On the Sum-of-Squares algorithm for bin packing

In this article we present a theoretical analysis of the online <i>Sum-of-Squares</i> algorithm (<i>SS</i>) for bin packing along with several new variants. <i>SS</i> is applicable to any instance of bin packing in which the bin capacity <i>B</i> and item sizes <i>s</i>(<i>a</i>) are integral (or can be scaled to be so), and runs in time <i>O</i>(<i>nB</i>). It performs remarkably well from an average case point of view: For any discrete distribution in which the optimal expected waste is sublinear, <i>SS</i> also has sublinear expected waste. For any discrete distribution where the optimal expected waste is bounded, <i>SS</i> has expected waste at most <i>O</i>(log <i>n</i>). We also discuss several interesting variants on <i>SS</i>, including a randomized <i>O</i>(<i>nB</i> log <i>B</i>)-time online algorithm <i>SS</i>* whose expected behavior is essentially optimal for all discrete distributions. Algorithm <i>SS</i>* depends on a new linear-programming-based pseudopolynomial-time algorithm for solving the NP-hard problem of determining, given a discrete distribution <i>F</i>, just what is the growth rate for the optimal expected waste.

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

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

[3]  W. Hoeffding.  Probability Inequalities for sums of Bounded Random Variables , 1963 .

[4]  Jeffrey D. Ullman,et al.  Worst-Case Performance Bounds for Simple One-Dimensional Packing Algorithms , 1974, SIAM J. Comput..

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

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

[7]  Martin Grötschel,et al.  The ellipsoid method and its consequences in combinatorial optimization , 1981, Comb..

[8]  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).

[9]  Richard M. Karp,et al.  On Linear Characterizations of Combinatorial Optimization Problems , 1982, SIAM J. Comput..

[10]  B. Hajek.  Hitting-time and occupation-time bounds implied by drift analysis with applications , 1982, Advances in Applied Probability.

[11]  Joseph Y.-T. Leung,et al.  On a Dual Version of the One-Dimensional Bin Packing Problem , 1984, J. Algorithms.

[12]  Wansoo T. Rhee.  Optimal Bin Packing with Items of Random Sizes , 1988, Math. Oper. Res..

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

[14]  R. Weber,et al.  STABILITY OF ON-LINE BIN PACKING WITH RANDOM ARRIVALS AND LONG-RUN-AVERAGE CONSTRAINTS , 1990 .

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

[16]  Mihalis Yannakakis,et al.  Fundamental discrepancies between average-case analyses under discrete and continuous distributions: a bin packing case study , 1991, STOC '91.

[17]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[18]  André van Vliet,et al.  An Improved Lower Bound for On-Line Bin Packing Algorithms , 1992, Inf. Process. Lett..

[19]  Wansoo T. Rhee,et al.  On-Line Bin Packing of Items of Random Sizes, II , 1993, SIAM J. Comput..

[20]  Edward G. Coffman,et al.  Markov chains, computer proofs, and average-case analysis of best fit bin packing , 1993, STOC '93.

[21]  Wansoo T. Rhee,et al.  On Line Bin Packing with Items of Random Size , 1993, Math. Oper. Res..

[22]  Yuval Rabani,et al.  Biased random walks, Lyapunov functions, and stochastic analysis of best fit bin packing , 1996, SODA '96.

[23]  E. G. Coffman,et al.  Bin packing with discrete item sizes, part II: Tight bounds on First Fit , 1997, Random Struct. Algorithms.

[24]  Susanne Albers,et al.  Average-case analyses of first fit and random fit bin packing , 2000, SODA '98.

[25]  Claire Mathieu,et al.  A Self Organizing Bin Packing Heuristic , 1999, ALENEX.

[26]  José M. Valério de Carvalho,et al.  Exact solution of bin-packing problems using column generation and branch-and-bound , 1999, Ann. Oper. Res..

[27]  Claire Mathieu,et al.  Linear waste of best fit bin packing on skewed distributions , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[28]  On the sum-of-squares algorithm for bin packing , 2000, STOC '00.

[29]  Mihalis Yannakakis,et al.  Bin Packing with Discrete Item Sizes, Part I: Perfect Packing Theorems and the Average Case Behavior of Optimal Packings , 2000, SIAM J. Discret. Math..

[30]  Claire Mathieu,et al.  Better approximation algorithms for bin covering , 2001, SODA '01.

[31]  Claire Mathieu,et al.  Linear waste of best fit bin packing on skewed distributions , 2002, Random Struct. Algorithms.

[32]  Mihalis Yannakakis,et al.  Perfect Packing Theorems and the Average-Case Behavior of Optimal and Online Bin Packing , 2002, SIAM Rev..

[33]  David S. Johnson,et al.  The Cutting-Stock Approach to Bin Packing: Theory and Experiments , 2003, ALENEX.

[34]  Claire Mathieu,et al.  On the Worst-case Performance of the Sum-of-Squares Algorithm for Bin Packing , 2005, ArXiv.

[35]  Dan Suciu,et al.  Journal of the ACM , 2006 .