Moran Feldman

A Tight Linear Time (1/2)-Approximation for Unconstrained Submodular Maximization

We consider the Unconstrained Submodular Maximization problem in which we are given a non-negative submodular function f : 2N → ℝ+, and the objective is to find a subset S ⊆ N maximizing f(S). This is one of the most basic submodular optimization problems, having a wide range of applications. Some well known problems captured by Unconstrained Submodular Maximization include MaxCut, Max-DiCut, and variants of Max-SAT and maximum facility location. We present a simple randomized linear time algorithm achieving a tight approximation guarantee of 1/2, thus matching the known hardness result of Feige et al. [11]. Our algorithm is based on an adaptation of the greedy approach which exploits certain symmetry properties of the problem. Our method might seem counterintuitive, since it is known that the greedy algorithm fails to achieve any bounded approximation factor for the problem.

Distributed Decision and Control for Cooperative UAVs Using Ad Hoc Communication

This study develops a novel distributed algorithm for task assignment (TA), coordination, and communication of multiple unmanned aerial vehicles (UAVs) engaging multiple targets and conceives an ad hoc routing algorithm for synchronization of target lists utilizing a distributed computing topology. Assuming limited communication bandwidth and range, coordination of UAV motion is achieved by implementing a simple behavioral flocking algorithm utilizing a tree topology for distributed flight coordination. Distributed TA is implemented by a relaxation process, wherein each node computes a temporary TA based on the union of the TAs of its neighbors in the tree. The computation of the temporary TAs at each node is based on weighted matching in the UAV-target distances graph. A randomized sampling mechanism is used to propagate TAs among different parts of the tree. Thus, changes in the location of the UAVs and targets do not pass through the root of the tree. Simulation experiments show that the combination of the flocking and the TA algorithms yields the best performance.

Nonmonotone submodular maximization via a structural continuous greedy algorithm

Consider a suboptimal solution S for a maximization problem. Suppose Ss value is small compared to an optimal solution OPT to the problem, yet S is structurally similar to OPT. A natural question in this setting is whether there is a way of improving S based solely on this information. In this paper we introduce the Structural Continuous Greedy Algorithm, answering this question affirmatively in the setting of the Nonmonotone Submodular Maximization Problem. We improve on the best approximation factor known for this problem. In the Nonmonotone Submodular Maximization Problem we are given a non-negative submodular function f, and the objective is to find a subset maximizing f. Our method yields an 0.42-approximation for this problem, improving on the current best approximation factor of 0.41 given by Gharan and Vondrak [5]. On the other hand, Feige et al. [4] showed a lower bound of 0.5 for this problem.

Hedonic clustering games

Clustering, the partitioning of objects with respect to a similarity measure, has been extensively studied as a global optimization problem. We investigate clustering from a game theoretic approach, and consider the class of hedonic clustering games. Here, a self organized clustering is obtained via decisions made by independent players, corresponding to the elements clustered. Being a hedonic setting, the utility of each player is determined by the identity of the other members of her cluster. This class of games seems to be quite robust, as it fits with rather different, yet commonly used, clustering criteria. Specifically, we investigate hedonic clustering games in two different models: fixed clustering, which subdivides into k-median and k-center, and correlation clustering. We provide a thorough and non-trivial analysis of these games, characterizing Nash equilibria, and proving upper and lower bounds on the price of anarchy and price of stability. For fixed clustering we focus on the existence of a Nash equilibrium, as it is a rather non-trivial issue in this setting. We study it both for general metrics and special cases, such as line and tree metrics. In the correlation clustering model, we study both minimization and maximization variants, and provide almost tight bounds on both price of anarchy and price of stability.

Deterministic algorithms for submodular maximization problems

Randomization is a fundamental tool used in many theoretical and practical areas of computer science. We study here the role of randomization in the area of submodular function maximization. In this area, most algorithms are randomized, and in almost all cases the approximation ratios obtained by current randomized algorithms are superior to the best results obtained by known deterministic algorithms. Derandomization of algorithms for general submodular function maximization seems hard since the access to the function is done via a value oracle. This makes it hard, for example, to apply standard derandomization techniques such as conditional expectations. Therefore, an interesting fundamental problem in this area is whether randomization is inherently necessary for obtaining good approximation ratios. In this work, we give evidence that randomization is not necessary for obtaining good algorithms by presenting a new technique for derandomization of algorithms for submodular function maximization. Our high level idea is to maintain explicitly a (small) distribution over the states of the algorithm, and carefully update it using marginal values obtained from an extreme point solution of a suitable linear formulation. We demonstrate our technique on two recent algorithms for unconstrained submodular maximization and for maximizing a submodular function subject to a cardinality constraint. In particular, for unconstrained submodular maximization we obtain an optimal deterministic 1/2-approximation showing that randomization is unnecessary for obtaining optimal results for this setting.

A Simple O(log log(rank))-Competitive Algorithm for the Matroid Secretary Problem

Only recently, progress has been made in obtaining o(log(rank))-competitive algorithms for the matroid secretary problem. More precisely, Chakraborty and Lachish (2012) presented a O((log(rank))1/2)-competitive procedure, and Lachish (2014) later presented a O(log log(rank))-competitive algorithm. Both these algorithms and their analyses are very involved, which is also reflected in the extremely high constants in their competitive ratios. Using different tools, we present a considerably simpler O(log log(rank))-competitive algorithm for the matroid secretary problem. Our algorithm can be interpreted as a distribution over a simple type of matroid secretary algorithms that are easy to analyze. Because of the simplicity of our procedure, we are also able to vastly improve on the hidden constant in the competitive ratio.

Improved Competitive Ratios for Submodular Secretary Problems (Extended Abstract)

The Classical Secretary Problem was introduced during the 60’s of the 20 th century, nobody is sure exactly when. Since its introduction, many variants of the problem have been proposed and researched. In the classical secretary problem, and many of its variant, the input (which is a set of secretaries, or elements) arrives in a random order. In this paper we apply to the secretary problem a simple observation which states that the random order of the input can be generated by independently choosing a random continuous arrival time for each secretary. Surprisingly, this simple observation enables us to improve the competitive ratio of several known and studied variants of the secretary problem. In addition, in some cases the proofs we provide assuming random arrival times are shorter and simpler in comparison to existing proofs. In this work we consider three variants of the secretary problem, all of which have the same objective of maximizing the value of the chosen set of secretaries given a monotone submodular function f. In the first variant we are allowed to hire a set of secretaries only if it is an independent set of a given partition matroid. The second variant allows us to choose any set of up to k secretaries. In the last and third variant, we can hire any set of secretaries satisfying a given knapsack constraint.

Improved approximations for k-exchange systems

Submodular maximization and set systems play a major role in combinatorial optimization. It is long known that the greedy algorithm provides a 1/(k + 1)-approximation for maximizing a monotone submodular function over a k-system. For the special case of k-matroid intersection, a local search approach was recently shown to provide an improved approximation of 1/(k +δ) for arbitrary δ > 0. Unfortunately, many fundamental optimization problems are represented by a k-system which is not a k-intersection. An interesting question is whether the local search approach can be extended to include such problems. We answer this question affirmatively. Motivated by the b-matching and k-set packing problems, as well as the more general matroid k-parity problem, we introduce a new class of set systems called k-exchange systems, that includes k-set packing, b-matching, matroid k-parity in strongly base orderable matroids, and additional combinatorial optimization problems such as: independent set in (k+1)-claw free graphs, asymmetric TSP, job interval selection with identical lengths and frequency allocation on lines. We give a natural local search algorithm which improves upon the current greedy approximation, for this new class of independence systems. Unlike known local search algorithms for similar problems, we use counting arguments to bound the performance of our algorithm. Moreover, we consider additional objective functions and provide improved approximations for them as well. In the case of linear objective functions, we give a non-oblivious local search algorithm, that improves upon existing local search approaches for matroid k-parity.

A simple O (log log(rank))-competitive algorithm for the matroid secretary problem

Only recently progress has been made in obtaining o(log(rank))-competitive algorithms for the matroid secretary problem. More precisely Chakraborty and Lachish (2012) presented a O([EQUATION]log(rank))-competitive procedure, and Lachish (2014) recently presented a O(log log (rank))-competitive algorithm. Both algorithms are involved with complex analyses. Using different tools, we present a considerably simpler O(log log(rank))-competitive algorithm. Our algorithm can be interpreted as a distribution over a simple type of matroid secretary algorithms which are easy to analyze. We are also able to vastly improve on the hidden constant in the competitive ratio.

Constrained Monotone Function Maximization and the Supermodular Degree

The problem of maximizing a constrained monotone set function has many practical applications and generalizes many combinatorial problems. Unfortunately, it is generally not possible to maximize a monotone set function up to an acceptable approximation ratio, even subject to simple constraints. One highly studied approach to cope with this hardness is to restrict the set function. An outstanding disadvantage of imposing such a restriction on the set function is that no result is implied for set functions deviating from the restriction, even slightly. A more flexible approach, studied by Feige and Izsak, is to design an approximation algorithm whose approximation ratio depends on the complexity of the instance, as measured by some complexity measure. Specifically, they introduced a complexity measure called supermodular degree, measuring deviation from submodularity, and designed an algorithm for the welfare maximization problem with an approximation ratio that depends on this measure. In this work, we give the first (to the best of our knowledge) algorithm for maximizing an arbitrary monotone set function, subject to a k-extendible system. This class of constraints captures, for example, the intersection of k-matroids (note that a single matroid constraint is sufficient to capture the welfare maximization problem). Our approximation ratio deteriorates gracefully with the complexity of the set function and k. Our work can be seen as generalizing both the classic result of Fisher, Nemhauser and Wolsey, for maximizing a submodular set function subject to a k-extendible system, and the result of Feige and Izsak for the welfare maximization problem. Moreover, when our algorithm is applied to each one of these simpler cases, it obtains the same approximation ratio as of the respective original work.