Fast approximation algorithms for fractional packing and covering problems

Fast algorithms that find approximate solutions for a general class of problems, which are called fractional packing and covering problems, are presented. The only previously known algorithms for solving these problems are based on general linear programming techniques. The techniques developed greatly outperform the general methods in many applications, and are extensions of a method previously applied to find approximate solutions to multicommodity flow problems. The algorithms are based on a Lagrangian relaxation technique, and an important result is a theoretical analysis of the running time of a Lagrangian relaxation based algorithm. Several applications of the algorithms are presented.<<ETX>>

[1]  Kurt Eisemann The Trim Problem , 1957 .

[2]  R. Gomory,et al.  A Linear Programming Approach to the Cutting-Stock Problem , 1961 .

[3]  Ralph E. Gomory,et al.  A Linear Programming Approach to the Cutting Stock Problem---Part II , 1963 .

[4]  Richard M. Karp,et al.  The Traveling-Salesman Problem and Minimum Spanning Trees , 1970, Oper. Res..

[5]  Eugene L. Lawler Fast approximation algorithms for knapsack problems , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[6]  Eugene L. Lawler,et al.  On Preemptive Scheduling of Unrelated Parallel Processors by Linear Programming , 1978, JACM.

[7]  Richard M. Karp,et al.  An efficient approximation scheme for the one-dimensional bin-packing problem , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[8]  Martin Dyer An O(n) algorithm for the multiple-choice knapsack linear program , 1984, Math. Program..

[9]  Robert E. Tarjan,et al.  Fibonacci Heaps and Their Uses in Improved Network Optimization Algorithms , 1984, FOCS.

[10]  Pravin M. Vaidya,et al.  Fast algorithms for convex quadratic programming and multicommodity flows , 1986, STOC '86.

[11]  P. Raghavan Probabilistic construction of deterministic algorithms: Approximating packing integer programs , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

[12]  Prabhakar Raghavan,et al.  Randomized rounding: A technique for provably good algorithms and algorithmic proofs , 1985, Comb..

[13]  Jan Karel Lenstra,et al.  Approximation Algorithms for Scheduling Unrelated Parallel Machines , 1987, FOCS.

[14]  Jan Karel Lenstra,et al.  Approximation algorithms for scheduling unrelated parallel machines , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[15]  Frank Thomson Leighton,et al.  An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[16]  Pravin M. Vaidya,et al.  Speeding-up linear programming using fast matrix multiplication , 1989, 30th Annual Symposium on Foundations of Computer Science.

[17]  Farhad Shahrokhi,et al.  The maximum concurrent flow problem , 1990, JACM.

[18]  Fillia Makedon,et al.  Fast approximation algorithms for multicommodity flow problems , 1991, STOC '91.

[19]  David B. Shmoys,et al.  Improved approximation algorithms for shop scheduling problems , 1991, SODA '91.

[20]  S. L. van de Velde Machine scheduling and Lagrangian relaxation , 1991 .

[21]  James B. Orlin,et al.  A faster algorithm for finding the minimum cut in a graph , 1992, SODA '92.

[22]  Andrew V. Goldberg A Natural Randomization Strategy for Multicommodity Flow and Related Algorithms , 1992, Inf. Process. Lett..

[23]  David R. Karger,et al.  An Õ(n2) algorithm for minimum cuts , 1993, STOC '93.