Partial set cover problem and set multi-cover problem are two generalizations of the set cover problem. In this paper, we consider the partial set multi-cover problem which is a combination of them: given an element set E, a collection of sets ( S 2^E ), a total covering ratio q, each set (S S ) is associated with a cost (c_S ), each element (e E ) is associated with a covering requirement (r_e ), the goal is to find a minimum cost sub-collection ( S ' S ) to fully cover at least q|E| elements, where element e is fully covered if it belongs to at least (r_e ) sets of ( S ' ). Denote by (r_ = r_e:e E ) the maximum covering requirement. We present an ((O (r_ ^2n(1+ ( 1 )+ 1-q q )),1- ) )-bicriteria approximation algorithm, that is, the output of our algorithm has cost (O(r_ ^2 n(1+ ( 1 )+ 1-q q )) ) times of the optimal value while the number of fully covered elements is at least ((1- )q|E| ).
From the above related work, it can be seen that both PSC and SMC admit performance ratios matching those best ratios for the classic set cover problem. However, combining partial set cover with set multi-cover has enormously increased the difficulty of studies. Ran @cite_6 were the first to study approximation algorithms for PSMC, using greedy strategy and dual-fitting analysis. However, their ratio is meaningful only when the covering percentage @math is very close to @math . In paper @cite_20 , the authors presented a simple greedy algorithm achieving performance ratio @math . They also presented a local ratio algorithm, which reveals a what they called shock wave'' phenomenon: their performance ratio is @math for both PSC and SMC , however, when @math is smaller than 1 by a very small constant, the ratio jumps abruptly to @math . In our recent paper @cite_21 , we proved that PSMC cannot have a better than polynomial performance ratio by a reduction from the well-known densest @math -subgraph problem .
@cite_20 @cite_21 @cite_6
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In this paper, we study the minimum partial set multi-cover problem (PSMC). Given an element set E, a collection of subsets ( S 2^E ), a cost (c_S ) on each set (S S ), a covering requirement (r_e ) for each element (e E ), and an integer k, the PSMC problem is to find a sub-collection ( F S ) to fully cover at least k elements such that the cost of ( F ) is as small as possible, where element e is fully covered by ( F ) if it belongs to at least (r_e ) sets of ( F ). This paper presents an approximation algorithm using local ratio method achieving performance ratio ( k ( 1 f-r_ + r_ r_ ) , 1 + f r_ + 1 r_ - 1 r_ -1, 1 ), where ( ) is the size of a maximum set, f is the maximum number of sets containing a common element, ( ) is the minimum percentage of elements required to be fully covered during iterations of the algorithm, and (r_ ) and (r_ ) are the maximum and the minimum covering requirement, respectively. In particular, when (r_ ) is a constant, the first term can be omitted. Notice that our ratio coincides with the classic ratio f for both the set multi-cover problem (in which case (k=|E| )) and the partial set single-cover problem (in which case (r_ =1 )). However, when (k 1 ), the ratio might be as large as ( (n) ). This result shows an interesting “shock wave like” feature of approximating PSMC. The purpose of this paper is trying to arouse some interest in such a feature and attract more work on this challenging problem. In a minimum partial set multi-cover problem (MinPSMC), given an element set E, a collection of subsets ( S 2^E ), a cost (w_S ) on each set (S S ), a covering requirement (r_e ) for each element (e E ), and an integer k, the goal is to find a sub-collection ( F S ) to fully cover at least k elements such that the cost of ( F ) is as small as possible, where element e is fully covered by ( F ) if it belongs to at least (r_e ) sets of ( F ). On the application side, the problem has its background in the seed selection problem in a social network. On the theoretical side, it is a natural combination of the minimum partial (single) set cover problem (MinPSC) and the minimum set multi-cover problem (MinSMC). Although both MinPSC and MinSMC admit good approximations whose performance ratios match those lower bounds for the classic set cover problem, previous studies show that theoretical study on MinPSMC is quite challenging. In this paper, we prove that MinPSMC cannot be approximated within factor (O(n^ 1 2( n)^c ) ) under the ETH assumption. A primal dual algorithm for MinPSMC is presented with a guaranteed performance ratio (O( n ) ) when (r_ ) and f are constants, where (r_ = _ e E r_e ) is the maximum covering requirement and f is the maximum frequency of elements (that is the maximum number of sets containing a common element). We also improve the ratio for a restricted version of MinPSMC which possesses a graph-type structure. We prove the first non-trivial performance ratios strictly above 0.5 for weighted versions of the oblivious matching problem. Even for the unweighted version, since Aronson, Dyer, Frieze, and Suen first proved a non-trivial ratio above 0.5 in the mid-1990s, during the next twenty years several attempts have been made to improve this ratio, until Chan, Chen, Wu and Zhao successfully achieved a significant ratio of 0.523 very recently (SODA 2014). To the best of our knowledge, our work is the first in the literature that considers the node-weighted and edge-weighted versions of the problem in arbitrary graphs (as opposed to bipartite graphs). (1) For arbitrary node weights, we prove that a weighted version of the Ranking algorithm has ratio strictly above 0.5. We have discovered a new structural property of the ranking algorithm: if a node has two unmatched neighbors at the end of algorithm, then it will still be matched even when its rank is demoted to the bottom. This property allows us to form LP constraints for both the node-weighted and the unweighted oblivious matching problems. As a result, we prove that the ratio for the node-weighted case is at least 0.501512. Interestingly via the structural property, we can also improve slightly the ratio for the unweighted case to 0.526823 (from the previous best 0.523166 in SODA 2014). (2) For a bounded number of distinct edge weights, we show that ratio strictly above 0.5 can be achieved by partitioning edges carefully according to the weights, and running the (unweighted) Ranking algorithm on each part. Our analysis is based on a new primal-dual framework known as , in which dual feasibility is bypassed. Instead, only dual constraints corresponding to edges in an optimal matching are satisfied. Using this framework we also design and analyze an algorithm for the edge-weighted online bipartite matching problem with free disposal. We prove that for the case of bounded online degrees, the ratio is strictly above 0.5. We consider positive covering integer programs, which generalize set cover and which have attracted a long line of research developing (randomized) approximation algorithms. Srinivasan (2006) gave a rounding algorithm based on the FKG inequality for systems which are "column-sparse." This algorithm may return an integer solution in which the variables get assigned large (integral) values; Kolliopoulos & Young (2005) modified this algorithm to limit the solution size, at the cost of a worse approximation ratio. We develop a new rounding scheme based on the Partial Resampling variant of the Lovasz Local Lemma developed by Harris & Srinivasan (2013). This achieves an approximation ratio of 1 + ln([EQUATION]), where amin is the minimum covering constraint and Δ1 is the maximum e1-norm of any column of the covering matrix (whose entries are scaled to lie in [0, 1]); we also show nearly-matching inapproximability and integrality-gap lower bounds. Our approach improves asymptotically, in several different ways, over known results. First, it replaces Δ0, the maximum number of nonzeroes in any column (from the result of Srinivasan) by Δ1 which is always - and can be much - smaller than Δ0; this is the first such result in this context. Second, our algorithm automatically handles multi-criteria programs; we achieve improved approximation ratios compared to the algorithm of Srinivasan, and give, for the first time when the number of objective functions is large, polynomial-time algorithms with good multi-criteria approximations. We also significantly improve upon the upper-bounds of Kolliopoulos & Young when the integer variables are required to be within (1 + e) of some given upper-bounds, and show nearly-matching inapproximability.