November.
    2. Jooken, Jorik & Leyman, Pieter & De Causmaecker, Patrick, 2023. "Features for the 0-1 knapsack problem based on inclusionwise maximal solutions," European Journal of Operational Research, Elsevier, vol. 311(1), pages 36-55.
    3. Maurizio Bruglieri & Roberto Cordone & Leo Liberti, 2022. "Maximum feasible subsystems of distance geometry constraints," Journal of Global Optimization, Springer, vol. 83(1), pages 29-47, May.
    4. Leo Lopes & Kate Smith-Miles, 2013. "Generating Applicable Synthetic Instances for Branch Problems," Operations Research, INFORMS, vol. 61(3), pages 563-577, June.
    5. Jooken, Jorik & Leyman, Pieter & De Causmaecker, Patrick, 2022. "A new class of hard problem instances for the 0–1 knapsack problem," European Journal of Operational Research, Elsevier, vol. 301(3), pages 841-854.
    6. Jing Shen & Yaofeng Ren, 2016. "Bounding the scaling window of random constraint satisfaction problems," Journal of Combinatorial Optimization, Springer, vol. 31(2), pages 786-801, February.
    7. Jeff Kahn & Gil Kalai, 2006. "Thresholds and expectation thresholds," Levine's Bibliography 122247000000001294, UCLA Department of Economics.
    8. Guy Kindler & Assaf Naor & Gideon Schechtman, 2010. "The UGC Hardness Threshold of the L p Grothendieck Problem," Mathematics of Operations Research, INFORMS, vol. 35(2), pages 267-283, May.

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