We consider the Unconstrained Submodular Maximization problem in which we are given a non-negative submodular function f : 2 N → 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 Max-Cut, 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.

No abstract

In this paper, we propose the first continuous optimization algorithms that achieve a constant factor approximation guarantee for the problem of monotone continuous submodular maximization subject to a linear constraint. We first prove that a simple variant of the vanilla coordinate ascent, called Coordinate-Ascent+, achieves a ( e−1 2e−1 − ε)-approximation guarantee while performing O(n/ε) iterations, where the computational complexity of each iteration is roughly O(n/ √ ε+n log n) (here, n denotes the dimension of the optimization problem). We then propose Coordinate-Ascent++, that achieves the tight (1 − 1/e − ε)-approximation guarantee while performing the same number of iterations, but at a higher computational complexity of roughly O(n 3 /ε 2.5 + n 3 log n/ε 2 ) per iteration. However, the computation of each round of Coordinate-Ascent++ can be easily parallelized so that the computational cost per machine scales as O(n/ √ ε + n log n).complexity of each iteration is O(n B/ε + n log n). Here, n and B denote the dimension of the optimization problem and the ℓ 1 radius of the constraint set, respectively.• We then develop Coordinate-Ascent++, that achieves the tight (1 − 1/e − ε) approximation guarantee while performing O(n/ǫ) iterations, where the computational complexity of each iteration is O(n 3 √ B/ε 2.5 + n 3 log n/ε 2 ). Moreover, Coordinate-Ascent++ can be easily parallelized so that the computational complexity per machine in each round scales as O(n B/ǫ + n log n).Notably, to establish these results, we do not assume that the continuous submodular function satisfies the diminishing returns condition.

We study graph partitioning problems from a min-max perspective, in which an input graph on n vertices should be partitioned into k parts, and the objective is to minimize the maximum number of edges leaving a single part. The two main versions we consider are where the k parts need to be of equal-size, and where they must separate a set of k given terminals. We consider a common generalization of these two problems, and design for it an O( √ log n log k)approximation algorithm. This improves over an O(log 2 n) approximation for the second version due to Svitkina and Tardos [ST04], and roughly O(k log n) approximation for the first version that follows from other previous work. We also give an improved O(1)-approximation algorithm for graphs that exclude any fixed minor.Our algorithm uses a new procedure for solving the Small-Set Expansion problem. In this problem, we are given a graph G and the goal is to find a non-empty set S ⊆ V of size |S| ≤ ρn with minimum edge-expansion. We give an O( log n log (1/ρ)) bicriteria approximation algorithm for the general case of Small-Set Expansion, and O(1) approximation algorithm for graphs that exclude any fixed minor.

Datacenter WAN tra c consists of high priority transfers that have to be carried as soon as they arrive, alongside large transfers with preassigned deadlines on their completion. e ability to o er guarantees to large transfers is crucial for business needs and impacts overall cost-of-business. State-of-the-art tra c engineering solutions only consider the current time epoch or minimize maximum utilization and hence cannot provide pre-facto promises to long-lived transfers. We present T , an online temporal planning scheme that appropriately packs long-running transfers across network paths and future timesteps, while leaving capacity slack for future changes. T builds on a tailored approximate solution to a mixed packing-covering linear program, which is parallelizable and scales well in both running time and memory usage. Consequently, T can quickly and e ectively update the promised future ow allocation when new transfers arrive or unexpected changes happen. Our experiments on traces from a large production WAN show, T can o er and keep promises to longlived transfers well in advance of their actual deadlines; the promise on minimal transfer size is comparable with an o ine optimal solution and outperforms state-of-the-art solutions by -X.

No abstract

We consider the k-balanced partitioning problem, where the goal is to partition the vertices of an input graph G into k equally sized components, while minimizing the total weight of the edges connecting different components. We allow k to be part of the input and denote the cardinality of the vertex set by n. This problem is a natural and important generalization of well-known graph partitioning problems, including minimum bisection and minimum balanced cut.We present a (bi-criteria) approximation algorithm achieving an approximation of O( √ log n log k), which matches or improves over previous algorithms for all relevant values of k. Our algorithm uses a semidefinite relaxation which combines 2 2 metrics with spreading metrics. Surprisingly, we show that the integrality gap of the semidefinite relaxation is Ω(log k) even for large values of k (e.g., k = n Ω(1) ), implying that the dependence on k of the approximation factor is necessary. This is in contrast to previous approximation algorithms for k-balanced partitioning, which are based on linear programming relaxations and their approximation factor is independent of k.

Submodular function maximization has been studied extensively in recent years under various constraints and models. The problem plays a major role in various disciplines. We study a natural online variant of this problem in which elements arrive one-by-one and the algorithm has to maintain a solution obeying certain constraints at all times. Upon arrival of an element, the algorithm has to decide whether to accept the element into its solution and may preempt previously chosen elements. The goal is to maximize a submodular function over the set of elements in the solution.We study two special cases of this general problem and derive upper and lower bounds on the competitive ratio. Specifically, we design a 1/e-competitive algorithm for the unconstrained case in which the algorithm may hold any subset of the elements, and constant competitive ratio algorithms for the case where the algorithm may hold at most k elements in its solution.