Probabilistic Analysis of Algorithms

Rather than analyzing the worst case performance of algorithms, one can investigate their performance on typical instances of a given size. This is the approach we investigate in this paper. Of course, the first question we must answer is: what do we mean by a typical instance of a given size?

[1]  N. Biggs THE TRAVELING SALESMAN PROBLEM A Guided Tour of Combinatorial Optimization , 1986 .

[2]  Alan M. Frieze,et al.  When is the Assignment Bound Tight for the Asymmetric Traveling Salesman Problem? , 1992, IPCO.

[3]  Alan M. Frieze,et al.  Optimal construction of edge-disjoint paths in random regular graphs , 2000, SODA '99.

[4]  Benjamin Yakir,et al.  The Differencing Algorithm LDM for Partitioning: A Proof of a Conjecture of Karmarkar and Karp , 1996, Math. Oper. Res..

[5]  Bruce A. Reed,et al.  Mick gets some (the odds are on his side) (satisfiability) , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[6]  M. Sipser,et al.  Maximum matching in sparse random graphs , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[7]  Leslie G. Valiant,et al.  Universal schemes for parallel communication , 1981, STOC '81.

[8]  Alan M. Frieze,et al.  An efficient algorithm for the vertex-disjoint paths problem in random graphs , 1996, SODA '96.

[9]  B. Bollobás,et al.  An algorithm for finding hamilton paths and cycles in random graphs , 1987 .

[10]  Martin E. Dyer,et al.  On patching algorithms for random asymmetric travelling salesman problems , 1990, Math. Program..

[11]  Paul Erdös,et al.  On the chromatic index of almost all graphs , 1977, J. Comb. Theory, Ser. B.

[12]  Donald L. Miller,et al.  Exact Solution of Large Asymmetric Traveling Salesman Problems , 1991, Science.

[13]  M. R. Rao,et al.  Odd Minimum Cut-Sets and b-Matchings , 1982, Math. Oper. Res..

[14]  Alan M. Frieze,et al.  Analysis of Two Simple Heuristics on a Random Instance of k-SAT , 1996, J. Algorithms.

[15]  N. Megiddo,et al.  New results on the average behavior of simplex algorithms , 1984 .

[16]  Stephen Smale,et al.  On the average number of steps of the simplex method of linear programming , 1983, Math. Program..

[17]  Endre Szemerédi,et al.  Many hard examples for resolution , 1988, JACM.

[18]  Alan M. Frieze,et al.  Optimal construction of edge-disjoint paths in random graphs , 1994, SODA '94.

[19]  Hector J. Levesque,et al.  Hard and Easy Distributions of SAT Problems , 1992, AAAI.

[20]  Ming-Te Chao,et al.  Probabilistic analysis of a generalization of the unit-clause literal selection heuristics for the k satisfiability problem , 1990, Inf. Sci..

[21]  Edsger W. Dijkstra,et al.  A note on two problems in connexion with graphs , 1959, Numerische Mathematik.

[22]  Karl-Heinz Borgwardt,et al.  The Average number of pivot steps required by the Simplex-Method is polynomial , 1982, Z. Oper. Research.

[23]  Dorit S. Hochbaum An Exact Sublinear Algorithm for the Max-Flow, Vertex Disjoint Paths and Communication Problems on Random Graphs , 1992, Oper. Res..

[24]  Alan M. Frieze,et al.  Probabilistic Analysis of a Relaxation for the k-Median Problem , 1986, Math. Oper. Res..

[25]  Ming-Te Chao,et al.  Probabilistic Analysis of Two Heuristics for the 3-Satisfiability Problem , 1986, SIAM J. Comput..

[26]  B. Pittel,et al.  Maximum matchings in sparse random graphs: Karp-Sipser revisited , 1998 .

[27]  Noga Alon,et al.  A spectral technique for coloring random 3-colorable graphs (preliminary version) , 1994, STOC '94.

[28]  George S. Lueker,et al.  On the Average Difference between the Solutions to Linear and Integer Knapsack Problems , 1982 .

[29]  Andrew V. Goldberg,et al.  On finding the exact solution of a zero-one knapsack problem , 1984, STOC '84.

[30]  Wojciech Szpankowski,et al.  Combinatorial optimization problems for which almost every algorithm is asymptotically optimal , 1995 .

[31]  Alan M. Frieze,et al.  An algorithm for finding Hamilton cycles in random graphs , 1985, STOC '85.

[32]  Edward G. Coffman,et al.  Approximation algorithms for bin packing: a survey , 1996 .

[33]  Béla Bollobás,et al.  The chromatic number of random graphs , 1988, Comb..

[34]  László Babai,et al.  Canonical labelling of graphs in linear average time , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[35]  Sanjeev Arora,et al.  Polynomial time approximation schemes for Euclidean TSP and other geometric problems , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[36]  M. Held,et al.  A dynamic programming approach to sequencing problems , 1962, ACM National Meeting.

[37]  Vasek Chvátal,et al.  Hard Knapsack Problems , 1980, Oper. Res..

[38]  Eugene L. Lawler,et al.  The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization , 1985 .

[39]  Neil Robertson,et al.  Graph Minors .XIII. The Disjoint Paths Problem , 1995, J. Comb. Theory B.

[40]  Alexander I. Barvinok,et al.  Measure concentration in optimization , 1997, Math. Program..

[41]  Colin McDiarmid,et al.  Determining the Chromatic Number of a Graph , 1979, SIAM J. Comput..

[42]  Kenneth Schilling,et al.  On the growth of random knapsacks , 1990, Discret. Appl. Math..

[43]  Edward G. Coffman,et al.  Probabilistic analysis of packing and partitioning algorithms , 1991, Wiley-Interscience series in discrete mathematics and optimization.

[44]  Leonid A. Levin,et al.  Average Case Complete Problems , 1986, SIAM J. Comput..

[45]  Ravi B. Boppana,et al.  Eigenvalues and graph bisection: An average-case analysis , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[46]  Alan M. Frieze An Algorithm for Finding Hamilton Cycles in Random Directed Graphs , 1988, J. Algorithms.

[47]  Kurt Mehlhorn,et al.  On the all-pairs shortest-path algorithm of Moffat and Takaoka , 1997, Random Struct. Algorithms.

[48]  Martin E. Dyer,et al.  Probabilistic analysis of the generalised assignment problem , 1990, IPCO.

[49]  Andrew Thomason A simple linear expected time algorithm for finding a hamilton path , 1989, Discret. Math..

[50]  Alistair Moffat,et al.  An All Pairs Shortest Path Algorithm with Expected Time O(n² log n) , 1987, SIAM J. Comput..

[51]  Alan M. Frieze,et al.  On the satisfiability and maximum satisfiability of random 3-CNF formulas , 1993, SODA '93.

[52]  K. Borgwardt The Simplex Method: A Probabilistic Analysis , 1986 .

[53]  Martin E. Dyer,et al.  On linear programs with random costs , 1986, Math. Program..

[54]  Nimrod Megiddo,et al.  A simplex algorithm whose average number of steps is bounded between two quadratic functions of the smaller dimension , 1985, JACM.

[55]  Richard M. Karp,et al.  Probabilistic Analysis of Partitioning Algorithms for the Traveling-Salesman Problem in the Plane , 1977, Math. Oper. Res..

[56]  B. Pittel,et al.  The average performance of the greedy matching algorithm , 1993 .

[57]  K. Borgwardt The Simplex Method: A Probabilistic Analysis , 1986 .

[58]  Alistair Moffat,et al.  An all pairs shortest path algorithm with expected running time O(n 2logn) , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[59]  Ian Holyer,et al.  The NP-Completeness of Edge-Coloring , 1981, SIAM J. Comput..

[60]  Rajeev Motwani,et al.  Stable husbands , 1990, SODA '90.

[61]  Alan M. Frieze,et al.  On the Lagarias-Odlyzko Algorithm for the Subset Sum Problem , 1986, SIAM J. Comput..

[62]  Rainer E. Burkard,et al.  The asymptotic probabilistic behaviour of quadratic sum assignment problems , 1983, Z. Oper. Research.

[63]  Steve Smale,et al.  The Problem of the Average Speed of the Simplex Method , 1982, ISMP.

[64]  Alan M. Frieze,et al.  Average-Case Complexity of Shortest-Paths Problems in the Vertex-Potential Model , 1997, RANDOM.

[65]  Alan M. Frieze,et al.  The shortest-path problem for graphs with random arc-lengths , 1985, Discret. Appl. Math..

[66]  Saharon Shelah,et al.  Expected Computation Time for Hamiltonian Path Problem , 1987, SIAM J. Comput..

[67]  Mark Jerrum,et al.  Simulated annealing for graph bisection , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[68]  G. Grimmett,et al.  On colouring random graphs , 1975 .

[69]  Richard M. Karp,et al.  Maximum Matchings in Sparse Random Graphs , 1981, FOCS 1981.

[70]  Yumi K. Tsuji,et al.  EVIDENCE FOR A SATISFIABILITY THRESHOLD FOR RANDOM 3CNF FORMULAS , 1992 .

[71]  Vasek Chvátal,et al.  Determining the Stability Number of a Graph , 1976, SIAM J. Comput..

[72]  Richard M. Karp,et al.  A Patching Algorithm for the Nonsymmetric Traveling-Salesman Problem , 1979, SIAM J. Comput..

[73]  V. Klee,et al.  HOW GOOD IS THE SIMPLEX ALGORITHM , 1970 .

[74]  Philippe Flajolet,et al.  An introduction to the analysis of algorithms , 1995 .

[75]  Gottfried Tinhofer A probabilistic analysis of some greedy cardinality matching algorithms , 1984, Ann. Oper. Res..

[76]  B. Pittel On the probable behaviour of some algorithms for finding the stability number of a graph , 1982, Mathematical Proceedings of the Cambridge Philosophical Society.

[77]  Paul Erdös,et al.  Random Graph Isomorphism , 1980, SIAM J. Comput..

[78]  Alan M. Frieze,et al.  Coloring Bipartite Hypergraphs , 1996, IPCO.

[79]  Jan Karel Lenstra,et al.  Probabilistic analysis of combinatorial algorithms: an annotated bibliography , 1984 .

[80]  Alan M. Frieze,et al.  Edge-colouring random graphs , 1988, J. Comb. Theory, Ser. B.

[81]  Eli Upfal,et al.  A fast parallel construction of disjoint paths in networks , 1985 .

[82]  Martin E. Dyer,et al.  Probabilistic Analysis of the Multidimensional Knapsack Problem , 1989, Math. Oper. Res..

[83]  Andreas Goerdt,et al.  A Threshold for Unsatisfiability , 1992, MFCS.

[84]  Clifford Stein,et al.  Finding Real-Valued Single-Source Shortest Paths , 1996, IPCO.

[85]  Peter A. Bloniarz A Shortest-Path Algorithm with Expected Time O(n2 log n log* n) , 1983, SIAM J. Comput..

[86]  Clifford Stein,et al.  Finding Real-Valued Single-Source Shortest Paths in o(n3) Expected Time , 1998, J. Algorithms.

[87]  G. S. Lueker,et al.  Probabilistic analysis of optimum partitioning , 1986, Journal of Applied Probability.

[88]  Richard M. Karp,et al.  The Differencing Method of Set Partitioning , 1983 .

[89]  Evangelos Kranakis,et al.  Approximating the Unsatisfiability Threshold of Random Formulas (Extended Abstract) , 1996, ESA.

[90]  Alan M. Frieze,et al.  Algorithmic theory of random graphs , 1997, Random Struct. Algorithms.

[91]  J. Steele Probability theory and combinatorial optimization , 1987 .

[92]  Hilary Putnam,et al.  A Computing Procedure for Quantification Theory , 1960, JACM.

[93]  D. Welsh,et al.  A Spectral Technique for Coloring Random 3-Colorable Graphs , 1994 .

[94]  Jeffrey C. Lagarias,et al.  Solving low density subset sum problems , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[95]  Philip M. Spira,et al.  A New Algorithm for Finding all Shortest Paths in a Graph of Positive Arcs in Average Time 0(n2 log2n) , 1973, SIAM J. Comput..

[96]  Martin E. Dyer,et al.  The Solution of Some Random NP-Hard Problems in Polynomial Expected Time , 1989, J. Algorithms.

[97]  Charles E. Blair,et al.  Random linear programs with many variables and few constraints , 1986, Math. Program..

[98]  J. Beardwood,et al.  The shortest path through many points , 1959, Mathematical Proceedings of the Cambridge Philosophical Society.

[99]  Frank Thomson Leighton,et al.  Graph bisection algorithms with good average case behavior , 1984, Comb..

[100]  Alan M. Frieze,et al.  Maximum matchings in sparse random graphs: Karp-Sipser revisited , 1998, Random Struct. Algorithms.

[101]  Alan M. Frieze,et al.  Analysis of a simple greedy matching algorithm on random cubic graphs , 1995, SODA '93.

[102]  Greg N. Frederickson,et al.  Probabilistic Analysis for Simple One- and Two-Dimensional Bin Packing Algorithms , 1980, Inf. Process. Lett..

[103]  Paul G. Spirakis,et al.  Tail bounds for occupancy and the satisfiability threshold conjecture , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[104]  Mark Jerrum,et al.  Large Cliques Elude the Metropolis Process , 1992, Random Struct. Algorithms.

[105]  Yannis C. Stamatiou,et al.  Approximating the unsatisfiability threshold of random formulas , 1998, Random Struct. Algorithms.