Replica Placement on Bounded Treewidth Graphs

We consider the replica placement problem: given a graph with clients and nodes, place replicas on a minimum set of nodes to serve all the clients; each client is associated with a request and maximum distance that it can travel to get served and there is a maximum limit (capacity) on the amount of request a replica can serve. The problem falls under the general framework of capacitated set covering. It admits an O(\log n)-approximation and it is NP-hard to approximate within a factor of $o(\log n)$. We study the problem in terms of the treewidth $t$ of the graph and present an O(t)-approximation algorithm.Comment: An abridged version of this paper is to appear in the proceedings of WADS'1

Approximation Algorithm for Vertex Cover with Multiple Covering Constraints

We consider the vertex cover problem with multiple coverage constraints in hypergraphs. In this problem, we are given a hypergraph G=(V,E) with a maximum edge size f, a cost function w: V - > Z^+, and edge subsets P_1,P_2,...,P_r of E along with covering requirements k_1,k_2,...,k_r for each subset. The objective is to find a minimum cost subset S of V such that, for each edge subset P_i, at least k_i edges of it are covered by S. This problem is a basic yet general form of classical vertex cover problem and a generalization of the edge-partitioned vertex cover problem considered by Bera et al. We present a primal-dual algorithm yielding an (f * H_r + H_r)-approximation for this problem, where H_r is the r^{th} harmonic number. This improves over the previous ratio of (3cf log r), where c is a large constant used to ensure a low failure probability for Monte-Carlo randomized algorithms. Compared to previous result, our algorithm is deterministic and pure combinatorial, meaning that no Ellipsoid solver is required for this basic problem. Our result can be seen as a novel reinterpretation of a few classical tight results using the language of LP primal-duality

Tight Approximation for Partial Vertex Cover with Hard Capacities

We consider the partial vertex cover problem with hard capacity constraints (Partial VC-HC) on hypergraphs. In this problem we are given a hypergraph G=(V,E) with a maximum edge size f and a covering requirement R. Each edge is associated with a demand, and each vertex is associated with a capacity and an (integral) available multiplicity. The objective is to compute a minimum vertex multiset such that at least R units of demand from the edges are covered by the capacities of the vertices in the multiset and the multiplicity of each vertex does not exceed its available multiplicity. In this paper we present an f-approximation for this problem, improving over a previous result of (2f+2)(1+epsilon) by Cheung et al to the tight extent possible. Our new ingredient of this work is a generalized analysis on the extreme points of the natural LP, developed from previous works, and a strengthened LP lower-bound obtained for the optimal solutions

O(f) Bi-Approximation for Capacitated Covering with Hard Capacities

We consider capacitated vertex cover with hard capacity constraints (VC-HC) on hypergraphs. In this problem we are given a hypergraph G = (V, E) with a maximum edge size f. Each edge is associated with a demand and each vertex is associated with a weight (cost), a capacity, and an available multiplicity. The objective is to find a minimum-weight vertex multiset such that the demands of the edges can be covered by the capacities of the vertices and the multiplicity of each vertex does not exceed its available multiplicity. In this paper we present an O(f) bi-approximation for VC-HC that gives a trade-off on the number of augmented multiplicity and the cost of the resulting cover. In particular, we show that, by augmenting the available multiplicity by a factor of k geq 2, a cover with a cost ratio of (1+ frac{1}{k - 1})(f - 1) to the optimal cover for the original instance can be obtained. This improves over a previous result, which has a cost ratio of f^2 via augmenting the available multiplicity by a factor of f

Dynamic programming based algorithms for set multicover and multiset multicover problems

Given a universe N containing n elements and a collection of multisets or sets over N, the multiset multicover (MSMC) problem or the set multicover (SMC) problem is to cover all elements at least a number of times as specified in their coverage requirements with the minimum number of multisets or sets. In this paper, we give various exact algorithms for these two problems with or without constraints on the number of times a multiset or set may be chosen. First, we show that the MSMC without multiplicity constraints problem can be solved in O* ((b + 1)n | F |) time and polynomial space, where b is the maximum coverage requirement and | F | denotes the total number of given multisets over N. (The O* notation suppresses a factor polynomial in n.) To our knowledge, this is the first known exact algorithm for the MSMC without multiplicity constraints problem. Second, by combining dynamic programming and the inclusion-exclusion principle, we can exactly solve the SMC without multiplicity constraints problem in O ((b + 2)n) time. Compared with two recent results, in [Q.-S. Hua, Y. Wang, D. Yu, F.C.M. Lau, Set multi-covering via inclusion-exclusion, Theoretical Computer Science, 410 (38-40) (2009) 3882-3892] and [J. Nederlof, Inclusion exclusion for hard problems, Master Thesis, Utrecht University, The Netherlands, 2008], respectively, ours is the fastest exact algorithm for the SMC without multiplicity constraints problem. Finally, by directly using dynamic programming, we give the first known exact algorithm for the MSMC or the SMC with multiplicity constraints problem in O ((b + 1)n | F |) time and O* ((b + 1)n) space. This algorithm can also be easily adapted as a constructive algorithm for the MSMC without multiplicity constraints problem. © 2010 Elsevier B.V. All rights reserved.postprin

Replica Placement via Capacitated Vertex Cover

In this paper, we study the replica placement problem on trees and present a constant factor approximation algorithm (with an additional additive constant factor). This improves the best known previous algorithm having an approximation ratio dependent on the maximum degree of the tree. Our techniques also extend to the partial cover version. Our algorithms are based on the LP rounding technique. The core component of our algorithm exploits a connection between the natural LP solutions of the replica placement problem and the capacitated vertex cover problem

A Local-Ratio-Based Power Control Approach for Capacitated Access Points in Mobile Edge Computing

Terminal devices (TDs) connect to networks through access points (APs) integrated into the edge server. This provides a prerequisite for TDs to upload tasks to cloud data centers or offload them to edge servers for execution. In this process, signal coverage, data transmission, and task execution consume energy, and the energy consumption of signal coverage increases sharply as the radius increases. Lower power leads to less energy consumption in a given time segment. Thus, power control for APs is essential for reducing energy consumption. Our objective is to determine the power assignment for each AP with same capacity constraints such that all TDs are covered, and the total power is minimized. We define this problem as a \emph{minimum power capacitated cover } (MPCC) problem and present a \emph{minimum local ratio} (MLR) power control approach for this problem to obtain accurate results in polynomial time. Power assignments are chosen in a sequence of rounds. In each round, we choose the power assignment that minimizes the ratio of its power to the number of currently uncovered TDs it contains. In the event of a tie, we pick an arbitrary power assignment that achieves the minimum ratio. We continue choosing power assignments until all TDs are covered. Finally, various experiments verify that this method can outperform another greedy-based way