Abstract
Relational algebra has been shown to be a powerful tool for solving a wide range of combinatorial optimization problems with small computational and programming effort. The problems considered in recent years are single- objective ones where one single objective function has to be optimized. With this paper we start considerations on the use of relational algebra for multi-objective problems. In contrast to single-objective optimization multiple objective functions have to be optimized at the same time usually resulting in a set of different trade-offs with respect to the different functions. On the one hand, we examine how to solve the mentioned problem exactly by using relational algebraic programs. On the other hand, we address the problem of objective reduction that has recently been shown to be NP-hard. We propose an exact algorithm for this problem based on relational algebra. Our experimental results show that this algorithm drastically outperforms the currently best one.
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
Beier, R., Vöcking, B.: Random knapsack in expected polynomial time. J. Comput. Syst. Sci. 69(3), 306–329 (2004)
Berghammer, R.: Solving algorithmic problems on orders and lattices by relation algebra and RelView. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2006. LNCS, vol. 4194, pp. 49–63. Springer, Heidelberg (2006)
Berghammer, R., Leoniuk, B., Milanese, U.: Implementation of relational algebra using binary decision diagrams. In: de Swart, H. (ed.) RelMiCS 2001. LNCS, vol. 2561, pp. 241–257. Springer, Heidelberg (2002)
Berghammer, R., Milanese, U.: Relational approach to boolean logic problems. In: MacCaull, W., Winter, M., Düntsch, I. (eds.) RelMiCS 2005. LNCS, vol. 3929, pp. 48–59. Springer, Heidelberg (2006)
Berghammer, R., Neumann, F.: RelView – an OBDD-based computer algebra system for relations. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2005. LNCS, vol. 3718, pp. 40–51. Springer, Heidelberg (2005)
Berghammer, R., Rusinowska, A., de Swart, H.C.M.: Applying relational algebra and RelView to coalition formation. European Journal of Operational Research 178(2), 530–542 (2007)
Beyer, D., Noack, A., Lewerentz, C.: Efficient relational calculation for software analysis. IEEE Transactions on Software Engineering 31(2), 137–149 (2005)
Brockhoff, D., Zitzler, E.: Dimensionality reduction in multiobjective optimization: The minimum objective subset problem. In: Waldmann, K.H., Stocker, U.M. (eds.) Operations Research Proceedings 2006, pp. 423–430. Springer, Heidelberg (2007)
Ehrgott, M.: Multicriteria Optimization, 2nd edn. Springer, Berlin (2005)
Goemans, M.X.: Minimum bounded degree spanning trees. In: Proc. of FOCS 2006, pp. 273–282. IEEE Computer Society Press, Los Alamitos (2006)
Kehden, B.: Evaluating sets of search points using relational algebra. In: Schmidt, R.A. (ed.) RelMiCS/AKA 2006. LNCS, vol. 4136, pp. 266–280. Springer, Heidelberg (2006)
Kellerer, H., Pferschy, U., Pisinger, D.: Knapsack Problems. Springer, Heidelberg (2004)
Könemann, J., Ravi, R.: Primal-dual meets local search: Approximating MSTs with nonuniform degree bounds. SIAM J. Comput. 34(3), 763–773 (2005)
Martello, S., Toth, P.: Knapsack Problems: Algorithms and Computer Implementations. Wiley, Chichester (1990)
Nemhauser, G., Ullmann, Z.: Discrete dynamic programming and capital allocation. Management Sci. 15(9), 494–505 (1969)
Ravi, R., Marathe, M.V., Ravi, S.S., Rosenkrantz, D.J., Hunt III, H.B.: Many birds with one stone: multi-objective approximation algorithms. In: Proc. of STOC 1993, pp. 438–447 (1993)
Schmidt, G., Ströhlein, T.: Relations and Graphs – Discrete Mathematics for Computer Scientists. Springer, Heidelberg (1993)
Singh, M., Lau, L.C.: Approximating minimum bounded degree spanning trees to within one of optimal. In: Proc. of STOC 2007, pp. 661–670. ACM Press, New York (2007)
Zitzler, E., Laumanns, M., Thiele, L.: SPEA2: Improving the Strength Pareto Evolutionary Algorithm for Multiobjective Optimization. In: Giannakoglou, K.C., et al. (eds.) Proc. of EUROGEN 2001, pp. 95–100. CIMNE (2002)
Zitzler, E., Thiele, L.: Multiobjective Evolutionary Algorithms: A Comparative Case Study and the Strength Pareto Approach. IEEE Transactions on Evolutionary Computation 3(4), 257–271 (1999)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Diedrich, F., Kehden, B., Neumann, F. (2008). Multi-objective Problems in Terms of Relational Algebra. In: Berghammer, R., Möller, B., Struth, G. (eds) Relations and Kleene Algebra in Computer Science. RelMiCS 2008. Lecture Notes in Computer Science, vol 4988. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78913-0_8
Download citation
DOI: https://doi.org/10.1007/978-3-540-78913-0_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-78912-3
Online ISBN: 978-3-540-78913-0
eBook Packages: Computer ScienceComputer Science (R0)