Abstract
The valued constraint satisfaction problem (VCSP) is an optimization framework originating from artificial intelligence which generalizes the classical constraint satisfaction problem (CSP). In this paper, we are interested in structural properties that can make problems from the VCSP framework, as well as other CSP variants, solvable to optimality in polynomial time. So far, the largest structural class that is known to be polynomial-time solvable to optimality is the class of bounded hypertree width instances introduced by Gottlob et al. Here, larger classes of tractable instances are singled out by using dynamic programming and structural decompositions based on a hypergraph invariant proposed by Grohe and Marx. In the second part of the paper, we take a different view on our optimization problems; instead of considering fixed arbitrary values for some structural invariant of the (hyper)graph structure of the constraints, we consider the problems parameterized by the tree-width of primal, dual, and incidence graphs, combined with several other basic parameters such as domain size and arity. Such parameterizations of plain CSPs have been studied by Samer and Szeider. Here, we extend their framework to encompass our optimization problems, by coupling it with further non-trivial machinery and new reductions. By doing so, we are able to determine numerous combinations of the considered parameters that make our optimization problems admit fixed-parameter algorithms.
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References
Atserias, A., Grohe, M., Marx, D.: Size bounds and query plans for relational joins. In: 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2008), pp. 739–748 (2008)
Beeri, C., Fagin, R., Maier, D., Yannakakis, M.: On the desirability of acyclic database schemes. Journal of the ACM 30, 479–513 (1983)
Chen, H., Grohe, M.: Constraint satisfaction with succinctly specified relations. Journal of Computer and System Sciences 76(8), 847–860 (2010)
Chung, F.R., Graham, R.L., Frankl, P., Shearer, J.B.: Some intersection theorems for ordered sets and graphs. Journal of Combinatorial Theory Series A 43, 23–37 (1986)
Cohen, D., Jeavons, P., Gyssens, M.: A unified theory of structural tractability for constraint satisfaction problems. Journal of Computer and System Sciences 74(5), 721–743 (2008)
Dechter, R.: Constraint Processing. Morgan Kaufmann (2003)
Dechter, R., Pearl, J.: Tree clustering for constraint networks (research note). Artificial Intelligence 38, 353–366 (1989)
Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness I: Basic results. SIAM Journal on Computing 24(4), 873–921 (1995)
Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness II: On completeness for W[1]. Theoretical Computer Science 141(1-2), 109–131 (1995)
Fagin, R.: Degrees of acyclicity for hypergraphs and relational database schemes. Journal of the ACM 30, 514–550 (1983)
Feder, T., Vardi, M.Y.: The computational structure of monotone monadic SNP and constraint satisfaction: a study through datalog and group theory. SIAM Journal on Computing 28(1), 57–104 (1998)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer (2006)
Freuder, E.C., Wallace, R.J.: Partial constraint satisfaction. Artificial Intelligence 58(1-3), 21–70 (1992)
Freuder, E.C.: A sufficient condition for backtrack-bounded search. Journal of the ACM 32, 755–761 (1985)
Färnqvist, T., Jonsson, P.: Bounded Tree-Width and CSP-Related Problems. In: Tokuyama, T. (ed.) ISAAC 2007. LNCS, vol. 4835, pp. 632–643. Springer, Heidelberg (2007)
de Givry, S., Schiex, T., Verfaillie, G.: Exploiting tree decompositions and soft local consistency in weighted CSP. In: Proceedings of the 21st National Conference on Artificial Intelligence (AAAI 2006), pp. 22–27 (2006)
Gottlob, G., Greco, G.: On the complexity of combinatorial auctions: structured item graphs and hypertree decomposition. In: Proceedings of the 8th ACM Conference on Electronic Commerce (EC 2007), pp. 152–161 (2007)
Gottlob, G., Greco, G., Scarcello, F.: Tractable Optimization Problems through Hypergraph-Based Structural Restrictions. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part II. LNCS, vol. 5556, pp. 16–30. Springer, Heidelberg (2009)
Gottlob, G., Leone, N., Scarcello, F.: Hypertree decompositions and tractable queries. Journal of Computer and System Sciences 64(3), 579–627 (2002)
Grohe, M.: The complexity of homomorphism and constraint satisfaction problems seen from the other side. Journal of the ACM 54(1), 1–24 (2007)
Grohe, M., Marx, D.: Constraint solving via fractional edge covers. In: Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2006), pp. 289–298 (2006)
Gutin, G., Rafiey, A., Yeo, A., Tso, M.: Level of repair analysis and minimum cost homomorphisms of graphs. Discrete Applied Mathematics 154(6), 881–889 (2006)
Kroon, L.G., Sen, A., Roy, H.D.: The Optimal Cost Chromatic Partition Problem for Trees and Interval Graphs. In: D’Amore, F., Marchetti-Spaccamela, A., Franciosa, P.G. (eds.) WG 1996. LNCS, vol. 1197, pp. 279–292. Springer, Heidelberg (1997)
Marx, D.: Approximating fractional hypertree width. ACM Transactions on Algorithms 6, 1–17 (2010)
Marx, D.: Tractable hypergraph properties for constraint satisfaction and conjunctive queries. In: Proceedings of the 42nd ACM Symposium on Theory of Computing (STOC 2010), pp. 735–744 (2010)
Marx, D.: Tractable structures for constraint satisfaction with truth tables. Theory of Computing Systems 48, 444–464 (2011)
Ndiaye, S., Jégou, P., Terrioux, C.: Extending to soft and preference constraints a framework for solving efficiently structured problems. In: Proceedings of the 2008 20th IEEE International Conference on Tools with Artificial Intelligence (ICTAI 2008), pp. 299–306 (2008)
Reed, B.A.: Finding approximate separators and computing tree width quickly. In: Proceedings of the Twenty-Fourth Annual ACM Symposium on Theory of Computing (STOC 1992), pp. 221–228 (1992)
Samer, M., Szeider, S.: Constraint satisfaction with bounded treewidth revisited. Journal of Computer and System Sciences 76, 103–114 (2010)
Schiex, T., Fargier, H., Verfaillie, G.: Valued constraint satisfaction problems: hard and easy problems. In: Proceedings of the 14th International Joint Conference on Artificial Intelligence (IJCAI 1995), pp. 631–637 (1995)
Takhanov, R.: A dichotomy theorem for the general minimum cost homomorphism problem. In: Proceedings of the 27th International Symposium on Theoretical Aspects of Computer Science (STACS 2010), pp. 657–668 (2010)
Terrioux, C., Jégou, P.: Bounded Backtracking for the Valued Constraint Satisfaction Problems. In: Rossi, F. (ed.) CP 2003. LNCS, vol. 2833, pp. 709–723. Springer, Heidelberg (2003)
Yannakakis, M.: Algorithms for acyclic database schemes. In: Proceedings of the Seventh International Conference on Very Large Data Bases, VLDB 1981, vol. 7, pp. 82–94 (1981)
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Färnqvist, T. (2012). Constraint Optimization Problems and Bounded Tree-Width Revisited. In: Beldiceanu, N., Jussien, N., Pinson, É. (eds) Integration of AI and OR Techniques in Contraint Programming for Combinatorial Optimzation Problems. CPAIOR 2012. Lecture Notes in Computer Science, vol 7298. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29828-8_11
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