Provide Approximation Algorithms Research Articles

Randomized Strategies for Robust Combinatorial Optimization with Approximate Separation

In this paper, we study the following robust optimization problem. Given a set family representing feasibility and candidate objective functions, we choose a feasible set, and then an adversary chooses one objective function, knowing our choice. The goal is to find a randomized strategy (i.e., a probability distribution over the feasible sets) that maximizes the expected objective value in the worst case. This problem is fundamental in wide areas such as artificial intelligence, machine learning, game theory, and optimization. To solve the problem, we provide a general framework based on the dual linear programming problem. In the framework, we utilize the ellipsoid algorithm with the approximate separation algorithm. We prove that there exists an α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-approximation algorithm for our robust optimization problem if there exists an α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-approximation algorithm for finding a (deterministic) feasible set that maximizes a nonnegative linear combination of the candidate objective functions. Using our result, we provide approximation algorithms for the max–min fair randomized allocation problem and the maximum cardinality robustness problem with a knapsack constraint.

Single machine robust scheduling with budgeted uncertainty

We consider robust single machine scheduling problems. First, we prove that with uncertain processing times, minimizing the number of tardy jobs is NP-hard. Second, we show that the weighted variant of the problem has the same complexity as the nominal counterpart whenever only the weights are uncertain. Last, we provide approximation algorithms for the problems minimizing the weighted sum of completion times. Noticeably, our algorithms extend to more general robust combinatorial optimization problems with cost uncertainty, such as max-cut.

Pricing Problems with Buyer Preselection

We investigate the problem of preselecting a subset of buyers (also called agents) participating in a market so as to optimize the performance of stable outcomes. We consider four scenarios arising from the combination of two stability notions, namely market envy-freeness and agent envy-freeness, with the two state-of-the-art objective functions of social welfare and seller’s revenue. When insisting on market envy-freeness, we prove that the problem cannot be approximated within n 1−ε (with n being the number of buyers) for any ε > 0, under both objective functions; we also provide approximation algorithms with an approximation ratio tight up to subpolynomial multiplicative factors for social welfare and the seller’s revenue. The negative result, in particular, holds even for markets with single-minded buyers. We also prove that maximizing the seller’s revenue is NP-hard even for a single buyer, thus closing a previous open question. Under agent envy-freeness and for both objective functions, instead, we design a polynomial time algorithm transforming any stable outcome for a market involving any subset of buyers into a stable outcome for the whole market without worsening its performance. This result creates an interesting middle-ground situation where, if on the one hand buyer preselection cannot improve the performance of agent envy-free outcomes, on the other one it can be used as a tool for simplifying the combinatorial structure of the buyers’ valuation functions in a given market. Finally, we consider the restricted case of multi-unit markets, where all items are of the same type and are assigned the same price. For these markets, we show that preselection may improve the performance of stable outcomes in all of the four considered scenarios, and design corresponding approximation algorithms.

New approximation algorithms for the heterogeneous weighted delivery problem

We study the heterogeneous weighted delivery (HWD) problem introduced in [Bärtschi et al., STACS'17] where k heterogeneous mobile agents (e.g., robots, vehicles, etc.), initially positioned on vertices of an n-vertex edge-weighted graph G, have to deliver m messages. Each message is initially placed on a source vertex of G and needs to be delivered to a target vertex of G. Each agent can move along the edges of G and carry at most one message at any time. Each agent has a rate of energy consumption per unit of traveled distance and the goal is that of delivering all messages using the minimum overall amount of energy.This problem has been shown to be NP-hard even when k=1, and is 4ρ-approximable where ρ is the ratio between the maximum and minimum energy consumption of the agents. In this paper, we provide approximation algorithms with approximation ratios independent of the energy consumption rates. First, we design a polynomial-time 8-approximation algorithm for k=O(log⁡n), closing a problem left open in [Bärtschi et al., ATMOS'17]. This algorithm can be turned into an O(k)-approximation algorithm that always runs in polynomial-time, regardless of the values of k. Then, we show that HWD problem is 36-approximable in polynomial-time when each agent has one of two possible consumption rates. Finally, we design a polynomial-time O˜(log3⁡n)-approximation algorithm for the general case.

Inequity aversion pricing over social networks: Approximation algorithms and hardness results

We study a revenue maximization problem in the context of social networks. Namely, we generalize a model introduced by Alon, Mansour, and Tennenholtz [2] that captures inequity aversion, i.e., it captures the fact that prices offered to neighboring nodes should not differ significantly. We first provide approximation algorithms for a natural class of instances, where the total revenue is the sum of single-value revenue functions. Our results improve on the current state of the art, especially when the number of distinct prices is small. This applies, for instance, to settings where the seller will only consider a fixed number of discount types or special offers. To complement our positive results, we resolve one of the open questions posed in [2] by establishing APX-hardness for the problem. Surprisingly, we further show that the problem is NP-complete even when the price differences are allowed to be large, or even when the number of allowed distinct prices is as small as three. Finally, we study extensions of the model regarding the demand type of the clients.

Randomized Strategies for Robust Combinatorial Optimization

In this paper, we study the following robust optimization problem. Given an independence system and candidate objective functions, we choose an independent set, and then an adversary chooses one objective function, knowing our choice. The goal is to find a randomized strategy (i.e., a probability distribution over the independent sets) that maximizes the expected objective value in the worst case. This problem is fundamental in wide areas such as artificial intelligence, machine learning, game theory and optimization. To solve the problem, we propose two types of schemes for designing approximation algorithms. One scheme is for the case when objective functions are linear. It first finds an approximately optimal aggregated strategy and then retrieves a desired solution with little loss of the objective value. The approximation ratio depends on a relaxation of an independence system polytope. As applications, we provide approximation algorithms for a knapsack constraint or a matroid intersection by developing appropriate relaxations and retrievals. The other scheme is based on the multiplicative weights update (MWU) method. The direct application of the MWU method does not yield a strict multiplicative approximation algorithm but yield one with an additional additive error term. A key technique to overcome the issue is to introduce a new concept called (η,γ)-reductions for objective functions with parameters η and γ. We show that our scheme outputs a nearly α-approximate solution if there exists an α-approximation algorithm for a subproblem defined by (η,γ)-reductions. This improves approximation ratios in previous results. Using our result, we provide approximation algorithms when the objective functions are submodular or correspond to the cardinality robustness for the knapsack problem.

Approximating maximum uniquely restricted matchings in bipartite graphs

A matching in a graph is uniquely restricted if no other matching covers exactly the same set of vertices. This notion was defined by Golumbic, Hirst, and Lewenstein (2001) and studied in a number of articles. We provide approximation algorithms for computing a uniquely restricted matching of maximum size in some bipartite graphs, namely those excluding a C4 or with maximum degree at most three. In particular, we achieve a ratio of 5∕9 for subcubic bipartite graphs, improving over a 1∕2-approximation algorithm proposed by Mishra (2011).

Robust scheduling with budgeted uncertainty

In this work we study minmax robust scheduling problems assuming that the processing times can take any value in the budgeted uncertainty set introduced by Bertsimas and Sim (2003, 2004). We consider problems on a single machine that minimize the (weighted and unweighted) sum of completion times and problems that minimize the makespan on parallel and unrelated machines. We provide approximation algorithms: constant factor, average non-constant factor, (fully or not) polynomial time approximation schemes. In addition, we prove that the robust version of minimizing the weighted completion time on a single machine is NP-hard in the strong sense.

Improvements and Generalizations of Stochastic Knapsack and Markovian Bandits Approximation Algorithms

We study the multi-armed bandit problem with arms which are Markov chains with rewards. In the finite-horizon setting, the celebrated Gittins indices do not apply, and the exact solution is intractable. We provide approximation algorithms for the general model of Markov decision processes with nonunit transition times. When preemption is allowed, we provide a (1/2 − ε)-approximation, along with an example showing this is tight. When preemption isn’t allowed, we provide a 1/12-approximation, which improves to a 4/27-approximation when transition times are unity. Our model captures the Markovian Bandits model of Gupta et al., the Stochastic Knapsack model of Dean et al., and the Budgeted Learning model of Guha and Munagala. Our algorithms improve existing results in all three areas. In our analysis, we encounter and overcome to our knowledge a new obstacle: an algorithm that provably exists via analytical arguments, but cannot be found in polynomial time.

A generalized approximation framework for fractional network flow and packing problems

We generalize the fractional packing framework of Garg and Koenemann (SIAM J Comput 37(2):630–652, 2007) to the case of linear fractional packing problems over polyhedral cones. More precisely, we provide approximation algorithms for problems of the form $$\max \{c^T x : Ax \le b, x \in C \}$$ , where the matrix A contains no negative entries and C is a cone that is generated by a finite set S of non-negative vectors. While the cone is allowed to require an exponential-sized representation, we assume that we can access it via one of three types of oracles. For each of these oracles, we present positive results for the approximability of the packing problem. In contrast to other frameworks, the presented one allows the use of arbitrary linear objective functions and can be applied to a large class of packing problems without much effort. In particular, our framework instantly allows to derive fast and simple fully polynomial-time approximation algorithms (FPTASs) for a large set of network flow problems, such as budget-constrained versions of traditional network flows, multicommodity flows, or generalized flows. Some of these FPTASs represent the first ones of their kind, while others match existing results but offer a much simpler proof.

On the rectangle escape problem

Motivated by the bus escape routing problem in printed circuit boards, we study the following rectangle escape problem: given a set S of n axis-aligned rectangles inside an axis-aligned rectangular region R, extend each rectangle in S toward one of the four borders of R so that the maximum density over the region R is minimized. The density of each point p∈R is defined as the number of extended rectangles containing p. We show that the problem is hard to approximate to within a factor better than 3/2 in general. When the optimal density is sufficiently large, we provide a randomized algorithm that achieves an approximation factor of 1+ε with high probability improving over the current best 4-approximation algorithm available for the problem. When the optimal density is one, we develop an exact algorithm that finds an optimal solution efficiently. We also provide approximation algorithms and inapproximability results for a restricted version of the problem where rectangles are allowed to escape toward only a subset of directions.

Orthology Correction for Gene Tree Reconstruction: Theoretical and Experimental Results

We consider how the orthology/paralogy information can be corrected in order to represent a gene tree, a problem that has recently gained interest in phylogenomics. Interestingly, the problem is related to the Minimum CoGraph Editing problem on the relation graph that represents orthology/paralogy information, where we want to minimize the number of edit operations on the given relation graph in order to obtain a cograph. In this paper we provide both theoretical and experimental results on the Minimum CoGraph Editing problem. On the theoretical side, we provide approximation algorithms for bounded degree relation graphs, for the general problem and for the problem restricted to deletion of edges. On the experimental side, we present a genetic algorithm for Minimum CoGraph Editing and we provide an experimental evaluation of the genetic algorithm on synthetic data.

Algorithms to Find Two-Hop Routing Policies in Multiclass Delay Tolerant Networks

Most of the literature on delay tolerant networks (DTNs) focuses on optimal routing policies exploiting a priori knowledge about nodes mobility traces. For the case in which no a priori knowledge is available (very common in practice), apart from basic epidemic routing, the main approaches focus on controlling two-hop routing policies. However, these latter results commonly employ fluid approximation techniques, which, in principle, do not provide any theoretical bound over the approximation ratio. In our work, we focus on the case without a priori mobility knowledge and we provide approximation algorithms with theoretical guarantees that can be applied to cases where the number of hops allowed in the routing process is arbitrary. Our approach is rather flexible allowing us to address heterogeneous mobility patterns and transmission technologies, to consider explicitly the signaling and transmission costs, and to include also nodes discarding packets after a local timeout. We then provide a comprehensive performance evaluation of our algorithms, showing that two-hop routing provides the best tradeoff between delay and energy and that, in this case, they find solutions very close to the optimal ones with a low overhead. Finally, we compare our methods against some state-of-the-art approaches by means of a DTN simulation environment in realistic settings.

Auctioning Time

We consider the problem of auctioning time - a one-dimensional continuously-divisible heterogeneous good - among multiple agents. Applications include auctioning time for using a shared device, auctioning TV commercial slots, and more. Different agents may have different valuations for the different possible intervals; the goal is to maximize the aggregate utility. Agents are self-interested and may misrepresent their true valuation functions if this benefits them. Thus, we seek auctions that are truthful . Considering the case that each agent may obtain a single interval, the challenge is twofold, as we need to determine both where to slice the interval, and who gets what slice. We consider two settings: discrete and continuous. In the discrete setting, we are given a sequence of m indivisible elements ( e 1 , …, e m ), and the auction must allocate each agent a consecutive subsequence of the elements. In the continuous setting, we are given a continuous, infinitely divisible interval, and the auction must allocate each agent a subinterval. The agents’ valuations are nonatomic measures on the interval. We show that, for both settings, the associated computational problem is NP-complete even under very restrictive assumptions. Hence, we provide approximation algorithms. For the discrete case, we provide a truthful auctioning mechanism that approximates the optimal welfare to within a log m factor. The mechanism works for arbitrary monotone valuations. For the continuous setting, we provide a truthful auctioning mechanism that approximates the optimal welfare to within an O (log n ) factor (where n is the number of agents). Additionally, we provide a truthful 2-approximation mechanism for the case that all pieces must be of some fixed size.

Using data mules for sensor network data recovery

In this paper, we study the problem of efficient data recovery using the data mules approach, where a set of mobile sensors with advanced mobility capabilities re-acquire lost data by visiting the neighbors of failed sensors, thereby avoiding permanent data loss in the network. Our approach involves defining the optimal communication graph and mules’ placements such that the overall traveling time and distance is minimized regardless to which sensors crashed. We explore this problem under different practical network topologies such as arbitrary graphs, grids and random linear networks and provide approximation algorithms based on multiple combinatorial techniques. Simulation experiments demonstrate that our algorithms outperform various competitive solutions for different network models, and that they are applicable for practical scenarios.

Region‐based fault‐tolerant distributed file storage system design in networks

Distributed storage of data files in different nodes of a network enhances its fault tolerance capability by offering protection against node and link failures. Reliability is often achieved through redundancy in one of the following two ways: (i) storage of multiple copies of the entire file at different locations (nodes) or (ii) storage of file segments (not entire files) at different node locations. In the file distribution scheme, file segments from a file are created in such a way that it is possible to reconstruct the entire file, just by accessing any segments. For the reconstruction scheme to work, it is essential that the segments of the file are stored in nodes that are connected in the network. However, in the event of node/link failures, the network might become disconnected (i.e., split into several connected components). We focus on node failures that are spatially correlated or region based. Such failures are often encountered in disaster situations or natural calamities where only the nodes in the disaster zone are affected. The first goal of this research is to design a least cost file storage scheme to ensure that no matter which region is destroyed; resulting in fragmentation of the network, a largest connected component of the residual network will have enough file segments with which to reconstruct the entire file. In case the least cost to ensure this objective is within the allocated budget, the storage design will be all region fault‐tolerant (ARFT). In case the least cost exceeds the allocated budget, design of an ARFT file storage system design is impossible. The second goal of this research is to design file storage schemes that will be maximum region fault‐tolerant within the allocated budget. The third goal of this research is to investigate the impact of the coding parameters and on storage requirements for ensuring all region or \textit{maximum region} fault‐tolerant design. We provide maximum region fault‐tolerant design. We provide approximation algorithms for the problems and evaluate their performance through simulation using two real networks and compare their results to the optimal solutions obtained using Integer Linear Program. The simulation results demonstrate that the approximation algorithms almost always produce near optimal results in a fraction of the time needed to find the optimal solution. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 66(4), 380–395 2015

Fault-tolerant design of wireless sensor networks with directional antennas

A tree structure is often used in wireless sensor networks to deliver sensor data to a sink node. Such a tree can be built using directional antennas as they offer considerable advantage over the omni-directional ones. A tree is adequate for data gathering from all sensor nodes if no node fails. We study the problem of enhancing the fault tolerance of a data gathering tree by adding additional links so that failure of a sensor or a pair of adjacent sensors would not disconnect the tree. We prove that the least-cost tree augmentation problem is NP-complete and provide approximation algorithms one for single node failure and the other for a pair of adjacent node failure, with performance bounds of two and four respectively.

A dynamic edge covering and scheduling problem: complexity results and approximation algorithms

We consider a new dynamic edge covering and scheduling problem that focuses on assigning resources to nodes in a network to minimize the amount of time required to process all edges in it. Resources need to be co-located at the endpoints of an edge for it to be processed and, therefore, this problem contains both edge covering and scheduling decisions. These new problems have motivating applications in traffic systems and military intelligence operations. We provide complexity results for the dynamic edge covering and scheduling problem over different types of networks. We then show that existing approximation algorithms for parallel machine scheduling problems can be leveraged to provide approximation algorithms for this new class of problems over certain types of networks.

TOWARDS MORE EFFICIENT INFECTION AND FIRE FIGHTING

The firefighter problem models the situation where an infection, a computer virus, an idea or fire etc. is spreading through a network and the goal is to save as many as possible nodes of the network through targeted vaccinations. The number of nodes that can be vaccinated at a single time-step is typically one, or more generally O(1). In a non-standard model, the so called spreading model, the vaccinations also spread in contrast to the standard model. Our main results are concerned with general graphs in the spreading model. We provide a very simple exact [Formula: see text]-time algorithm. In the special case of trees, where the standard and spreading model are equivalent, our algorithm is substantially simpler than that exact subexponential algorithm for trees presented in Ref. 2. On the other hand, we show that the firefighter problem on weighted directed graphs in the spreading model cannot be approximated within a constant factor better than 1 − 1/e unless NP ⊆ DTIME (nO( log log n)). We also present several results in the standard model. We provide approximation algorithms for planar graphs in case when at least two vaccinations can be performed at a time-step. We also derive trade-offs between approximation factors for polynomial-time solutions and the time complexity of exact or nearly exact solutions for instances of the firefighter problem for the so called directed layered graphs.

WEIGHTED GEOMETRIC SET COVER PROBLEMS REVISITED

We study several set cover problems in low dimensional geometric settings. Specifically, we describe a PTAS for the problem of computing a minimum cover of given points by a set of weighted fat objects. Here, we allow the objects to expand by some prespecified δ-fraction of their diameter. Next, we show that the problem of computing a minimum weight cover of points by weighted halfplanes (without expansion) can be solved exactly in the plane. We also study the problem of covering ℝ d by weighted halfspaces, and provide approximation algorithms and hardness results. We also investigate the dual settings of computing minimum weight simplex that covers a given target point. Finally, we provide a near linear time algorithm for the problem of solving a LP minimizing the total weight of violated constraints needed to be removed to make it feasible.