In this paper, we consider the problem of choosing a minimum cost set of resources for executing a specified set of jobs. Each input job is an interval, determined by its start-time and end-time. Each resource is also an interval determined by its start-time and end-time; moreover, every resource has a capacity and a cost associated with it. We consider two versions of this problem. In the partial covering version, we are also given as input a number k, specifying the number of jobs that must be performed. The goal is to choose k jobs and find a minimum cost set of resources to perform the chosen k jobs (at any point of time the capacity of the chosen set of resources should be sufficient to execute the jobs active at that time). We present an O(log n)-factor approximation algorithm for this problem. We also consider the prize collecting version, wherein every job also has a penalty associated with it. The feasible solution consists of a subset of the jobs, and a set of resources, to perform the chosen subset of jobs. The goal is to find a feasible solution that minimizes the sum of the costs of the selected resources and the penalties of the jobs that are not selected. We present a constant factor approximation algorithm for this problem
In the setting where resources and jobs are embodied as intervals, the objective of finding a minimum cost collection of resources that fulfill the jobs is typically called the full cover problem. Full cover problems in the interval context have been dealt with earlier, in various earlier works @cite_6 @cite_7 @cite_4 . Partial cover problems in the interval context have been considered earlier in @cite_0 . The work in existing literature that is closest in spirit to our result is that of Bar- @cite_6 , and @cite_0 . In @cite_6 , the authors consider the full cover version, and present a @math -approximation algorithm. In this case, all the jobs are to be covered, and therefore the demand profile to be covered is fixed. The goal is to find the minimum cost set of resources, for covering this profile. In our setting, we need to cover only @math of the jobs. A solution needs to select @math jobs to be covered in such a manner that the resources required to cover the resulting demand profile has minimum cost.
@cite_0 @cite_4 @cite_7 @cite_6
2175530190 2901780026 45352664 2509865736
An instance of the generalized partial cover problem consists of a ground set U and a family of subsets @math . Each element e∈U is associated with a profit p(e), whereas each subset @math has a cost c(S). The objective is to find a minimum cost subcollection @math such that the combined profit of the elements covered by @math is at least P, a specified profit bound. In the prize-collecting version of this problem, there is no strict requirement to cover any element; however, if the subsets we pick leave an element e∈U uncovered, we incur a penalty of π(e). The goal is to identify a subcollection @math that minimizes the cost of @math plus the penalties of uncovered elements. Although problem-specific connections between the partial cover and the prize-collecting variants of a given covering problem have been explored and exploited, a more general connection remained open. The main contribution of this paper is to establish a formal relationship between these two variants. As a result, we present a unified framework for approximating problems that can be formulated or interpreted as special cases of generalized partial cover. We demonstrate the applicability of our method on a diverse collection of covering problems, for some of which we obtain the first non-trivial approximability results. 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 consider the problem of allocating resources to satisfy demand requirements varying over time. The input specifies a demand for each timeslot. Each resource is specified by a start-time, end-time, an associated cost and a capacity. A feasible solution is a multiset of resources such that at any point of time, the sum of the capacities offered by the resources is at least the demand requirement at that point of time. The goal is to minimize the total cost of the resources included in the solution. This problem arises naturally in many scenarios such as workforce management, sensor networks, cloud computing, energy management and distributed computing. We study this problem under the partial cover setting and the zero-one setting. In the former scenario, the input also includes a number k and the goal is to choose a minimum cost solution that satisfies the demand requirements of at least k timeslots. For this problem, we present a 16-approximation algorithm; we show that there exist "well-structured" near-optimal solutions and that such a solution can be found in polynomial time via dynamic programming. In the zero-one setting, a feasible solution is allowed to pick at most one copy of any resource. For this case, we present a 4-approximation algorithm; our algorithm uses a novel LP relaxation involving flow-cover inequalities. 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.