Abstract
Sums-of-products provide a basis for describing certain computational problems, particularly problems related to constraint satisfaction including SAT, MAX SAT, and #SAT. They also can be used to describe many problems arising from graph theory. By modeling a problem as a sum-of-products problem, the concept of “subproblem independence” takes on a clear meaning. Subproblem independence has immediate computational implications since it can be used to create programs with reduced levels of nesting and programs which exploit memoization. The concept of subproblem independence also extends to quantified sums.
Subproblem independence can be linked directly to structural concepts associated with tree decompositions for graphs and the closely related structure trees for algebraic problems. Thus methods of finding tree decompositions apply directly to finding independent subproblems.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
A. Arnborg and A. Proskurowski. Linear time algorithms for np-hard problems on graphs embedded in k-trees. Technical Report TRITA-NA-8404, Department of Numerical Analysis and Computer Science, Royal Institute of Technology, Stockholm, Sweden, 1984.
A. Arnborg, D. G. Corneil, and A. Proskurowski. Complexity of finding embeddings in a k-tree. SIAM J. Alg. and Discr. Methods, 8:277–284, 1987.
U. Bertele and F. Brioschi. Nonserial Dynamic Programming. Academic, New York, 1972.
H. L. Bodlaender. A linear-time algorithm for finding tree-compositions of small treewidth. SIAM Journal on Computing, 25:1305–1317, 1996.
M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco, 1979.
R. M. Karp. Reducibility among combinatorial problems. In R. E. Miller and J. W. Thatcher, editors, Complexity of Computer Computations, pages 85–103. Plenum, New York, 1972.
R. L. Lipton and R. E. Tarjan. Applications of a planar separator theorem. SIAM J. Comput., 9:615–629, 1980.
R. L. Lipton, D. J. Rose, and R. E. Tarjan. Generalized nested dissection. SIAM J. Numer. Analysis, 16:346–358, 1979.
D. Michie. ‘Memo’ functions and machine learning. Nature, 218:19–22, 1968.
C. H. Papadimitriou. Games against Nature. J. Comput. System Sci., 31:288–301, 1985.
A. Rosenthal. Dynamic programming is optimal for non-serial optimization problems. SIAM J. Comput., 11:47–59, 1982.
R. E. Stearns and H. B. Hunt III. Power indices and easier NP-complete problems. Mathematical Systems Theory, 23:209–225, 1990.
R. E. Stearns and H. B. Hunt III. An algebraic model for combinatorial problems. SIAM J. Comput., 25(2):448–476, 1996.
R. E. Stearns and H. B. Hunt III. Exploiting structure in quantified formulas. J. Algorithms, 43:220–263, 2002.
R. E. Stearns and H. B. Hunt III. Resource bounds and subproblem independence. Theory Comput. Systems, 38:731–761, 2005.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer Science + Business Media B.V.
About this chapter
Cite this chapter
Stearns, R.E., Hunt, H.B. (2009). Sums-of-Products and Subproblem Independence. In: Ravi, S.S., Shukla, S.K. (eds) Fundamental Problems in Computing. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-9688-4_11
Download citation
DOI: https://doi.org/10.1007/978-1-4020-9688-4_11
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-9687-7
Online ISBN: 978-1-4020-9688-4
eBook Packages: Computer ScienceComputer Science (R0)