D. Achlioptas. Threshold phenomena in random graph colouring and satisfiability. PhD thesis, Department of Computer Science, University of Toronto, 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....minimal unsolvable subproblem would thus be a very good heuristic for search algorithm. However, this problem is in general as difficult as finding the optimum solution itself. In recent years, there has been quite a lot work on determining better exact bounds for the phase transition, see, e.g. [86, 87, 88, 89, 90]. The best bound for # COL is #:## c #:##, while for # SAT it is known that #:### c #:##. 3.2 Nature of the phase transition The k SAT problem has been studied in detail by Monasson and Zecchina [91, 92, 93, 94, 95] and others [96, 97, 98] who have found interesting analogies between ....

D. Achlioptas, Threshold Phenomena in Random Graph Colouring and Satisfiability. PhD thesis, University of Toronto, 1999. Available at http://www.research.microsoft.com/#optas/.

....problems. For the k col problem on random graphs with average degree #, Achlioptas and Friedgut [7] have shown rigorously that there is a # # where the fraction of colourable graphs jumps from 1 to 0. Even though there is currently no exact analytical result for the value of # # , Achlioptas [8] has bound it rigorously by #### ## # # ####. Small world graphs have recently been introduced in an attempt to capture the topological properties of real graphs. Regular lattices have long average distance between nodes, but show a high degree of clustering (i.e. if we remove a node from the ....

D. Achlioptas, "Threshold Phenomena in Random Graph Colouring and Satisfiability ", Ph.D. Thesis, 1999. Available at http://www.research.microsoft.com/#optas/.

....affect on the probability of satisfiability. Modifying this argument to the case where we have R(k Gamma 1) additional requirements uses the same ideas, but is is technically a little complicated. The proofs of Steps (1) and (2) are identical to the corresponding steps found in [3] 10] and [1]. We refer the reader to either of those for more details. 2 4 Generating Instances In practical terms, generating random CSP s from CSP (P) can sometimes be awkward. In this section, we will discuss how to do this fairly simply for some distributions P with transitions and partial transitions. ....

D. Achlioptas. Threshold Phenomena in Random Graph Colouring and Satisfiability. Ph.D. Thesis, Dept. of Computer Science, University of Toronto, 1999.

....In [FRI99] Friedgut introduced the concept of the sharp threshold for a graph property and established a very general sucient conditional for a threshold to be sharp. Friedgut also used his sucient condition to show the existence of a sharp threshold for the random SAT problem. In his PhD thesis [ACH99b], Achlioptas used Friedgut s sucient condition to show the existence of a sharp threshold for the k colorability graph property. Although the above results are interesting in their own right, they do not give explicit expressions for the threshold functions. Locating the exact thresholds for ....

....problem for graph colorability is much more dicult than those for connectivity and Hamiltonicity. We still do not know the exact value of d k except that d 2 = 1. In recent years, researchers have come up with many approaches to establish upper and lower bounds for d k . In his PhD thesis [ACH99b], Achlioptas proved that the threshold for the k colorability is sharp for k 3 and showed that there exists a function t k (n) such that for any 0, we have lim n 1 P rfG(n; t k (n) n ) is k colorableg = 1 and lim n 1 P rfG(n; t k (n) n ) is k colorableg = 0: The problem ....

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D.Achlioptas, Threshold Phenomena in Random Graph Colouring and Satisability, 1999.

....is (2(7=8) m=n ) n , almost certainly the number of satisfying assignments will be substantially below the expected value. The beautifully simple argument of Kirousis et al. 1998) demonstrates that this is not the only source of wastefulness in these calculations. See the thesis of Achlioptas (1999) for more details. What can be said of a lower bound on m as a function of t Here again, very little seems to be known. Assume that almost certainly, there are l disjoint clauses, C 1 ; C 2 ; C l in a random Majsat formula with parameters k; m; n; t. Let k = 3. There are 2 3l total ....

Achlioptas, D. 1999. Threshold phenomena in random graph colouring and satisfiability. Ph.D. Dissertation, University of Toronto.

....is (2(7=8) m=n ) n , almost certainly the number of satisfying assignments will be substantially below the expected value. The beautifully simple argument of Kirousis et al. 1998) demonstrates that this is not the only source of wastefulness in these calculations. See the thesis of Achlioptas (1999) for more details. What can be said of a lower bound on m as a function of t Here again, very little seems to be known. Assume that almost certainly, there are l disjoint clauses, C 1 ; C 2 ; C l in a random Majsat formula with parameters k; m; n; t. Let k = 3. There are 2 3l total ....

Achlioptas, D. 1999. Threshold phenomena in random graph colouring and satisfiability. Ph.D. Dissertation, University of Toronto.

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D. Achlioptas. Threshold phenomena in random graph colouring and satisfiability. PhD thesis, Department of Computer Science, University of Toronto, 1999.

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D. Achlioptas. Threshold Phenomena in Random Graph Colouring and Satis ability. Ph.D. Thesis, Dept. of Computer Science, University of Toronto, 1999.

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