Priority algorithms for the subset-sum problem

Greedy algorithms are simple, but their relative power is not well understood. The priority framework [5] captures a key notion of "greediness" in the sense that it processes (in some locally optimal manner) one data item at a time, depending on and only on the current knowledge of the input. This algorithmic model provides a tool to assess the computational power and limitations of greedy algorithms, especially in terms of their approximability. In this paper, we study priority algorithm approximation ratios for the Subset-Sum Problem, focusing on the power of revocable decisions. We first provide a tight bound of α ≈ 0.657 for irrevocable priority algorithms. We then show that the approximation ratio of fixed order revocable priority algorithms is between β; ≈ 0.780 and γ ≈ 0.852, and the ratio of adaptive order revocable priority algorithms is between 0.8 and δ ≈ 0.893.

[1]  J. M. Moore.  An n Job, One Machine Sequencing Algorithm for Minimizing the Number of Late Jobs , 1968 .

[2]  Deeparnab Chakrabarty,et al.  Knapsack Problems , 2008 .

[3]  Hans Kellerer,et al.  An efficient fully polynomial approximation scheme for the Subset-Sum Problem , 2003, J. Comput. Syst. Sci..

[4]  K. Arrow.  Social Choice and Individual Values , 1951 .

[5]  Allan Borodin,et al.  Priority Algorithms for Graph Optimization Problems , 2004, WAOA.

[6]  Eugene Levner,et al.  A Fast Approximation Algorithm For The Subset-Sum Problem , 1994 .

[7]  Rafail Ostrovsky,et al.  Approximation Algorithms for the Job Interval Selection Problem and Related Scheduling Problems , 2006, Math. Oper. Res..

[8]  Christelle Guéret,et al.  A new lower bound for the open‐shop problem , 1999, Ann. Oper. Res..

[9]  Allan Borodin,et al.  Priority algorithms for the subset-sum problem , 2008, J. Comb. Optim..

[10]  David Pisinger.  An exact algorithm for large multiple knapsack problems , 1999, Eur. J. Oper. Res..

[11]  Laureano F. Escudero,et al.  Coefficient reduction for knapsack-like constraints in 0-1 programs with variable upper bounds , 1990 .

[12]  Amos Fiat,et al.  On-line routing of virtual circuits with applications to load balancing and machine scheduling , 1997, JACM.

[13]  Sudipto Guha,et al.  Approximating the throughput of multiple machines under real-time scheduling , 1999, STOC '99.

[14]  Paolo Toth,et al.  A Framework for Tightening 0-1 Programs Based on Extensions of Pure 0-1 KP and SS Problems , 1995, IPCO.

[15]  Alberto Caprara,et al.  Worst-case analysis of the subset sum algorithm for bin packing , 2004, Oper. Res. Lett..

[16]  K. Fine,et al.  Social Choice and Individual Rankings II , 1974 .

[17]  Allan Borodin,et al.  Toward a Model for Backtracking and Dynamic Programming , 2005, Computational Complexity Conference.

[18]  Russell Impagliazzo,et al.  Models of Greedy Algorithms for Graph Problems , 2004, SODA '04.

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

[20]  Rafail Ostrovsky,et al.  Approximation algorithms for the job interval selection problem and related scheduling problems , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[21]  Kazuo Iwama,et al.  Removable Online Knapsack Problems , 2002, ICALP.

[22]  Eugene Levner,et al.  Fast approximation algorithms for knapsack type problems , 1980 .

[23]  Allan Borodin,et al.  On the Power of Priority Algorithms for Facility Location and Set Cover , 2002, APPROX.

[24]  Rudolf Fleischer,et al.  On‐line scheduling revisited , 2000 .

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

[26]  Han Hoogeveen,et al.  New Lower and Upper Bounds for Scheduling Around a Small Common Due Date , 1994, Oper. Res..

[27]  Oded Regev.  Priority algorithms for makespan minimization in the subset model , 2002, Inf. Process. Lett..

[28]  Philippe Baptiste,et al.  Polynomial time algorithms for minimizing the weighted number of late jobs on a single machine with equal processing times , 1999 .

[29]  Paolo Toth,et al.  Worst-case analysis of greedy algorithms for the subset-sum problem , 1984, Math. Program..

[30]  Marek Chrobak,et al.  A Note on Scheduling Equal-Length Jobs to Maximize Throughput , 2006, J. Sched..

[31]  Paolo Toth,et al.  Knapsack Problems: Algorithms and Computer Implementations , 1990 .

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

[33]  Allan Borodin,et al.  (Incremental) Priority Algorithms , 2002, SODA '02.

[34]  Sudipto Guha,et al.  Approximating the Throughput of Multiple Machines in Real-Time Scheduling , 2002, SIAM J. Comput..

[35]  Frits C. R. Spieksma,et al.  Interval selection: applications, algorithms and lower bounds , 2002 .