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Register a new userGeneral and Fractional Hypertree Decompositions: Hard and Easy Cases
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Added by
Reinhard Pichler
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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Hypertree decompositions, as well as the more powerful generalized hypertree decompositions (GHDs), and the yet more general fractional hypertree decompositions (FHD) are hypergraph decomposition methods successfully used for answering conjunctive queries and for the solution of constraint satisfaction problems. Every hypergraph H has a width relative to each of these decomposition 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 any k greater than or equal to 3. The complexity of checking fhw(H) <= k for a fixed k has been open for more than 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 proving these hardness results, we identify meaningful restrictions, for which checking for bounded ghw or fhw becomes tractable.
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The problem of publishing personal data without giving up privacy is becoming increasingly important. An interesting formalization recently proposed is the k-anonymity. This approach requires that the rows in a table are clustered in sets of size at least k and that all the rows in a cluster become the same tuple, after the suppression of some records. The natural optimization problem, where the goal is to minimize the number of suppressed entries, is known to be NP-hard when the values are over a ternary alphabet, k = 3 and the rows length is unbounded. In this paper we give a lower bound on the approximation factor that any polynomial-time algorithm can achive on two restrictions of the problem,namely (i) when the records values are over a binary alphabet and k = 3, and (ii) when the records have length at most 8 and k = 4, showing that these restrictions of the problem are APX-hard.
Our main result is that every graph $G$ on $nge 10^4r^3$ vertices with minimum degree $delta(G) ge (1 - 1 / 10^4 r^{3/2} ) n$ has a fractional $K_r$-decomposition. Combining this result with recent work of Barber, Kuhn, Lo and Osthus leads to the best known minimum degree thresholds for exact (non-fractional) $F$-decompositions for a wide class of graphs~$F$ (including large cliques). For general $k$-uniform hypergraphs, we give a short argument which shows that there exists a constant $c_k>0$ such that every $k$-uniform hypergraph $G$ on $n$ vertices with minimum codegree at least $(1- c_k /r^{2k-1}) n $ has a fractional $K^{(k)}_r$-decomposition, where $K^{(k)}_r$ is the complete $k$-uniform hypergraph on $r$ vertices. (Related fractional decomposition results for triangles have been obtained by Dross and for hypergraph cliques by Dukes as well as Yuster.) All the above new results involve purely combinatorial arguments. In particular, this yields a combinatorial proof of Wilsons theorem that every large $F$-divisible complete graph has an $F$-decomposition.
We prove that for any integer $kgeq 2$ and $varepsilon>0$, there is an integer $ell_0geq 1$ such that any $k$-uniform hypergraph on $n$ vertices with minimum codegree at least $(1/2+varepsilon)n$ has a fractional decomposition into tight cycles of length $ell$ ($ell$-cycles for short) whenever $ellgeq ell_0$ and $n$ is large in terms of $ell$. This is essentially tight. This immediately yields also approximate integral decompositions for these hypergraphs into $ell$-cycles. Moreover, for graphs this even guarantees integral decompositions into $ell$-cycles and solves a problem posed by Glock, Kuhn and Osthus. For our proof, we introduce a new method for finding a set of $ell$-cycles such that every edge is contained in roughly the same number of $ell$-cycles from this set by exploiting that certain Markov chains are rapidly mixing.
The hypervolume indicator is an increasingly popular set measure to compare the quality of two Pareto sets. The basic ingredient of most hypervolume indicator based optimization algorithms is the calculation of the hypervolume contribution of single solutions regarding a Pareto set. We show that exact calculation of the hypervolume contribution is #P-hard while its approximation is NP-hard. The same holds for the calculation of the minimal contribution. We also prove that it is NP-hard to decide whether a solution has the least hypervolume contribution. Even deciding whether the contribution of a solution is at most $(1+eps)$ times the minimal contribution is NP-hard. This implies that it is neither possible to efficiently find the least contributing solution (unless $P = NP$) nor to approximate it (unless $NP = BPP$). Nevertheless, in the second part of the paper we present a fast approximation algorithm for this problem. We prove that for arbitrarily given $eps,delta>0$ it calculates a solution with contribution at most $(1+eps)$ times the minimal contribution with probability at least $(1-delta)$. Though it cannot run in polynomial time for all instances, it performs extremely fast on various benchmark datasets. The algorithm solves very large problem instances which are intractable for exact algorithms (e.g., 10000 solutions in 100 dimensions) within a few seconds.
By a well known result the treewidth of k-outerplanar graphs is at most 3k-1. This paper gives, besides a rigorous proof of this fact, an algorithmic implementation of the proof, i.e. it is shown that, given a k-outerplanar graph G, a tree decomposition of G of width at most 3k-1 can be found in O(kn) time and space. Similarly, a branch decomposition of a k-outerplanar graph of width at most 2k+1 can be also obtained in O(kn) time, the algorithm for which is also analyzed.
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