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Efficient Approximation of Fractional Hypertree Width | IEEE Conference Publication | IEEE Xplore

Efficient Approximation of Fractional Hypertree Width


Abstract:

We give two new approximation algorithms to compute the fractional hypertree width of an input hypergraph. The first algorithm takes as input n-vertex m-edge hypergraph $...Show More

Abstract:

We give two new approximation algorithms to compute the fractional hypertree width of an input hypergraph. The first algorithm takes as input n-vertex m-edge hypergraph H of fractional hypertree width at most \omega, runs in polynomial time and produces a tree decomposition of H of fractional hypertree width \mathcal{O}(\omega\log n\log\omega), i.e., it is an \mathcal{O}(\log n\log\omega)-approximation algorithm. As an immediate corollary this yields poly-nomial time \mathcal{O}(\log^{2}n\log\omega)-approximation algorithms for (generalized) hypertree width as well. To the best of our knowledge our algorithm is the first non-trivial polynomial-time approximation algorithm for fractional hypertree width and (generalized) hypertree width, as opposed to algorithms that run in polynomial time only when \omega is considered a constant. For hypergraphs where every pair of hyperedges have at most \eta vertices in common, the al-gorithm outputs a hypertree decomposition with fractional hypertree width \mathcal{O}(\eta\omega^{2}\log\omega) and generalized hypertree width \mathcal{O}(\eta\omega^{2}\log\omega(\log\eta+\text{log}\omega)). This ratio is comparable with the recent algorithm of Lanzinger and Razgon [STACS 2024], which produces a hypertree decomposition with generalized hypertree width {\mathcal{O}}(\omega^{2}(\omega+\eta)), but uses time (at least) exponential in \eta and \omega. The second algorithm runs in time n^{\omega}m^{\mathcal{O}(1)} and pro-duces a tree decomposition of H of fractional hypertree width \mathcal{O}(\omega{\mathrm{l}}\text{og}^{2}\omega). This significantly improves over the (n+m)^{\mathcal{O}(\omega^{3})} time algorithm of Marx [ACM TALG 2010], which produces a tree decomposition of fractional hyper-tree width \mathcal{O}(\omega^{3}), both in terms of running time and the approximation ratio. Our main technical contribution, and the key insight behind both algorithms, is a variant of the classic Menger's ...
Date of Conference: 27-30 October 2024
Date Added to IEEE Xplore: 29 November 2024
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Conference Location: Chicago, IL, USA

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