Using dual approximation algorithms for scheduling problems: Theoretical and practical results

The problem of scheduling a set of n jobs on m identical machines so as to minimize the makespan time is perhaps the most well-studied problem in the theory of approximation algorithms for NP-hard optimization problems. In this paper we present the strongest possible type of result for this problem, a polynomial approximation scheme. More precisely, for each ε, we give an algorithm that runs in time O((n/ε)1/ε2) and has relative error at most ε. For algorithms that are polynomial in n and m, the strongest previously-known result was that the MULTIFIT algorithm delivers a solution with no worse than 20% relative error. In addition, we present a refinement of our scheme in the case where the performance guarantee is equal to that of MUL-TIFIT, that yields an algorithm that is both more efficient and easier to analyze than MULTIFIT. In this case, in order to guarantee a maximum relative error of 1/5+2-k, the algorithm runs in O(n(k+logn)) time. The scheme is based on a new approach to constructing approximation algorithms, which we call dual approximation algorithms, where the aim is find superoptimal, but infeasible solutions, and the performance is measured by the degree of infeasibility allowed. This notion should find wide applicability in its own right, and should be considered for any optimization problem where traditional approximation algorithms have been particularly elusive.

[1]  Ronald L. Graham,et al.  Bounds on Multiprocessing Timing Anomalies , 1969, SIAM Journal of Applied Mathematics.

[2]  Donald K. Friesen,et al.  Tighter Bounds for the Multifit Processor Scheduling Algorithm , 1984, SIAM J. Comput..

[3]  Michael Allen Langston,et al.  Processor scheduling with improved heuristic algorithms , 1981 .

[4]  David B. Shmoys,et al.  A packing problem you can almost solve by sitting on your suitcase , 1986 .

[5]  G. S. Lueker,et al.  Bin packing can be solved within 1 + ε in linear time , 1981 .

[6]  D. K. Friesen,et al.  Sensitivity Analysis for Heuristic Algorithms , 1978 .

[7]  Eugene L. Lawler,et al.  Parameterized Approximation Scheme for the Multiple Knapsack Problem , 2009, SIAM J. Comput..

[8]  Oscar H. Ibarra,et al.  Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems , 1975, JACM.

[9]  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).

[10]  Edward G. Coffman,et al.  An Application of Bin-Packing to Multiprocessor Scheduling , 1978, SIAM J. Comput..

[11]  Journal of the Association for Computing Machinery , 1961, Nature.

[12]  Sartaj Sahni,et al.  Algorithms for Scheduling Independent Tasks , 1976, J. ACM.

[13]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[14]  Ronald L. Graham,et al.  Bounds for certain multiprocessing anomalies , 1966 .