Computational Complexity and Statistical Physics最新文献

{"title":"The Easiest Hard Problem: Number Partitioning","authors":"S. Mertens","doi":"10.1093/oso/9780195177374.003.0012","DOIUrl":"https://doi.org/10.1093/oso/9780195177374.003.0012","url":null,"abstract":"Number partitioning is one of the classical NP-hard problems of combinatorial optimization. It has applications in areas like public key encryption and task scheduling. The random version of number partitioning has an \"easy-hard\" phase transition similar to the phase transitions observed in other combinatorial problems like $k$-SAT. In contrast to most other problems, number partitioning is simple enough to obtain detailled and rigorous results on the \"hard\" and \"easy\" phase and the transition that separates them. We review the known results on random integer partitioning, give a very simple derivation of the phase transition and discuss the algorithmic implications of both phases.","PeriodicalId":156167,"journal":{"name":"Computational Complexity and Statistical Physics","volume":"106 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2003-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131995564","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Constraint Satisfaction by Survey Propagation","authors":"A. Braunstein, M. Mézard, M. Weigt, R. Zecchina","doi":"10.1093/oso/9780195177374.003.0011","DOIUrl":"https://doi.org/10.1093/oso/9780195177374.003.0011","url":null,"abstract":"Survey Propagation is an algorithm designed for solving typical instances of random constraint satisfiability problems. It has been successfully tested on random 3-SAT and random $G(n,frac{c}{n})$ graph 3-coloring, in the hard region of the parameter space. Here we provide a generic formalism which applies to a wide class of discrete Constraint Satisfaction Problems.","PeriodicalId":156167,"journal":{"name":"Computational Complexity and Statistical Physics","volume":"107 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2002-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124150539","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Scalability, Random Surfaces, and Synchronized Computing Networks","authors":"Z. Toroczkai, G. Korniss, M. Novotny, H. Guclu","doi":"10.1093/oso/9780195177374.003.0020","DOIUrl":"https://doi.org/10.1093/oso/9780195177374.003.0020","url":null,"abstract":"","PeriodicalId":156167,"journal":{"name":"Computational Complexity and Statistical Physics","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121781248","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Analyzing Search Algorithms with Physical Methods","authors":"S. Cocco, R. Monasson, A. Montanari, G. Semerjian","doi":"10.1093/oso/9780195177374.003.0010","DOIUrl":"https://doi.org/10.1093/oso/9780195177374.003.0010","url":null,"abstract":"","PeriodicalId":156167,"journal":{"name":"Computational Complexity and Statistical Physics","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114310349","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Phase Transitions for Quantum Search Algorithms","authors":"T. Hogg","doi":"10.1093/oso/9780195177374.003.0019","DOIUrl":"https://doi.org/10.1093/oso/9780195177374.003.0019","url":null,"abstract":"","PeriodicalId":156167,"journal":{"name":"Computational Complexity and Statistical Physics","volume":"58 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127974823","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Ground States, Energy Landscape, and Low-Temperature Dynamics of ±J Spin Glasses","authors":"S. Kobe, J. Krawczyk","doi":"10.1093/oso/9780195177374.003.0013","DOIUrl":"https://doi.org/10.1093/oso/9780195177374.003.0013","url":null,"abstract":"The previous three chapters have focused on the analysis of computational problems using methods from statistical physics. This chapter largely takes the reverse approach. We turn to a problem from the physics literature, the spin glass, and use the branch-and-bound method from combinatorial optimization to analyze its energy landscape. The spin glass model is a prototype that combines questions of computational complexity from the mathematical point of view and of glassy behavior from the physical one. In general, the problem of finding the ground state, or minimal energy configuration, of such model systems belongs to the class of NP-hard tasks. The spin glass is defined using the language of the Ising model, the fundamental description of magnetism at the level of statistical mechanics. The Ising model contains a set of n spins, or binary variables Si, each of which can take on the value “up” (Si = 1) or “down” (Si = −1). Finding the ground state means finding the spin variable values minimizing the Ising Hamiltonian energy (cost) function, written in general as","PeriodicalId":156167,"journal":{"name":"Computational Complexity and Statistical Physics","volume":"262 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121885435","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Phase Transition in the Random HornSAT Problem","authors":"Demetrios D. Demopoulos, Moshe Y. Vardi","doi":"10.1093/oso/9780195177374.003.0017","DOIUrl":"https://doi.org/10.1093/oso/9780195177374.003.0017","url":null,"abstract":"","PeriodicalId":156167,"journal":{"name":"Computational Complexity and Statistical Physics","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114939155","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Satisfiability Threshold Conjecture: Techniques Behind Upper Bound Improvements","authors":"L. Kirousis, Y. Stamatiou, M. Zito","doi":"10.1093/oso/9780195177374.003.0015","DOIUrl":"https://doi.org/10.1093/oso/9780195177374.003.0015","url":null,"abstract":"One of the most challenging problems in probability and complexity theory concerns the establishment and the determination of the satisfiability threshold for random Boolean formulas consisting of clauses with exactly k literals, or k-SAT formulas with emphasis on the case k = 3, or 3-SAT. According to many experimental observations, there exists a critical value rk of the number of clauses to the number of variables ratio r = m/n such that almost all randomly generated formulas with r > rk are unsatisfiable while almost all randomly generated formulas with r < rk are satisfiable. The statement that such a crossover point really exists is called the “satisfiability threshold conjecture”. While experiments hint at such a direction, as far as theoretical work is concerned, progress has been difficult. Up to now, there are rigorous proofs of only successively better upper and lower bounds for the value of the (conjectured) threshold although, in an important advance, Friedgut proved that the phase transition is sharp (without showing the existence of a fixed transition point). In this work, our goal is to review the series of improvements of the upper bounds for 3-SAT and the techniques from which the improvements resulted. We give only a passing reference to the improvements of the lower bounds, as they rely on significantly different techniques that would require much more space to present.","PeriodicalId":156167,"journal":{"name":"Computational Complexity and Statistical Physics","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132577512","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Introduction: Where Statistical Physics Mects Computation","authors":"A. Percus, Gabriel Istrate, Cristopher Moore","doi":"10.1093/oso/9780195177374.003.0007","DOIUrl":"https://doi.org/10.1093/oso/9780195177374.003.0007","url":null,"abstract":"Computer science and physics have been closely linked since the birth of modern computing. This book is about that link. John von Neumann’s original design for digital computing in the 1940s was motivated by applications in ballistics and hydrodynamics, and his model still underlies today’s hardware architectures. Within several years of the invention of the first digital computers, the Monte Carlo method was developed, putting these devices to work simulating natural processes using the principles of statistical physics. It is difficult to imagine how computing might have evolved without the physical insights that nurtured it. It is impossible to imagine how physics would have evolved without computation. While digital computers quickly became indispensable, a true theoretical understanding of the efficiency of the computation process did not occur until twenty years later. In 1965, Hartmanis and Stearns [30] as well as Edmonds [20, 21] articulated the notion of computational complexity, categorizing algorithms according to how rapidly their time and space requirements grow with input size. The qualitative distinctions that computational complexity draws between algorithms form the foundation of theoretical computer science. Chief among these distinctions is that of polynomial versus exponential time. A combinatorial problem belongs in the complexity class P (polynomial time) if there exists an algorithm guaranteeing a solution in a computation time, or number of elementary steps of the algorithm, that grows at most polynomially with input size. Loosely speaking, such problems are considered computationally feasible. An example might be sorting a list of n numbers: even a particularly naive and inefficient algorithm for this will run in a number of steps that grows as O(n), and so sorting is in the class P. A problem belongs in the complexity class NP (non-deterministic polynomial time) if it is merely possible to test, in polynomial time, whether a specific presumed solution is correct. Of course, P ⊆ NP: for any problem whose solution can be found in polynomial time, one can surely verify the validity of a presumed solution in polynomial time.","PeriodicalId":156167,"journal":{"name":"Computational Complexity and Statistical Physics","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117061191","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Proving Conditional Randomness using The Principle of Deferred Decisions","authors":"A. Kaporis, L. Kirousis, Y. Stamatiou","doi":"10.1093/oso/9780195177374.003.0016","DOIUrl":"https://doi.org/10.1093/oso/9780195177374.003.0016","url":null,"abstract":"In order to prove that a certain property holds asymptotically for a restricted class of objects such as formulas or graphs, one may apply a heuristic on a random element of the class, and then prove by probabilistic analysis that the heuristic succeeds with high probability. This method has been used to establish lower bounds on thresholds for desirable properties such as satisfiability and colorability: lower bounds for the 3-SAT threshold were discussed briefly in the previous chapter. The probabilistic analysis depends on analyzing the mean trajectory of the heuristic—as we have seen in Cocco et al. [3]—and in parallel, showing that in the asymptotic limit the trajectory’s properties are strongly concentrated about their mean. However, the mean trajectory analysis requires that certain random characteristics of the heuristic’s starting sample are retained throughout the trajectory. We propose a methodology in this chapter to determine the conditional that should be imposed on a random object, such as a conjunctive normal form (CNF)","PeriodicalId":156167,"journal":{"name":"Computational Complexity and Statistical Physics","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133558583","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}