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Register a new userApproximate the k-Set Packing Problem by Local Improvements
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Huiwen Yu
Publication date
2013
fields
Informatics Engineering
and research's language is
English
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We study algorithms based on local improvements for the $k$-Set Packing problem. The well-known local improvement algorithm by Hurkens and Schrijver has been improved by Sviridenko and Ward from $frac{k}{2}+epsilon$ to $frac{k+2}{3}$, and by Cygan to $frac{k+1}{3}+epsilon$ for any $epsilon>0$. In this paper, we achieve the approximation ratio $frac{k+1}{3}+epsilon$ for the $k$-Set Packing problem using a simple polynomial-time algorithm based on the method by Sviridenko and Ward. With the same approximation guarantee, our algorithm runs in time singly exponential in $frac{1}{epsilon^2}$, while the running time of Cygans algorithm is doubly exponential in $frac{1}{epsilon}$. On the other hand, we construct an instance with locality gap $frac{k+1}{3}$ for any algorithm using local improvements of size $O(n^{1/5})$, here $n$ is the total number of sets. Thus, our approximation guarantee is optimal with respect to results achievable by algorithms based on local improvements.
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We present a packing-based approximation algorithm for the $k$-Set Cover problem. We introduce a new local search-based $k$-set packing heuristic, and call it Restricted $k$-Set Packing. We analyze its tight approximation ratio via a complicated combinatorial argument. Equipped with the Restricted $k$-Set Packing algorithm, our $k$-Set Cover algorithm is composed of the $k$-Set Packing heuristic cite{schrijver} for $kgeq 7$, Restricted $k$-Set Packing for $k=6,5,4$ and the semi-local $(2,1)$-improvement cite{furer} for 3-Set Cover. We show that our algorithm obtains a tight approximation ratio of $H_k-0.6402+Theta(frac{1}{k})$, where $H_k$ is the $k$-th harmonic number. For small $k$, our results are 1.8667 for $k=6$, 1.7333 for $k=5$ and 1.5208 for $k=4$. Our algorithm improves the currently best approximation ratio for the $k$-Set Cover problem of any $kgeq 4$.
We study emph{parallel} online algorithms: For some fixed integer $k$, a collective of $k$ parallel processes that perform online decisions on the same sequence of events forms a $k$-emph{copy algorithm}. For any given time and input sequence, the overall performance is determined by the best of the $k$ individual total results. Problems of this type have been considered for online makespan minimization; they are also related to optimization with emph{advice} on future events, i.e., a number of bits available in advance. We develop textsc{Predictive Harmonic}$_3$ (PH3), a relatively simple family of $k$-copy algorithms for the online Bin Packing Problem, whose joint competitive factor converges to 1.5 for increasing $k$. In particular, we show that $k=6$ suffices to guarantee a factor of $1.5714$ for PH3, which is better than $1.57829$, the performance of the best known 1-copy algorithm textsc{Advanced Harmonic}, while $k=11$ suffices to achieve a factor of $1.5406$, beating the known lower bound of $1.54278$ for a single online algorithm. In the context of online optimization with advice, our approach implies that 4 bits suffice to achieve a factor better than this bound of $1.54278$, which is considerably less than the previous bound of 15 bits.
We consider the class of packing integer programs (PIPs) that are column sparse, i.e. there is a specified upper bound k on the number of constraints that each variable appears in. We give an (ek+o(k))-approximation algorithm for k-column sparse PIPs, improving on recent results of $k^2cdot 2^k$ and $O(k^2)$. We also show that the integrality gap of our linear programming relaxation is at least 2k-1; it is known that k-column sparse PIPs are $Omega(k/ log k)$-hard to approximate. We also extend our result (at the loss of a small constant factor) to the more general case of maximizing a submodular objective over k-column sparse packing constraints.
Several algorithms with an approximation guarantee of $O(log n)$ are known for the Set Cover problem, where $n$ is the number of elements. We study a generalization of the Set Cover problem, called the Partition Set Cover problem. Here, the elements are partitioned into $r$ emph{color classes}, and we are required to cover at least $k_t$ elements from each color class $mathcal{C}_t$, using the minimum number of sets. We give a randomized LP-rounding algorithm that is an $O(beta + log r)$ approximation for the Partition Set Cover problem. Here $beta$ denotes the approximation guarantee for a related Set Cover instance obtained by rounding the standard LP. As a corollary, we obtain improved approximation guarantees for various set systems for which $beta$ is known to be sublogarithmic in $n$. We also extend the LP rounding algorithm to obtain $O(log r)$ approximations for similar generalizations of the Facility Location type problems. Finally, we show that many of these results are essentially tight, by showing that it is NP-hard to obtain an $o(log r)$-approximation for any of these problems.
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