Set partitioning via inclusion–exclusion

Andreas Björklund, Thore Husfeldt, Mikko Koivisto

    Research output: Journal Article or Conference Article in JournalJournal articleResearchpeer-review

    Abstract

    Given a set N with n elements and a family F of subsets, we show how to partition N into k such subsets in 2nnO(1) time. We also consider variations of this problem where the subsets may overlap or are weighted, and we solve the decision, counting, summation, and optimization versions of these problems. Our algorithms are based on the principle of inclusion-exclusion and the zeta transform. In effect we get exact algorithms in 2nnO(1) time for several well-studied partition problems including domatic number, chromatic number, maximum k-cut, bin packing, list coloring, and the chromatic polynomial. We also have applications to Bayesian learning with decision graphs and to model-based data clustering. If only polynomial space is available, our algorithms run in time 3 nnO(1) if membership in F can be decided in polynomial time. We solve chromatic number in O(2.2461n) time and domatic number in O(2.8718n) time. Finally, we present a family of polynomial space approximation algorithms that find a number between χ(G) and (1 e+ )χ(G) in time O(1.2209n + 2.2461e−n).
    Original languageEnglish
    JournalSIAM Journal on Computing
    Volume39
    Issue number2
    Pages (from-to)546-563
    ISSN0097-5397
    Publication statusPublished - 2009

    Keywords

    • set partition
    • graph coloring
    • exact algorithm
    • zeta transform
    • inclusion-exclusion

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