Distributed Approximation of k-Service Assignment

We consider the k-Service Assignment problem (k-SA), defined as follows. The input consists of a network that contains servers and clients, and an integer k. Each server has a finite capacity, and each client is associated with a demand and a profit. A feasible solution is an assignment of clients to neighboring servers such that (i) the total demand assigned to a server is at most its capacity, and (ii) a client is assigned either to k servers or to none. The profit of an assignment is the total profit of clients that are assigned to k servers, and the goal is to find a maximum profit assignment. In the r-restricted version of k-SA, no client requires more than an r-fraction of the capacity of any adjacent server. The k-SA problem is motivated by backup placement in networks and by resource allocation in 4G cellular networks. It can also be viewed as machine scheduling on related machines with assignment restrictions. We present a centralized polynomial time greedy (k+1-r)/(1-r)-approximation algorithm for r-restricted k-SA. We then show that a variant of this algorithm achieves an approximation ratio of k+1 using a resource augmentation factor of 1+r. We use the latter to present a (k+1)^2-approximation algorithm for k-SA. In the distributed setting, we present: (i) a (1+epsilon)*(k +1-r)/(1-r)-approximation algorithm for r-restricted k-SA, (ii) a (1+epsilon)(k+1)-approximation algorithm that uses a resource augmentation factor of 1+r for r-restricted k-SA, both for any constant epsilon>0, and (iii) an O{k^2}-approximation algorithm for k-SA (in expectation). The three distributed algorithms compute a solution with high probability and terminate in O(k^2 *log^3(n)) rounds

Online Budgeted Maximum Coverage

We study the Online Budgeted Maximum Coverage (OBMC) 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 reject the arriving set, and it may also drop previously accepted sets (preemption). Rejecting or dropping a set is irrevocable. The goal is to maximize the total weight of the elements covered by the sets in the chosen collection. We present a deterministic 4/(1-r)-competitive algorithm for OBMC, where r is the maximum ratio between the cost of a set and the total budget. Building on that algorithm, we then present a randomized O(1)-competitive algorithm for OBMC. On the other hand, we show that the competitive ratio of any deterministic online algorithm is Omega(1/(sqrt{1-r})). We also give a deterministic O(Delta)-competitive algorithm, where Delta is the maximum weight of a set (given that the minimum element weight is 1), and if the total weight of all elements, w(U), is known in advance, we show that a slight modification of that algorithm is O(min{Delta,sqrt{w(U)}})-competitive. A matching lower bound of Omega(min{Delta,sqrt{w(U)}}) is also given. Previous to the present work, only the unit cost version of OBMC was studied under the online setting, giving a 4-competitive algorithm [Saha, Getoor, 2009]. Finally, our results, including the lower bounds, apply to Removable Online Knapsack which is the preemptive version of the Online Knapsack problem

Set It and Forget It: Approximating the Set Once Strip Cover Problem

We consider the Set Once Strip Cover problem, in which n wireless sensors are deployed over a one-dimensional region. Each sensor has a fixed battery that drains in inverse proportion to a radius that can be set just once, but activated at any time. The problem is to find an assignment of radii and activation times that maximizes the length of time during which the entire region is covered. We show that this problem is NP-hard. Second, we show that RoundRobin, the algorithm in which the sensors simply take turns covering the entire region, has a tight approximation guarantee of 3/2 in both Set Once Strip Cover and the more general Strip Cover problem, in which each radius may be set finitely-many times. Moreover, we show that the more general class of duty cycle algorithms, in which groups of sensors take turns covering the entire region, can do no better. Finally, we give an optimal O(n^2 log n)-time algorithm for the related Set Radius Strip Cover problem, in which all sensors must be activated immediately.Comment: briefly announced at SPAA 201

Maximizing Barrier Coverage Lifetime with Mobile Sensors

Sensor networks are ubiquitously used for detection and tracking and as a result covering is one of the main tasks of such networks. We study the problem of maximizing the coverage lifetime of a barrier by mobile sensors with limited battery powers, where the coverage lifetime is the time until there is a breakdown in coverage due to the death of a sensor. Sensors are first deployed and then coverage commences. Energy is consumed in proportion to the distance traveled for mobility, while for coverage, energy is consumed in direct proportion to the radius of the sensor raised to a constant exponent. We study two variants which are distinguished by whether the sensing radii are given as part of the input or can be optimized, the fixed radii problem and the variable radii problem. We design parametric search algorithms for both problems for the case where the final order of the sensors is predetermined and for the case where sensors are initially located at barrier endpoints. In contrast, we show that the variable radii problem is strongly NP-hard and provide hardness of approximation results for fixed radii for the case where all the sensors are initially co-located at an internal point of the barrier

Efficiently Realizing Interval Sequences

We consider the problem of realizable interval-sequences. An interval sequence comprises of $n$ integer intervals $[a_i,b_i]$ such that $0\leq a_i \leq b_i \leq n-1$, and is said to be graphic/realizable if there exists a graph with degree sequence, say, $D=(d_1,\ldots,d_n)$ satisfying the condition $a_i \leq d_i \leq b_i$, for each $i \in [1,n]$. There is a characterisation (also implying an $O(n)$ verifying algorithm) known for realizability of interval-sequences, which is a generalization of the Erdos-Gallai characterisation for graphic sequences. However, given any realizable interval-sequence, there is no known algorithm for computing a corresponding graphic certificate in $o(n^2)$ time. In this paper, we provide an $O(n \log n)$ time algorithm for computing a graphic sequence for any realizable interval sequence. In addition, when the interval sequence is non-realizable, we show how to find a graphic sequence having minimum deviation with respect to the given interval sequence, in the same time. Finally, we consider variants of the problem such as computing the most regular graphic sequence, and computing a minimum extension of a length $p$ non-graphic sequence to a graphic one.Comment: 19 pages, 1 figur

The Generalized Microscopic Image Reconstruction Problem

This paper presents and studies a generalization of the microscopic image reconstruction problem (MIR) introduced by Frosini and Nivat [Andrea Frosini and Maurice Nivat, 2007; Nivat, 2002]. Consider a specimen for inspection, represented as a collection of points typically organized on a grid in the plane. Assume each point x has an associated physical value l_x, which we would like to determine. However, it might be that obtaining these values precisely (by a surgical probe) is difficult, risky, or impossible. The alternative is to employ aggregate measuring techniques (such as EM, CT, US or MRI), whereby each measurement is taken over a larger window, and the exact values at each point are subsequently extracted by computational methods. In this paper we extend the MIR framework in a number of ways. First, we consider a generalized setting where the inspected object is represented by an arbitrary graph G, and the vector l in R^n assigns a value l_v to each node v. A probe centered at a vertex v will capture a window encompassing its entire neighborhood N[v], i.e., the outcome of a probe centered at v is P_v = sum_{w in N[v]} l_w. We give a criterion for the graphs for which the extended MIR problem can be solved by extracting the vector l from the collection of probes, P^- = {P_v | v in V}. We then consider cases where such reconstruction is impossible (namely, graphs G for which the probe vector P is inconclusive, in the sense that there may be more than one vector l yielding P). Let us assume that surgical probes (whose outcome at vertex v is the exact value of l_v) are technically available to us (yet are expensive or risky, and must be used sparingly). We show that in such cases, it may still be possible to achieve reconstruction based on a combination of a collection of standard probes together with a suitable set of surgical probes. We aim at identifying the minimum number of surgical probes necessary for a unique reconstruction, depending on the graph topology. This is referred to as the Minimum Surgical Probing problem (MSP). Besides providing a solution for the above problems for arbitrary graphs, we also explore the range of possible behaviors of the Minimum Surgical Probing problem by determining the number of surgical probes necessary in certain specific graph families, such as perfect k-ary trees, paths, cycles, grids, tori and tubes

Minimum Neighboring Degree Realization in Graphs and Trees

We study a graph realization problem that pertains to degrees in vertex neighborhoods. The classical problem of degree sequence realizability asks whether or not a given sequence of n positive integers is equal to the degree sequence of some n-vertex undirected simple graph. While the realizability problem of degree sequences has been well studied for different classes of graphs, there has been relatively little work concerning the realizability of other types of information profiles, such as the vertex neighborhood profiles. In this paper we introduce and explore the minimum degrees in vertex neighborhood profile as it is one of the most natural extensions of the classical degree profile to vertex neighboring degree profiles. Given a graph G = (V,E), the min-degree of a vertex v ? V, namely MinND(v), is given by min{deg(w) ? w ? N[v]}. Our input is a sequence ? = (d_?^{n_?}, ?d?^{n?}), where d_{i+1} > d_i and each n_i is a positive integer. We provide some necessary and sufficient conditions for ? to be realizable. Furthermore, under the restriction that the realization is acyclic, i.e., a tree or a forest, we provide a full characterization of realizable sequences, along with a corresponding constructive algorithm. We believe our results are a crucial step towards understanding extremal neighborhood degree relations in graphs