Toward a Model for Backtracking and Dynamic Programming

We consider a model (BT) for backtracking algorithms. Our model generalizes both the priority model of Borodin, Nielson and Rackoff, as well as a simple dynamic programming model due to Woeginger, and hence spans a wide spectrum of algorithms. After witnessing the strength of the model, we then show its limitations by providing lower bounds for algorithms in this model for several classical problems such as interval scheduling, knapsack and satisfiability.

[1]  P. Helman,et al.  A Comprehensive Model of Dynamic Programming , 1985 .

[2]  M. Held,et al.  Finite-State Processes and Dynamic Programming , 1967 .

[3]  Gerhard J. Woeginger,et al.  When Does a Dynamic Programming Formulation Guarantee the Existence of a Fully Polynomial Time Approximation Scheme (FPTAS)? , 2000, INFORMS J. Comput..

[4]  Rajeev Motwani,et al.  On syntactic versus computational views of approximability , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[5]  Jan Vondrák,et al.  Approximating the stochastic knapsack problem: the benefit of adaptivity , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

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

[7]  Allan Borodin,et al.  The Power of Priority Algorithms for Facility Location and Set Cover , 2004, Algorithmica.

[8]  Donald W. Loveland,et al.  A machine program for theorem-proving , 2011, CACM.

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

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

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

[12]  Magnús M. Halldórsson,et al.  Online independent sets , 2002, Theor. Comput. Sci..

[13]  Claude E. Shannon,et al.  The synthesis of two-terminal switching circuits , 1949, Bell Syst. Tech. J..

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

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

[16]  Vasek Chvátal,et al.  Hard Knapsack Problems , 1980, Oper. Res..

[17]  Ronald L. Rivest,et al.  Introduction to Algorithms, Second Edition , 2001 .

[18]  Clifford Stein,et al.  Introduction to Algorithms, 2nd edition. , 2001 .

[19]  Allan Borodin,et al.  How Well Can Primal-Dual and Local-Ratio Algorithms Perform? , 2005, ICALP.

[20]  Johan Håstad,et al.  Some optimal inapproximability results , 2001, JACM.

[21]  Alexander A. Razborov,et al.  Natural Proofs , 1997, J. Comput. Syst. Sci..

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

[23]  Rajeev Motwani,et al.  Randomized algorithms , 1996, CSUR.

[24]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .

[25]  David G. Mitchell,et al.  Finding hard instances of the satisfiability problem: A survey , 1996, Satisfiability Problem: Theory and Applications.

[26]  Béla Bollobás,et al.  Proving Integrality Gaps without Knowing the Linear Program , 2006, Theory Comput..

[27]  Jun Gu,et al.  Algorithms for the satisfiability (SAT) problem: A survey , 1996, Satisfiability Problem: Theory and Applications.

[28]  Hilary Putnam,et al.  A Computing Procedure for Quantification Theory , 1960, JACM.

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

[30]  Frits C. R. Spieksma,et al.  Interval selection: Applications, algorithms, and lower bounds , 2003, J. Algorithms.

[31]  Vijay V. Vazirani,et al.  Approximation Algorithms , 2001, Springer Berlin Heidelberg.

[32]  Michael Alekhnovich,et al.  Lower bounds for polynomial calculus: non-binomial case , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[33]  Lefteris M. Kirousis,et al.  Selecting Complementary Pairs of Literals , 2003, Electron. Notes Discret. Math..

[34]  Paul Helman,et al.  A common schema for dynamic programming and branch and bound algorithms , 1989, JACM.

[35]  Esther M. Arkin,et al.  Scheduling jobs with fixed start and end times , 1987, Discret. Appl. Math..

[36]  Michael Alekhnovich,et al.  Exponential Lower Bounds for the Running Time of DPLL Algorithms on Satisfiable Formulas , 2004, SODA '04.

[37]  Sanjeev Arora.  Proving Integrality Gaps without Knowing the Linear Program , 2003, FCT.

[38]  Dimitris Achlioptas,et al.  Optimal myopic algorithms for random 3-SAT , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[39]  Donald E. Knuth,et al.  Optimum binary search trees , 1971, Acta Informatica.