Propose Approximation Algorithms Research Articles

Optimal Pricing for Submodular Valuations with Bounded Curvature

The optimal pricing problem is a fundamental problem that arises in combinatorial auctions. Suppose that there is one seller who has indivisible items and multiple buyers who want to purchase a combination of the items. The seller wants to sell his items for the highest possible prices, and each buyer wants to maximize his utility (i.e., valuation minus payment) as long as his payment does not exceed his budget. The optimal pricing problem seeks a price of each item and an assignment of items to buyers such that every buyer achieves the maximum utility under the prices. The goal of the problem is to maximize the total payment from buyers. In this paper, we consider the case that the valuations are submodular. We show that the problem is computationally hard even if there exists only one buyer. Then we propose approximation algorithms for the unlimited budget case. We also extend the algorithm for the limited budget case when there exists one buyer and multiple buyers collaborate with each other.

Approximation algorithms on consistent dynamic map labeling

We consider the dynamic map labeling problem: given a set of rectangular labels on the map, the goal is to appropriately select visible ranges for all the labels such that no two consistent labels overlap at every scale and the sum of total visible ranges is maximized. This is also called the active range optimization (ARO) problem defined by Been et al. (2006) [2]. We propose approximation algorithms for several variants of this problem. For the simple ARO problem, we provide a 3clog⁡n-approximation algorithm for unit-width rectangular labels if there is a c-approximation algorithm for the unit-width label placement problem in the plane; and a randomized polynomial-time O(log⁡nlog⁡log⁡n)-approximation algorithm for arbitrary rectangular labels. For the general ARO problem, we prove that it remains NP-complete even for congruent square labels with equal selectable scale range. Moreover, we contribute 12-approximation algorithms for both arbitrary square labels and unit-width rectangular labels, and a 6-approximation algorithm for congruent square labels, and show that the bounds are tight.

Approximation Algorithms for Energy, Reliability, and Makespan Optimization Problems

We consider the problem of scheduling an application on a parallel computational platform. The application is a particular task graph, either a linear chain of tasks, or a set of independent tasks. The platform is made of identical processors, whose speed can be dynamically modified. It is also subject to failures: if a processor is slowed down to decrease the energy consumption, it has a higher chance to fail. Therefore, the scheduling problem requires us to re-execute or replicate tasks (i.e., execute twice the same task, either on the same processor, or on two distinct processors), in order to increase the reliability. It is a tri-criteria problem: the goal is to minimize the energy consumption, while enforcing a bound on the total execution time (the makespan), and a constraint on the reliability of each task. Our main contribution is to propose approximation algorithms for linear chains of tasks and independent tasks. For linear chains, we design a fully polynomial-time approximation scheme. However, we show that there exists no constant factor approximation algorithm for independent tasks, unless P=NP, and we propose in this case an approximation algorithm with a relaxation on the makespan constraint.

Improved approximation algorithms for some min-max and minimum cycle cover problems

Given an undirected weighted graph G=(V,E), a set {C1,C2,…,Ck} of cycles is called a cycle cover of the vertex subset V′ if V′⊆∪i=1kV(Ci) and its cost is given by the maximum weight of the cycles. The Min-Max Cycle Cover Problem (MMCCP) is to find a minimum cost cycle cover of V with at most k cycles. The Rooted Min-Max Cycle Cover Problem (RMMCCP) is to find a minimum cost cycle cover of V∖D with at most k cycles, each of which contains one vertex in D. The Minimum Cycle Cover Problem (MCCP) aims to find a cycle cover of V of cost at most λ with the minimum number of cycles. We propose approximation algorithms for MMCCP and RMMCCP with performance ratios 5 and 6, respectively. These results improve the previous algorithms in term of both approximation ratios and running times. For MCCP we obtain a 143-approximation algorithm that has the same time complexity as the previous best 5-approximation algorithm. Moreover, we transform a ρ-approximation algorithm for TSP into approximation algorithms for MMCCP, RMMCCP and MCCP with ratios 4ρ, 4ρ+1 and 4ρ, respectively.

Scheduling a single machine with parallel batching to minimize makespan and total rejection cost

We consider the problem of scheduling a set of n jobs on a single machine with parallel batching and with rejection being allowed. Two bi-criteria problems are considered: (a) minimize the makespan subject to the constraint that the total rejection cost does not exceed a given threshold, and (b) minimize the total rejection cost subject to the constraint that the makespan does not exceed a given threshold. For the case of a batching machine with infinite capacity (i.e., the batch size allowed on the machine is larger than or equal to the number of jobs), we assume that the jobs have release dates. We present an O(n2)-time 2-approximation algorithm for problem (a) and, in addition, we present dynamic programming algorithms and fully polynomial-time approximation schemes for both problems (a) and (b). For the case of a batching machine with finite capacity (i.e., the batch size allowed on the machine is less than the number of jobs), we assume that the jobs have identical release dates. We propose approximation algorithms for (a) and present dynamic programming algorithms and fully polynomial-time approximation schemes for both problems (a) and (b).

Optimal Association in Wireless Mesh Networks

The wireless multihop backhaul is a unique feature of wireless mesh networks (WMNs), which necessitates redesign of association control algorithms. In this paper, we formulate and propose approximation algorithms for the problem of optimal joint association and bandwidth allocation in WMNs, considering max–min fairness (MM) and proportional fairness (PF) objectives. We first relax the integral association constraint and get an optimal fractional association solution. Then, we propose two rounding algorithms, namely, largest fraction rounding and bipartite graph rounding, to get an integral solution and analyze their theoretical approximation ratios. Finally, we propose two approximation ratio improvement algorithms so that the improved approximation ratio can more accurately reflect the true performance gap between the produced solution and the optimal one. Our simulation results show that the proposed algorithms achieve performance that is close to the optimal and outperform popular heuristic algorithms. We also compare the performance of PF and MM in WMNs in terms of network throughput and fairness in user bandwidth. Finally, we compare the performances of the proposed rounding algorithms and show that the approximation ratio can be reduced to 1–2 by the proposed ratio improvement algorithms. Therefore, our proposed algorithm is able to achieve nearly optimal association control, as well as bandwidth allocation, considering MM or PF, with small approximation ratios, in WMNs.

Minimizing the number of mobile chargers for large-scale wireless rechargeable sensor networks

Traditional wireless sensor networks (WSNs) are constrained by limited battery energy that powers the sensor nodes, which impedes the large-scale deployment of WSNs. Wireless power transfer technology provides a promising way to solve this problem. With such novel technology, recent works propose to use a single mobile charger (MC) traveling through the network fields to replenish energy to every sensor node so that none of the nodes will run out of energy. These algorithms work well in small-scale networks. In large-scale networks, these algorithms, however, do not work efficiently, especially when the amount of energy the MC can provide is limited. To address this issue, multiple MCs can be used. In this paper, we investigate the minimum MCs problem (MinMCP) for two-dimensional (2D) wireless rechargeable sensor networks (WRSNs), i.e., how to find the minimum number of energy-constrained MCs and design their recharging routes in a 2D WRSN such that each sensor node in the network maintains continuous work, assuming that the energy consumption rate for all sensor nodes are identical. By reduction from the Distance Constrained Vehicle Routing Problem (DVRP), we prove that MinMCP is NP-hard. Then we propose approximation algorithms for this problem. Finally, we conduct extensive simulations to validate the effectiveness of our algorithms.

The Approximability of Shortest Path-Based Graph Orientations of Protein–Protein Interaction Networks

The graph orientation problem calls for orienting the edges of an undirected graph so as to maximize the number of prespecified source-target vertex pairs that admit a directed path from the source to the target. Most algorithmic approaches to this problem share a common preprocessing step, in which the input graph is reduced to a tree by repeatedly contracting its cycles. Although this reduction is valid from an algorithmic perspective, the assignment of directions to the edges of the contracted cycles becomes arbitrary and, consequently, the connecting source-target paths may be arbitrarily long. In the context of biological networks, the connection of vertex pairs via shortest paths is highly motivated, leading to the following variant: Given an undirected graph and a collection of source-target vertex pairs, assign directions to the edges so as to maximize the number of pairs that are connected by a shortest (in the original graph) directed path. Here we study this variant, provide strong inapproximability results for it, and propose approximation algorithms for the problem, as well as for relaxations where the connecting paths need only be approximately shortest.

On Maximizing the Lifetime of Wireless Sensor Networks Using Virtual Backbone Scheduling

Wireless Sensor Networks (WSNs) are key for various applications that involve long-term and low-cost monitoring and actuating. In these applications, sensor nodes use batteries as the sole energy source. Therefore, energy efficiency becomes critical. We observe that many WSN applications require redundant sensor nodes to achieve fault tolerance and Quality of Service (QoS) of the sensing. However, the same redundancy may not be necessary for multihop communication because of the light traffic load and the stable wireless links. In this paper, we present a novel sleep-scheduling technique called Virtual Backbone Scheduling (VBS). VBS is designed for WSNs has redundant sensor nodes. VBS forms multiple overlapped backbones which work alternatively to prolong the network lifetime. In VBS, traffic is only forwarded by backbone sensor nodes, and the rest of the sensor nodes turn off their radios to save energy. The rotation of multiple backbones makes sure that the energy consumption of all sensor nodes is balanced, which fully utilizes the energy and achieves a longer network lifetime compared to the existing techniques. The scheduling problem of VBS is formulated as the Maximum Lifetime Backbone Scheduling (MLBS) problem. Since the MLBS problem is NP-hard, we propose approximation algorithms based on the Schedule Transition Graph (STG) and Virtual Scheduling Graph (VSG). We also present an Iterative Local Replacement (ILR) scheme as a distributed implementation. Theoretical analyses and simulation studies verify that VBS is superior to the existing techniques.

Fast algorithms for min independent dominating set

We first devise a branching algorithm that computes a minimum independent dominating set with running time O∗(1.3351n)=O∗(20.417n) and polynomial space. This improves upon the best state of the art algorithms for this problem. We then study approximation of the problem by moderately exponential time algorithms and show that it can be approximated within ratio 1+ϵ, for any ϵ>0, in a time smaller than the one of exact computation and exponentially decreasing with ϵ. We also propose approximation algorithms with better running times for ratios greater than 3 in general graphs and give improved moderately exponential time approximation results in triangle-free and bipartite graphs. These latter results are based upon a new bound on the number of maximal independent sets of a given size in these graphs, which is a result interesting per se.

Decentralized Opportunistic Navigation Strategies for Multi-agent Systems in the Presence of an Adversary

AbstractWe investigate a jamming attack on the communication network of a multi-agent system in a formation. We propose a communication and motion model for the agents. The communication model provides a relation in the spatial domain for effective jamming by an intruder. We formulate the problem as a zero-sum pursuit-evasion game. In our earlier work we used Isaacs’ approach to obtain motion strategies for a network of agents to evade the jamming attack.In this work, we imagine a scenario in which each agent has a knowledge about the value function under perfect state information, beforehand. Due to lack of information about all the agents in the team each agent is constrained to make a local decision based on the information about his neighbors. We propose on-line algorithms under decentralized information patterns which converge for each player. We propose approximation algorithms for the agents, based on averaging and provide some bounds on their performance.

Approximation algorithms for homogeneous polynomial optimization with quadratic constraints

In this paper, we consider approximation algorithms for optimizing a generic multi-variate homogeneous polynomial function, subject to homogeneous quadratic constraints. Such optimization models have wide applications, e.g., in signal processing, magnetic resonance imaging (MRI), data training, approximation theory, and portfolio selection. Since polynomial functions are non-convex, the problems under consideration are all NP-hard in general. In this paper we shall focus on polynomial-time approximation algorithms. In particular, we first study optimization of a multi-linear tensor function over the Cartesian product of spheres. We shall propose approximation algorithms for such problem and derive worst-case performance ratios, which are shown to be dependent only on the dimensions of the model. The methods are then extended to optimize a generic multi-variate homogeneous polynomial function with spherical constraint. Likewise, approximation algorithms are proposed with provable approximation performance ratios. Furthermore, the constraint set is relaxed to be an intersection of co-centered ellipsoids; namely, we consider maximization of a homogeneous polynomial over the intersection of ellipsoids centered at the origin, and propose polynomial-time approximation algorithms with provable worst-case performance ratios. Numerical results are reported, illustrating the effectiveness of the approximation algorithms studied.

IMPROPER COLORING OF WEIGHTED GRID AND HEXAGONAL GRAPHS

We study a weighted improper coloring problem motivated by a frequency allocation problem. It consists of associating to each vertex a set of p(v) (weight) distinct colors (frequencies), such that the set of vertices having a given color induces a graph of degree at most k (the case k = 0 corresponds to proper coloring). The objective is to minimize the number of colors. We propose approximation algorithms to compute such a coloring for general graphs. We apply these to obtain good approximation ratio for grid and hexagonal graphs. Furthermore we give exact results for the 2-dimensional grid and the triangular lattice when the weights are all the same.

Energy‐constrained mobile sensor with motion plans for monitoring stochastic events

AbstractWith the development of robotics and embedded system, utilizing mobile sensors to capture stochastic events is emerging as a promising method to monitor a region of interest (RoI). In previous work, the quality of monitoring (QoM) is evaluated based on the stochastic events capture without taking the energy of motion into consideration. Since sensor nodes are normally constrained by limited energy capability, it is desirable to guide the mobile sensor in an energy‐efficient motion to capture events information. In this paper, by analyzing the different kinds of surveillance that may result in different required QoM, we obtain the expected Information captured Per unit of Energy consumption (IPE), which is a function with multiple parameters including the event type, the event dynamics, and the velocity of the mobile sensor. Our analysis is based on a realistic energy model of motion, and can achieve suboptimal motion plans by adopting existing typical approximation algorithm, and thus enable the sensor velocity to be optimized for capturing stochastic events information. We propose approximation algorithms to enable the tradeoff between the computation and efficiency, which make motion plans more practical in some realistic scenarios. The efficiency and effectiveness of proposed algorithm are validated by the extensive simulations. Copyright © 2009 John Wiley & Sons, Ltd.

Reliable Routings in Networks With Generalized Link Failure Events

We study routing problems in networks that require guaranteed reliability against multiple correlated link failures. We consider two different routing objectives: The first ensures ldquolocal reliability,rdquo i.e., the goal is to route so that each connection in the network is as reliable as possible. The second ensures ldquoglobal reliability,rdquo i.e., the goal is to route so that as few as possible connections are affected by any possible failure. We exhibit a trade-off between the two objectives and resolve their complexity and approximability for several classes of networks. Furthermore, we propose approximation algorithms and heuristics. We perform experiments to evaluate the heuristics against optimal solutions that are obtained using an integer linear programming solver. We also investigate up to what degree the routing trade-offs occur in randomly generated instances.

Data replication optimisation in grid delivery network

In this paper, we examine the data replication problem in a particular Grid Delivery Network (GDN) which is a system that provides video services, among which is Video On Demand (VOD). In this system, the data are divided into fixed size blocks which must be replicated on hosts to decrease the total download time. We propose a probabilistic model to optimise the average download time of requests based on the host's availability and the document size distribution. The objective function induced by this model is a non-linear integer problem. It can be solved in real values by Lagrangian optimisation. We prove that in a particular case, this problem can be reduced to a knapsack problem. We propose approximation algorithms and validate them using simulations with varying characteristics.

Proportional fairness in multi-carrier system with multi-slot frames: upper bound and user multiplexing algorithms

Optimal Proportional Fair Scheduling (PFS) in a multi-carrier system is a prohibitively complex combinatorial problem. In this paper we consider practical time frames with multiple time slots, where this optimal allocation becomes even more complex. Therefore, we derive bounds for the optimal proportional fair allocation, by means of convex optimization, and propose approximation algorithms where several users can be time-multiplexed on a same subchannel. With a much lower complexity than the optimal allocation, these algorithms achieve an excellent tradeoff between throughput and proportional fairness, even with the increased signaling overhead.

ALGORITHMIC APPROACHES TO SELECTING CONTROL CLONES IN DNA ARRAY HYBRIDIZATION EXPERIMENTS

We study the problem of selecting control clones in DNA array hybridization experiments. The problem arises in the OFRG method for analyzing microbial communities. The OFRG method performs classification of rRNA gene clones using binary fingerprints created from a series of hybridization experiments, where each experiment consists of hybridizing a collection of arrayed clones with a single oligonucleotide probe. This experiment produces analog signals, one for each clone, which then need to be classified, that is, converted into binary values 1 and 0 that represent hybridization and non-hybridization events. In addition to the sample rRNA gene clones, the array contains a number of control clones needed to calibrate the classification procedure of the hybridization signals. These control clones must be selected with care to optimize the classification process. We formulate this as a combinatorial optimization problem called Balanced Covering. We prove that the problem is ℕℙ-hard, and we show some results on hardness of approximation. We propose approximation algorithms based on randomized rounding, and we show that, with high probability, our algorithms approximate well the optimum solution. The experimental results confirm that the algorithms find high quality control clones. The algorithms have been implemented and are publicly available as part of the software package called CloneTools.

Computing the Types of the Relationships Between Autonomous Systems

We investigate the problem of computing the types of the relationships between Internet Autonomous Systems. We refer to the model introduced by Gao [IEEE/ACM Transactions on Networking, 9(6):733-645, 2001] and Subramanian (IEEE Infocom, 2002) that bases the discovery of such relationships on the analysis of the AS paths extracted from the BGP routing tables. We characterize the time complexity of the above problem, showing both NP-completeness results and efficient algorithms for solving specific cases. Motivated by the hardness of the general problem, we propose approximation algorithms and heuristics based on a novel paradigm and show their effectiveness against publicly available data sets. The experiments provide evidence that our algorithms perform significantly better than state-of-the-art heuristics

Fixed topology alignment with recombination

In this paper, we study a new version of multiple sequence alignment, fixed topology alignment with recombination. We show that it cannot be approximated within any constant ratio unless P=NP. For a restricted version, we show that the problem is MAX-SNP-hard. This implies that there is no PTAS for this version unless P=NP. We also propose approximation algorithms for a special case, where each internal node has at most one recombination child and any two merge paths for different recombination nodes do not share any common node.