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

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

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

Online Dynamic Power Management with Hard Real-Time Guarantees

We consider the problem of online dynamic power management that provides hard real-time guarantees for multi-processor systems. In this problem, a set of jobs, each associated with an arrival time, a deadline, and an execution time, arrives to the system in an online fashion. The objective is to compute a non-migrative preemptive schedule of the jobs and a sequence of power on/off operations of the processors so as to minimize the total energy consumption while ensuring that all the deadlines of the jobs are met. We assume that we can use as many processors as necessary. In this paper we examine the complexity of this problem and provide online strategies that lead to practical energy-efficient solutions for real-time multi-processor systems. First, we consider the case for which we know in advance that the set of jobs can be scheduled feasibly on a single processor. We show that, even in this case, the competitive factor of any online algorithm is at least 2.06. On the other hand, we give a 4-competitive online algorithm that uses at most two processors. For jobs with unit execution times, the competitive factor of this algorithm improves to 3.59. Second, we relax our assumption by considering as input multiple streams of jobs, each of which can be scheduled feasibly on a single processor. We present a trade-off between the energy-efficiency of the schedule and the number of processors to be used. More specifically, for k given job streams and h processors with h>k, we give a scheduling strategy such that the energy usage is at most 4.k/(h-k) times that used by any schedule which schedules each of the k streams on a separate processor. Finally, we drop the assumptions on the input set of jobs. We show that the competitive factor of any online algorithm is at least 2.28, even for the case of unit job execution times for which we further derive an O(1)-competitive algorithm

Capacitated Domination Problem

我們考慮在圖論上廣為人知的支配集問題，並將它加以推廣。所謂「有容量的支配集問題」是指在給定的圖上，尋找滿足每個點的容量限制(capacity)與需求限制(demand)的最小支配集。在這個問題模型中，每個點都有各自的容量和需求。容量(capacity)指的是，每份此頂點的實體(copy)，可以容納多少單位來自閉鄰點(closedeighborhood)的需求。而需求(demand)指的則是，此頂點需要多少單位來自閉鄰點的容量供給。此論文中，我們從演算法的觀點探討「有容量的支配集問題」在樹狀結構中的表現。在需求不可分割的模型裡，我們提供了線性時間複雜度的演算法；而對於需求可分割的問題模型，我們則提供了虛擬多項式時間(pseudo-polynomial time)的演算法。此之外，我們也証明，當需求可分割時，這個問題在樹狀結構上也是NP-完備，並進一步提供可任意逼近最佳解(polynomialime approximation scheme)的演算法。對於一般的圖，我們則提供以線性規劃的primal-dual為基礎的近似演算法。We consider a generalization of the well-known domination problem on graphs. The soft capacitated domination problem with demand constraints is to find a dominating set D of minimum cardinality satisfying both the capacity and demand constraints. The capacity constraint specifies that each vertex has a capacity that it can use to meet the demand of dominated vertices in its closed neighborhood, and the number of copies of each vertex allowed in D is unbounded. The demand constraint specifies that the demand of each vertex in is met by the capacities of vertices in D dominating it.n this thesis, we study the capacitated domination problem on trees from an algorithmic point of view. We present a linear time algorithm for the unsplittable demand model, and a pseudo-polynomial time algorithm for the splittable demand model. In addition, we show that the capacitated domination problem on trees with splittable demand constraints is NP-complete and provide a polynomial time approximation scheme (PTAS). We also give a primal-dual approximation algorithm for the weighted capacitated domination problem with splittable demand constraints on general graphs.Contents試委員會審定書....................................icknowledgement...................................iihinese Abstract....................................iiinglish Abstract....................................iv Introduction.1 Preliminaries...................................5 Capacitated Domination on Trees.1 Unsplittable Demand Model...........................7.2 NP-completeness for Splittable Demand Model................13.3 A Pseudo-polynomial Time Algorithm for Splittable Demand Model....16.3.1 The Relaxed Knapsack Problem....................21 Approximation Algorithms for Splittable Demand Model.1 Approximation on Trees.............................23.2 Approximation on General Graphs.......................27 Conclusion.1 Discussion.....................................32.2 Future Work...................................34ibliography 34 Pseudo Codes 39.1 Algorithm MCDUT...............................39.2 Algorithm MCDST................................43.3 Algorithm MCDAWG..............................46 Implementation 4