ESA 2014: Algorithms - ESA 2014 pp 443-454

# Representative Sets of Product Families

• Fedor V. Fomin
• Daniel Lokshtanov
• Saket Saurabh
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8737)

## Abstract

A subfamily $${\cal F}'$$ of a set family $${\cal F}$$ is said to q-represent $${\cal F}$$ if for every $$A \in{\cal F}$$ and B of size q such that A ∩ B = ∅ there exists a set $$A' \in{\cal F}'$$ such that A′ ∩ B = ∅. In a recent paper [SODA 2014] three of the authors gave an algorithm that given as input a family $${\cal F}$$ of sets of size p together with an integer q, efficiently computes a q-representative family $${\cal F'}$$ of $${\cal F}$$ of size approximately $${p+q \choose p}$$, and demonstrated several applications of this algorithm. In this paper, we consider the efficient computation of q-representative sets for product families $${\cal F}$$. A family $${\cal F}$$ is a product family if there exist families $${\cal A}$$ and $${\cal B}$$ such that $${\cal F} = \{A \cup B~:~A \in{\cal A}, B \in{\cal B}, A \cap B = \emptyset\}$$. Our main technical contribution is an algorithm which given $${\cal A}$$, $${\cal B}$$ and q computes a q-representative family $${\cal F}'$$ of $${\cal F}$$. The running time of our algorithm is sublinear in $$|{\cal F}|$$ for many choices of $${\cal A}$$, $${\cal B}$$ and q which occur naturally in several dynamic programming algorithms. We also give an algorithm for the computation of q-representative sets for product families $${\cal F}$$ in the more general setting where q-representation also involves independence in a matroid in addition to disjointness. This algorithm considerably outperforms the naive approach where one first computes $${\cal F}$$ from $${\cal A}$$ and $${\cal B}$$, and then computes the q-representative family $${\cal F}'$$ from $${\cal F}$$.

We give two applications of our new algorithms for computing q-representative sets for product families. The first is a $$3.8408^kn^{{\mathcal{O}}(1)}$$ deterministic algorithm for the Multilinear Monomial Detection (k -MlD) problem. The second is a significant improvement of deterministic dynamic programming algorithms for “connectivity problems” on graphs of bounded treewidth.

## Keywords

Product Family Dynamic Programming Algorithm Query Time Deterministic Algorithm Arithmetic Circuit
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Alon, N., Yuster, R., Zwick, U.: Color-coding. J. Assoc. Comput. Mach. 42(4), 844–856 (1995)
2. 2.
Bellman, R., Karush, W.: Mathematical programming and the maximum transform. J. Soc. Indust. Appl. Math. 10, 550–567 (1962)
3. 3.
Bellman, R., Karush, W.: On the maximum transform and semigroup of transformations. Bull. Amer. Math. Soc. 68, 516–518 (1962)
4. 4.
Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Fourier meets Möbious: Fast subset convolution. In: Proceedings of the 39th Annual ACM Symposium on Theory of Computing (STOC 2007). ACM Press, New York (2007) (page to appear)Google Scholar
5. 5.
Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Narrow sieves for parameterized paths and packings. CoRR, abs/1007.1161 (2010)Google Scholar
6. 6.
Björklund, A., Kaski, P., Kowalik, L.: Probably optimal graph motifs. In: STACS. LIPIcs, vol. 20, pp. 20–31 (2013)Google Scholar
7. 7.
Bodlaender, H.L., Cygan, M., Kratsch, S., Nederlof, J.: Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part I. LNCS, vol. 7965, pp. 196–207. Springer, Heidelberg (2013)
8. 8.
Cygan, M., Nederlof, J., Pilipczuk, M., Pilipczuk, M., van Rooij, J.M.M., Wojtaszczyk, J.O.: Solving connectivity problems parameterized by treewidth in single exponential time. In: Proceedings of the 52nd Annual Symposium on Foundations of Computer Science (FOCS 2011). IEEE (2011)Google Scholar
9. 9.
Fomin, F.V., Lokshtanov, D., Raman, V., Saurabh, S., Rao, B.V.R.: Faster algorithms for finding and counting subgraphs. J. Comput. System Sci. 78(3), 698–706 (2012)
10. 10.
Fomin, F.V., Lokshtanov, D., Saurabh, S.: Efficient computation of representative sets with applications in parameterized and exact algorithms. In: SODA, pp. 142–151 (2014)Google Scholar
11. 11.
Guillemot, S., Sikora, F.: Finding and counting vertex-colored subtrees. Algorithmica 65(4), 828–844 (2013)
12. 12.
Koutis, I.: Faster algebraic algorithms for path and packing problems. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 575–586. Springer, Heidelberg (2008)
13. 13.
Koutis, I.: Constrained multilinear detection for faster functional motif discovery. Inf. Process. Lett. 112(22), 889–892 (2012)
14. 14.
Lovász, L.: Flats in matroids and geometric graphs. In: Combinatorial surveys (Proc. Sixth British Combinatorial Conf., Royal Holloway Coll., Egham), pp. 45–86. Academic Press, London (1977)Google Scholar
15. 15.
Marx, D.: Parameterized coloring problems on chordal graphs. Theor. Comput. Sci. 351(3), 407–424 (2006)
16. 16.
Marx, D.: A parameterized view on matroid optimization problems. Theor. Comput. Sci. 410(44), 4471–4479 (2009)
17. 17.
Monien, B.: How to find long paths efficiently. In: Analysis and design of algorithms for combinatorial problems, Udine, 1982. North-Holland Math. Stud., vol. 109, pp. 239–254. North-Holland, Amsterdam (1985)
18. 18.
Oxley, J.G.: Matroid theory, vol. 3. Oxford University Press (2006)Google Scholar
19. 19.
Shachnai, H., Zehavi, M.: Faster computation of representative families for uniform matroids with applications. CoRR, abs/1402.3547 (2014)Google Scholar
20. 20.
Williams, R.: Finding paths of length k in O *(2k) time. Inf. Process. Lett. 109(6), 315–318 (2009)
21. 21.
Williams, V.V.: Multiplying matrices faster than Coppersmith-Winograd. In: Proceedings of the 44th Symposium on Theory of Computing Conference (STOC 2012), pp. 887–898. ACM (2012)Google Scholar

## Authors and Affiliations

• Fedor V. Fomin
• 1
• Daniel Lokshtanov
• 1