Approximation Algorithms for Mixed Fractional Packing and Covering Problems

We study general mixed fractional packing and covering problems (MPCe ) of the following form: Given a vector $f: B \rightarrow {\rm IR}^{M}_{+}$ of M nonnegative continuous convex functions and a vector $g: B \rightarrow {\rm IR}^{M}_{+}$ of M nonnegative continuous concave functions, two M – dimensional nonnegative vectors a,b, a nonempty convex compact set B and a relative tolerance e ∈ (0,1), find an approximately feasible vector x ∈ B such that f(x) ≤ (1 + e) a and g(x) ≥ (1 – e) b or find a proof that no vector is feasible (that satisfies x ∈ B, f(x) ≤ a and g(x) ≥ b).

[1]  Jochen Könemann,et al.  Faster and simpler algorithms for multicommodity flow and other fractional packing problems , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[2]  Daniel Bienstock,et al.  Potential Function Methods for Approximately Solving Linear Programming Problems: Theory and Practice , 2002 .

[3]  Klaus Jansen,et al.  On Preemptive Resource Constrained Scheduling: Polynomial-Time Approximation Schemes , 2002, IPCO.

[4]  Michael D. Grigoriadis,et al.  Approximate Lagrangian Decomposition with a Modified Karmarkar Logarithmic Potential , 1997 .

[5]  Éva Tardos,et al.  Fast approximation algorithms for fractional packing and covering problems , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[6]  Klaus Jansen,et al.  Approximation Algorithms for General Packing Problems with Modified Logarithmic Potential Function , 2002, IFIP TCS.

[7]  Klaus Jansen.  Approximation Algorithm for the Mixed Fractional Packing and Covering Problem , 2006, SIAM J. Optim..

[8]  Leonid Khachiyan,et al.  Approximate Max-Min Resource Sharing for Structured Concave Optimization , 2000, SIAM J. Optim..

[9]  Claire Mathieu,et al.  Approximate strip packing , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[10]  Klaus Jansen,et al.  Approximation Algorithms for Mixed Fractional Packing and Covering Problems , 2004, IFIP TCS.

[11]  Leonid Khachiyan,et al.  Coordination Complexity of Parallel Price-Directive Decomposition , 1996, Math. Oper. Res..

[12]  Leonid Khachiyan,et al.  Fast Approximation Schemes for Convex Programs with Many Blocks and Coupling Constraints , 1994, SIAM J. Optim..

[13]  Klaus Jansen.  Approximation Algorithms for the General Max-min Resource Sharing Problem: Faster and Simpler , 2004, SWAT.

[14]  Klaus Jansen,et al.  On PreemptiveResource Constrained Scheduling: Polynomial-Time Approximation Schemes , 2006, SIAM J. Discret. Math..

[15]  Éva Tardos,et al.  Fast Approximation Algorithms for Fractional Packing and Covering Problems , 1995, Math. Oper. Res..

[16]  Neal E. Young,et al.  Randomized rounding without solving the linear program , 1995, SODA '95.

[17]  Neal E. Young,et al.  Sequential and parallel algorithms for mixed packing and covering , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[18]  Jochen Könemann,et al.  Fast Combinatorial Algorithms for Packing and Covering Problems , 2000 .

[19]  Lisa Fleischer,et al.  A fast approximation scheme for fractional covering problems with variable upper bounds , 2004, SODA '04.

[20]  Sudipto Guha,et al.  Approximating a finite metric by a small number of tree metrics , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[21]  Hervé Rivano,et al.  Fractional Path Coloring with Applications to WDM Networks , 2001, ICALP.