Y. Fu. The use and abuse of statistical mechanics in computational complexity. In D. Stein, editor, Lectures in the Sciences of Complexity, SFI Series in the Sciences of Complexity. AddisonWesley Longman, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... lack a clear phase boundary at which hard problems cluster [9, p. 1300] while Gent et al. have recently concluded that peaks of hardness for TSP are associated with phase transitions in the probability of existence of a solution smaller than a given size [6, 4] Similarly, according to Fu [3] no phase transition of any kind is found for Number Partitioning, while Gent and Walsh [5] again claim the contrary. 1 Automated Reasoning Project, Australian National University, Canberra, ACT 0200, Australia. e mail: John.Slaney anu.edu.au 2 CSIRO Math. Info. Sciences, PO Box 664, ....

W. Fu, The uses and abuses of statistical mechanics in computational complexity, 815--826, Addison-Wesley Longman, 1989.

....that this method can also give useful models of phase transitions in other combinatorial problems with a control parameter. Several authors have attempted to relate NP hardness or NP completeness to the characteristics of phase transitions in models of disordered systems. Fu and Anderson (see Fu 1989) have proposed spin glasses (magnets with 2 spin interactions of random sign) as having inherent exponential complexity. Huberman and colleagues (see Clearwater 1991) were first to focus on the diverging correlation length seen at continuous phase transitions as the root of computational ....

Fu, Y. (1989). The Uses and Abuses of Statistical Mechanics in Computational Complexity. in Lectures in the Sciences of Complexity, ed. D. Stein, pp. 815-826, Addison-Wesley, 1989.

....defined by transpositions, as long as s is smaller than the number of cities. 2 B. Krakhofer P.F. Stadler: Local Optima The Graph Bipartitioning Problem One of the combinatorial optimization problems which have been studied in great detail is the Graph Bipartitioning Problem (GBP) [1, 6, 7, 12, 19, 24]. Given a graph with an even number n of vertices and an associated matrix H of edge weights, the task is to find a partition X of the vertex set W of this graph into two equal sized subsets X and X such that the total edge weight f GBP (X ) def = X i2X X j2X h ij (2) connecting the ....

Y. Fu and P. W. Anderson. The use and abuse of statistical mechanics in computational complexity. In D. Stein, editor, Lectures in the Science of Complexity, SFI Studies in Science of Complexity. Addison-Wesley, Longman, 1989.

....composed of many interacting values can often be understood at the macroscopic level in terms of a few order parameters that are characteristic of the system as a whole. Summarizing the properties of complex systems through a small set of parameters is routine in statistical mechanics [Fu89] 2 ] Kirkpatrick85] 7 ] This is possible because a large number of local interactions can produce dramatic coordinated macroscopic behavior, such as phase transitions, that do not depend on the detailed interactions within the system. Examples of phase transitions in AI are given in [Karp83] ....

....orderings are a HC) and so an almost fully connected graph has a very high probability of containing a HC. In this region there are a very large number of HCs, and this number drops rapidly as the boundary is approached. At the other extreme, a random graph barely above an average connectivity of 2 is unlikely to even be connected, and so is very unlikely to contain a HC. For some critical value of the average connectivity between these two extremes, the probability of a HC changes steeply from almost 0 to almost 1. Theory predicts that the transition will occur at an average connectivity of ....

[Article contains additional citation context not shown here]

Fu, Y. "The Uses and Abuses of Statistical Mechanics in Computational Complexity", in " Lectures in the Sciences of Complexity", Ed. D. L. Stein, pp 815-826, Addison Wesley, 1989.

....multiprocessor scheduling, and the minimization of VLSI circuit size and delay. Our results correct the claim of Fu that . at least one NP complete problem, that of random number partitioning, can be solved exactly in statistical mechanics and no phase transition of any kind is found . [3]. By means of an annealed theory, we identify a simple phase transition for number partitioning. It remains an open question if there is any NP complete problem which lacks a phase transition. 2 Annealed theory We first compute the expected number of perfect partitions (see Section 6 for an ....

....found often with little search. 10 Conclusions We have outlined a technique for studying phase transition behaviour based upon annealed theories and finite size scaling. Using an annealed theory, we identified a constrainedness parameter for number partitioning. Contrary to the claims of Fu [3], a phase transition occurs at a the critical value of this parameter. Hard number partitioning problems are associated with this transition. Finite size scaling methods developed in statistical mechanics describe the behaviour around this critical value. We were able to identify phase transition ....

Y. Fu. The uses and abuses of statistical mechanics in computational complexity. In D. Stein, ed., Lectures in the Sciences of Complexity, 815--826. Addison-Wesley Longman, 1989.

....the traveling salesman problem [GW95] constraint satisfaction [Smi94, Pro94] independent set problems [GW94b] and Hamiltonian circuits [CKT91] In this paper, we continue our exploration of the ubiquity of phase transition behaviour in NP complete problems by considering number partitioning. Fu [Fu89] has claimed that a phase transition does not occur in number partitioning. The support for this claim comes from analysing the statistical mechanics of the problem. If we accept his argument, Fu then claims that . we must accept as a conclusion that no connection whatsoever exists between the ....

....if any, must be of a form very different from what has been imagined . We show here that phase transition behaviour does occur in number partitioning problems. As a consequence, we are not required to separate phase transition behaviour from computational complexity. 2 Number Partitioning Fu [Fu89] considers the optimization version of the number partioning problem. That is, given a bag, S of N positive integers (N even) find a partition of S into two disjoint bags, S 1 and S 2 both of size N=2 such that we minimize, j X i2S 1 a i Gamma X i2S 2 a i j Department of Computer Science ....

[Article contains additional citation context not shown here]

Y. Fu. The uses and abuses of statistical mechanics in computational complexity. In D. Stein, editor, Lectures in the Sciences of Complexity, pages 815--826. Addison-Wesley Longman, 1989.

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Y. Fu. The use and abuse of statistical mechanics in computational complexity. In D. Stein, editor, Lectures in the Sciences of Complexity, SFI Series in the Sciences of Complexity. AddisonWesley Longman, 1989.

No context found.

Y. Fu. The uses and abuses of statistical mechanics in computational complexity. In D. Stein, editor, Lectures in the Sciences of Complexity, pages 815--826. Addison-Wesley Longman,

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