Abstract
We study classes of Dynamic Programming (DP) algorithms which, due to their algebraic definitions, are closely related to coefficient extraction methods. DP algorithms can easily be modified to exploit sparseness in the DP table through memorization. Coefficient extraction techniques on the other hand are both space-efficient and parallelisable, but no tools have been available to exploit sparseness. We investigate the systematic use of homomorphic hash functions to combine the best of these methods and obtain improved space-efficient algorithms for problems including LINEAR SAT, SET PARTITION and SUBSET SUM. Our algorithms run in time proportional to the number of nonzero entries of the last segment of the DP table, which presents a strict improvement over sparse DP. The last property also gives an improved algorithm for CNF SAT and SET COVER with sparse projections.
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.
References
Alon, N., Gutin, G., Kim, E.J., Szeider, S., Yeo, A.: Solving MAX-r-SAT above a tight lower bound. Algorithmica 61(3), 638–655 (2011)
Björklund, A.: Determinant sums for undirected Hamiltonicity. In: FOCS, pp. 173–182. IEEE Computer Society (2010)
Björklund, A., Husfeldt, T., Koivisto, M.: Set partitioning via inclusion-exclusion. SIAM J. Comput. 39(2), 546–563 (2009)
Crowston, R., Gutin, G., Jones, M., Yeo, A.: Lower bound for Max-r-Lin2 and its applications in algorithmics and graph theory. CoRR, abs/1104.1135 (2011)
Eppstein, D., Galil, Z., Giancarlo, R., Italiano, G.F.: Sparse dynamic programming I: linear cost functions. J. ACM 39, 519–545 (1992)
Eppstein, D., Galil, Z., Giancarlo, R., Italiano, G.F.: Sparse dynamic programming II: convex and concave cost functions. J. ACM 39(3), 546–567 (1992)
Fomin, F.V., Kratsch, D.: Exact Exponential Algorithms, 1st edn. Springer-Verlag New York, Inc., New York (2010)
Gaspers, S., Szeider, S.: Strong backdoors to nested satisfiability. CoRR, abs/1202.4331 (2012)
Harnik, D., Naor, M.: On the compressibility of NP instances and cryptographic applications. SIAM Journal on Computing 39(5), 1667–1713 (2010)
Håstad, J.: Some optimal inapproximability results. J. ACM 48, 798–859 (2001)
Horowitz, E., Sahni, S.: Computing partitions with applications to the knapsack problem. J. ACM 21, 277–292 (1974)
Impagliazzo, R., Paturi, R.: On the complexity of k-SAT. J. Comput. Syst. Sci. 62(2), 367–375 (2001)
Koutis, I., Williams, R.: Limits and Applications of Group Algebras for Parameterized Problems. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part I. LNCS, vol. 5555, pp. 653–664. Springer, Heidelberg (2009)
Lipton, R.J.: Beating Bellman for the knapsack problem (2010), http://rjlipton.wordpress.com/2010/03/03/beating-bellman-for-the-knapsack-problem/ , http://rjlipton.wordpress.com
Lokshtanov, D., Nederlof, J.: Saving space by algebraization. In: Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, pp. 321–330. ACM, New York (2010)
Mansour, Y.: Randomized interpolation and approximation of sparse polynomials. SIAM J. Comput. 24(2), 357–368 (1995)
Nederlof, J.: Fast Polynomial-Space Algorithms Using Möbius Inversion: Improving on Steiner Tree and Related Problems. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part I. LNCS, vol. 5555, pp. 713–725. Springer, Heidelberg (2009)
Nishimura, N., Ragde, P., Szeider, S.: Solving #sat using vertex covers. Acta Inf. 44(7), 509–523 (2007)
Reed, B.A., Smith, K., Vetta, A.: Finding odd cycle transversals. Oper. Res. Lett. 32(4), 299–301 (2004)
Rosser, B.: Explicit bounds for some functions of prime numbers. American Journal of Mathematics 63(1), 211–232 (1941)
Schroeppel, R., Shamir, A.: A T = O(2n/2), S = O(2n/4) algorithm for certain NP-complete problems. SIAM J. Comput. 10(3), 456–464 (1981)
Solomon, L.: The burnside algebra of a finite group. Journal of Combinatorial Theory 2(4), 603–615 (1967)
Traxler, P.: Exponential Time Complexity of SAT and Related Problems. PhD thesis, ETH Zürich (2010)
Williams, R.: Finding paths of length k in \(\mathcal{O}^\star(2^{k})\) time. Inf. Process. Lett. 109(6), 315–318 (2009)
Williams, R., Gomes, C.P., Selman, B.: Backdoors to typical case complexity. In: IJCAI, pp. 1173–1178. Morgan Kaufmann (2003)
Woeginger, G.J.: Open problems around exact algorithms. Discrete Applied Mathematics 156(3), 397–405 (2008)
Zippel, R.: Probabilistic Algorithms for Sparse Polynomials. In: Ng, E.W. (ed.) EUROSAM 1979 and ISSAC 1979. LNCS, vol. 72, pp. 216–226. Springer, Heidelberg (1979)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kaski, P., Koivisto, M., Nederlof, J. (2012). Homomorphic Hashing for Sparse Coefficient Extraction. In: Thilikos, D.M., Woeginger, G.J. (eds) Parameterized and Exact Computation. IPEC 2012. Lecture Notes in Computer Science, vol 7535. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33293-7_15
Download citation
DOI: https://doi.org/10.1007/978-3-642-33293-7_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-33292-0
Online ISBN: 978-3-642-33293-7
eBook Packages: Computer ScienceComputer Science (R0)