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Representative Sets of Product Families

  • Fedor V. Fomin
  • Daniel Lokshtanov
  • Fahad Panolan
  • Saket Saurabh
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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Fedor V. Fomin
    • 1
  • Daniel Lokshtanov
    • 1
  • Fahad Panolan
    • 2
  • Saket Saurabh
    • 1
    • 2
  1. 1.University of BergenNorway
  2. 2.The Institute of Mathematical SciencesChennaiIndia

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