Budgeted Maximum Coverage Problem Research Articles

Occams: A Text Summarization Package

Extractive text summarization selects asmall subset of sentences from a document, which gives good “coverage” of a document. When given a set of term weights indicating the importance of the terms, the concept of coverage may be formalized into a combinatorial optimization problem known as the budgeted maximum coverage problem. Extractive methods in this class are known to beamong the best of classic extractive summarization systems. This paper gives a synopsis of thesoftware package occams, which is a multilingual extractive single and multi-document summarization package based on an algorithm giving an optimal approximation to the budgeted maximum coverage problem. The occams package is written in Python and provides an easy-to-use modular interface, allowing it to work in conjunction with popular Python NLP packages, such as nltk, stanza or spacy.

Iterated hyperplane search for the budgeted maximum coverage problem

We present an iterated hyperplane search approach for the budgeted maximum coverage problem. Our algorithm relies on the idea of searching on specific areas identified by cardinality-constrained hyperplanes. It combines three complementary procedures: a tabu search procedure to identify a promising hyperplane, a hyperplane search procedure to examine candidate solutions on cardinality-constrained hyperplanes and a dedicated perturbation procedure to ensure the diversification of the search. We show the competitiveness of the algorithm on 30 benchmark instances and present experiments to study the key components of the algorithm.

Effective Variable Depth Local Search for the Budgeted Maximum Coverage Problem

We address the Budgeted Maximum Coverage Problem (BMCP), which is a natural and more practical extension of the standard 0–1 knapsack problem and the set cover problem. Given m elements with nonnegative weights, n subsets of elements with nonnegative costs, and a total budget, BMCP aims to select some subsets such that the total cost of selected subsets does not exceed the budget, and the total weight of associated elements is maximized. In this paper, we propose a variable depth local search algorithm (VDLS) for the BMCP. VDLS first generates an initial solution by a greedy algorithm, then iteratively improves the solution through a partial depth-first search method, that can improve the solution by simultaneously changing the states (selected or not) of multiple subsets. Such method allows VDLS to explore the solution space widely and deeply, and to yield high-quality solutions. We further propose a neighbor structure to boost the algorithm performance, that is, both subsets have a neighbor relation if they share at least one common associated element. By applying the neighbor structure, VDLS can adjust the selected subsets while losing as few covered elements as possible. Since the existing BMCP benchmarks only have simple structures and small scales, we design 60 new instances with relatively large scales and complex structures to enrich the diversity of the BMCP instances. Experimental results on 30 public instances and 60 new instances we designed demonstrate that VDLS significantly outperforms the existing heuristic and the general CPLEX exact solver.

Efficient and Effective Local Search for the Set-Union Knapsack Problem and Budgeted Maximum Coverage Problem

The Set-Union Knapsack Problem (SUKP) and Budgeted Maximum Coverage Problem (BMCP) are two closely related variant problems of the popular knapsack problem. Given a set of weighted elements and a set of items with nonnegative values, where each item covers several distinct elements, these two problems both aim to find a subset of items that maximizes an objective function while satisfying a knapsack capacity (budget) constraint. We propose an efficient and effective local search algorithm called E2LS for these two problems. To our knowledge, this is the first time that an algorithm has been proposed for both of them. E2LS trade-offs the search region and search efficiency by applying a proposed novel operator ADD$^*$ to traverse the refined search region. Such a trade-off mechanism allows E2LS to explore the solution space widely and quickly. The tabu search method is also applied in E2LS to help the algorithm escape from local optima. Extensive experiments on a total of 168 public instances with various scales demonstrate the excellent performance of the proposed algorithm for both the SUKP and BMCP.

Online Budgeted Maximum Coverage

We study the Online Budgeted Maximum Coverage problem. Subsets of a weighted ground set U arrive one by one, where each set has a cost. The online algorithm has to select a collection of sets, under the constraint that their cost is at most a given budget. Upon arrival of a set the algorithm must decide whether to accept or to irrevocably reject the arriving set, and it may also irrevocably drop previously accepted sets. The goal is to maximize the total weight of the elements covered by the sets in the chosen collection. We give a deterministic \(\frac{4}{1-r}\)-competitive algorithm where r is the maximum ratio between the cost of a set and the total budget, and show that the competitive ratio of any deterministic online algorithm is \(\Omega (\frac{1}{1-r})\). We further give a randomized O(1)-competitive algorithm. We also give a deterministic \(O(\Delta )\)-competitive algorithm, where \(\Delta \) is the maximum weight of a set and a modified version of it with competitive ratio of \(O(\min \{\Delta ,\sqrt{w(U)}\})\) for the case that the total weight of the elements, w(U), is known in advance. A matching lower bound of \(\Omega (\min \{\Delta ,\sqrt{w(U)}\})\) is given. Finally, our results, including the lower bounds, apply also to Removable Online Knapsack.

Probability learning based tabu search for the budgeted maximum coverage problem

The Budgeted Maximum Coverage Problem (BMCP) is a general model with a number of real-world applications. Given n elements with nonnegative profits, m subsets of elements with nonnegative weights and a total budget, the BMCP aims to select some subsets such that the total weight of the selected subsets does not exceed the budget, while the total profit of the associated elements is maximized. BMCP is NP-hard and thus computationally challenging. We investigate for the first time an effective practical algorithm for solving this problem, which combines reinforcement learning and local search. The algorithm iterates through two distinct phases, namely a tabu search phase and a probability learning based perturbation phase. To assess the effectiveness of the proposed algorithm, we show computational results on a set of 30 benchmark instances introduced in this paper and present comparative studies with respect to the approximation algorithm, the genetic algorithm and the CPLEX solver.

Approximation algorithms for profit maximisation in multicast D2D networks

As the demand for mobile data services increases, telecom companies need to develop wise strategies to retain existing customers. For instance, in a multicast scenario, satisfying individual user's quality of service (QoS), data demand at varying rates, etc. are complicated tasks. Earlier works on device-to-device (D2D) multicast have majorly discussed the cases of throughput maximisation without considering the individual user's data request rates. In this paper, we propose two algorithms to maximise the telecom operator's profit collected from the users in a two-hop D2D multicast network, when users have different channel qualities and data request rates while receiving multicast data through a single transmission session. First, we model our multicast scheme for the proposed scenario as a budgeted maximum coverage problem. Later, we propose two approximation algorithms that guarantee approximation ratios of 1−1/ √e and 1 − 1 / e, respectively, where e denotes the base of the natural logarithm. Numerical results show that the proposed algorithms perform better than the other candidate algorithms and nearly approximates the optimal solution.

Budgeted video replacement policy in mobile crowdsensing

Mobile crowdsensing offers a new platform that recruits a suitable set of users to collectively complete an information collection/sensing task through users’ equipped devices. As a special case, video crowdsensing is to collect different video segments of the same event that are taken separately by the built-in cameras of mobile devices, and then combine them into a complete video. Mobile crowdsensing has attracted considerable attention recently due to the rich information that can be provided by videos. However, because of the limited caching space, a suitable video replacement policy is necessary. In this paper, we propose a Budgeted Video replaCement policy in mobile Video crowdsensing (BVCV), which first determines a video segment’s value according to its caching situation and natural attributes. Then, we formulate the video caching problem as a budgeted maximum coverage problem, which is a well-known NP-hard problem. Finally, we propose a practical greedy solution and also infer the approximate ratio, which could be regarded as the lower bound of BVCV to the optimal solution. Our experiments with the real mobility datasets (StudentLife dataset, Buffalo/phonelab-wifi dataset) show that, the proposed budgeted video replacement policy achieves a longer successfully delivered video length, compared with other general replacement policies.

Partial Vertex Cover and Budgeted Maximum Coverage in Bipartite Graphs

In this paper, we study two closely related problems on bipartite graphs, viz., the partial vertex cover problem and the budgeted maximum coverage problem. Both these problems arise in a number of ...

The Budgeted Maximum Coverage Problem in Partially Deployed Software Defined Networks

Due to the large installed base of distributed legacy networks, software defined networking (SDN) nodes may be required to coexist with legacy nodes to form hybrid networks. It has been shown that such a hybrid network has better performance than a pure legacy network due to smarter transmission scheduling. Despite such advantages, limited budgets continue to hinder the rapid adaptation of SDNs. Only a part of the network can be upgraded at a time especially for large-scale networks. In this paper, we define the minimum percentage of SDN nodes in a path, and paths with at least one SDN node, as the hop coverage and path coverage, respectively. We intend to evaluate the relationship between cost and coverage in the partially deployed SDNs. We formulate SDN node selection as four optimization problems with hop/path coverage and cost as objectives and constraints, respectively, and vice-versa. We propose two heuristic solutions: 1) maximum number of uncovered path first (MUcPF) and 2) maximum number of minimum hop covered path first (MMHcPF), to these NP-hard problems. Through a MATLAB experiment, we show that MUcPF is significantly better in terms of economy and efficiency to establish a hybrid path between every pair of hosts in the network. In particular, it required 5%–15% less investment to achieve 100% path coverage compared to other algorithms. The results show the coverage consistency of MMHcPF on each individual path along with gains in terms of cost and efficiency. It takes 5%–20% less investment to achieve certain hop coverage target compared to other existing algorithms.

A note on the set union knapsack problem

Recently, Khuller, Moss and Naor presented a greedy algorithm for the budgeted maximum coverage problem. In this note, we observe that this algorithm also approximates a special case of a set-union knapsack problem within a constant factor. In the special case, an element is a member of less than a constant number of subsets. This guarantee naturally extends to densest k-subgraph problem on graphs of bounded degree.

The Generalized Maximum Coverage Problem

We define a new problem called the Generalized Maximum Coverage Problem (GMC). GMC is an extension of the Budgeted Maximum Coverage Problem, and it has important applications in wireless OFDMA scheduling. We use a variation of the greedy algorithm to produce a ( ( 2 e − 1 ) ( e − 1 ) + ε ) -approximation for every ε > 0 , and then use partial enumeration to reduce the approximation ratio to e ( e − 1 ) + ε .

The budgeted maximum coverage problem

The budgeted maximum coverage problem is: given a collection S of sets with associated costs defined over a domain of weighted elements, and a budget L, find a subset of S′⫅ S such that the total cost of sets in S′ does not exceed L, and the total weight of elements covered by S′ is maximized. This problem is NP-hard. For the special case of this problem, where each set has unit cost, a (1−1/ e) -approximation is known. Yet, prior to this work, no approximation results were known for the general cost version. The contribution of this paper is a (1−1/ e) -approximation algorithm for the budgeted maximum coverage problem. We also argue that this approximation factor is the best possible, unless NP⫅ DTIME(n O( log logn) ) .