No abstract

The concept of submodularity plays a vital role in combinatorial optimization. In particular, many important optimization problems can be cast as submodular maximization problems, including maximum coverage, maximum facility location and max cut in directed/undirected graphs.In this paper we present the first known approximation algorithms for the problem of maximizing a nondecreasing submodular set function subject to multiple linear constraints. Given a d-dimensional budget vector L, for some d ≥ 1, and an oracle for a non-decreasing submodular set function f over a universe U , where each element e ∈ U is associated with a d-dimensional cost vector, we seek a subset of elements S ⊆ U whose total cost is at mostL, such that f (S) is maximized.We develop a framework for maximizing submodular functions subject to d linear constraints that yields a (1 − ε)(1 − e −1 )-approximation to the optimum for any ε > 0, where d > 1 is some constant. Our study is motivated by a variant of the classical maximum coverage problem that we call maximum coverage with multiple packing constraints. We use our framework to obtain the same approximation ratio for this problem. To the best of our knowledge, this is the first time the theoretical bound of 1 − e −1 is (almost) matched for both of these problems.

This paper studies an optimization problem that arises in the context of distributed resource allocation: Given a conflict graph that represents the competition of processors over resources, we seek an allocation under which no two jobs with conflicting requirements are executed simultaneously. Our objective is to minimize the average response time of the system. In alternative formulation this is known as the Minimum Color Sum (MCS) problem [25]. We show, that the algorithm based on finding iteratively a maximum independent set (MaxIS) is a 4-approximation to the MCS. This bound is tight to within a factor of 2. We give improved ratios for the classes of bipartite, bounded-degree, and line graphs. The bound generalizes to a 4ρ-approximation of MCS for classes of graphs for which the maximum independent set problem can be approximated within a factor of ρ. On the other hand, we show that an n 1−ϵ-approximation is NP-hard, for some ϵ > 0. For some instances of the resource allocation problem, such as the Dining Philosophers, an efficient solution requires edge coloring of the conflict graph. We introduce the Minimum Edge Color Sum (MECS) problem which is shown to be NP-hard. We show that a 2-approximation to MECS(G) can be obtained distributively using compact coloring within O(log 2 n) communication rounds.

Submodular maximization generalizes many fundamental problems in discrete optimization, including Max-Cut in directed/undirected graphs, maximum coverage, maximum facility location and marketing over social networks.In this paper we consider the problem of maximizing any submodular function subject to d knapsack constraints, where d is a fixed constant. We establish a strong relation between the discrete problem and its continuous relaxation, obtained through extension by expectation of the submodular function. Formally, we show that, for any non-negative submodular function, an α-approximation algorithm for the continuous relaxation implies a randomized (α − ε)-approximation algorithm for the discrete problem. We use this relation to improve the best known approximation ratio for the problem to 1/4 − ε, for any ε > 0, and to obtain a nearly optimal (1−e −1 −ε)−approximation ratio for the monotone case, for any ε > 0. We further show that the probabilistic domain defined by a continuous solution can be reduced to yield a polynomial size domain, given an oracle for the extension by expectation. This leads to a deterministic version of our technique.

a b s t r a c tWe consider a scheduling problem in which a bounded number of jobs can be processed simultaneously by a single machine. The input is a set of n jobs J = {J 1 , . . . , J n }. Each job, J j , is associated with an interval [s j , c j ] along which it should be processed. Also given is the parallelism parameter g ≥ 1, which is the maximal number of jobs that can be processed simultaneously by a single machine. Each machine operates along a contiguous time interval, called its busy interval, which contains all the intervals corresponding to the jobs it processes. The goal is to assign the jobs to machines so that the total busy time is minimized.The problem is known to be NP-hard already for g = 2. We present a 4-approximation algorithm for general instances, and approximation algorithms with improved ratios for instances with bounded lengths, for instances where any two intervals intersect, and for instances where no interval is properly contained in another. Our study has application in optimizing the switching costs of optical networks.

Abstract. We study two variants of the classic knapsack problem, in which we need to place items of different types in multiple knapsacks; each knapsack has a limited capacity, and a bound on the number of different types of items it can hold: in the class-constrained multiple knapsack problem (CMKP) we wish to maximize the total number of packed items; in the fair placement problem (FPP) our goal is to place the same (large) portion from each set. We look for a perfect placement, in which both problems are solved optimally. We first show that the two problems are NP-hard; we then consider some special cases, where a perfect placement exists and can be found in polynomial time. For other cases, we give approximate solutions. Finally, we give a nearly optimal solution for the CMKP. Our results for the CMKP and the FPP are shown to provide efficient solutions for two fundamental problems arising in multimedia storage subsystems.

For a video-on-demand computer system, we propose a scheme which balances the load on the disks, thereby helping to solve a performance problem crucial to achieving maximal video throughput. Our load-balancing scheme consists of two components. The static component determines good assignments of videos to groups of striped disks. The dynamic component uses these assignments, and features a "DASD dancing" algorithm which performs real-time disk scheduling in an effective manner. Our scheme works synergistically with disk striping. We examine the performance of the proposed algorithm via simulation experiments.

The transactional approach to contention management guarantees consistency by making sure that whenever two transactions have a conflict on a resource, only one of them proceeds. A major challenge in implementing this approach lies in guaranteeing progress, since transactions are often restarted.Inspired by the paradigm of non-clairvoyant job scheduling, we analyze the performance of a contention manager by comparison with an optimal, clairvoyant contention manager that knows the list of resource accesses that will be performed by each transaction, as well as its release time and duration. The realistic, nonclairvoyant contention manager is evaluated by the competitive ratio between the last completion time (makespan) it provides and the makespan provided by an optimal contention manager.Assuming that the amount of exclusive accesses to the resources is non-negligible, we present a simple proof that every work conserving contention manager guaranteeAlgorithmica (2010) 57: 44-61 45 ing the pending commit property achieves an O(s) competitive ratio, where s is the number of resources. This bound holds for the GREEDY contention manager studied by Guerraoui et al. (Proceedings of the 24th Annual ACM Symposium on Principles of Distributed Computing (PODC), pp. [258][259][260][261][262][263][264] 2005) and is a significant improvement over the O(s 2 ) bound they prove for the competitive ratio of GREEDY. We show that this bound is tight for any deterministic contention manager, and under certain assumptions about the transactions, also for randomized contention managers.When transactions may fail, we show that a simple adaptation of GREEDY has a competitive ratio of at most O(ks), assuming that a transaction may fail at most k times. If a transaction can modify its resource requirements when re-invoked, then any deterministic algorithm has a competitive ratio (ks). For the case of unit length jobs, we give (almost) matching lower and upper bounds.