广义和分数阶超树分解的复杂度分析

G. Gottlob, Matthias Lanzinger, R. Pichler, Igor Razgon
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引用次数: 14

摘要

超树分解(Hypertree decomposition, hd)以及更强大的广义超树分解(general Hypertree decomposition, GHDs)和更一般的分数阶超树分解(fractional Hypertree decomposition, fhd)是用于回答连接查询和解决约束满足问题的超图分解方法。每个超图H都有一个相对于这些方法的宽度:它的超树宽度hw(H),它的广义超树宽度ghw(H),和它的分数超树宽度fhw(H)。已知对于固定k,可以在多项式时间内检验hw(H)≤k,对于k≥3,检验ghw(H)≤k是np完全的。对于固定k,检查fhw(H)≤k的复杂性已经开放了十多年。通过证明检验fhw(H)≤k是np完全的,即使k=2,我们解决了这个开放问题。同样的构造还允许我们证明当k=2时检验ghw(H)≤k的np完备性。在此之后,我们确定了有意义的限制,使检查有界的ghw或fhw易于处理,或者允许有效的近似fhw。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Complexity Analysis of Generalized and Fractional Hypertree Decompositions
Hypertree decompositions (HDs), as well as the more powerful generalized hypertree decompositions (GHDs), and the yet more general fractional hypertree decompositions (FHDs) are hypergraph decomposition methods successfully used for answering conjunctive queries and for solving constraint satisfaction problems. Every hypergraph H has a width relative to each of these methods: its hypertree width hw(H), its generalized hypertree width ghw(H), and its fractional hypertree width fhw(H), respectively. It is known that hw(H)≤ k can be checked in polynomial time for fixed k, while checking ghw(H)≤ k is NP-complete for k ≥ 3. The complexity of checking fhw(H)≤ k for a fixed k has been open for over a decade. We settle this open problem by showing that checking fhw(H)≤ k is NP-complete, even for k=2. The same construction allows us to prove also the NP-completeness of checking ghw(H)≤ k for k=2. After that, we identify meaningful restrictions that make checking for bounded ghw or fhw tractable or allow for an efficient approximation of the fhw.
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