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.
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.
Abstract.We motivate and describe a new parameterized approximation paradigm which studies the interaction between performance ratio and running time for any parameterization of a given optimization problem. As a key tool, we introduce the concept of α-shrinking transformation, for α ≥ 1. Applying such transformation to a parameterized problem instance decreases the parameter value, while preserving approximation ratio of α (or α-fidelity). For example, it is well-known that Vertex Cover cannot be approximated within any constant factor better than 2 [22] (under usual assumptions). Our parameterized α-approximation algorithm for k-Vertex Cover, parameterized by the solution size, has a running time of 1.273(2−α)k , where the running time of the best FPT algorithm is 1.273 k [10]. Our algorithms define a continuous tradeoff between running times and approximation ratios, allowing practitioners to appropriately allocate computational resources. Moving even beyond the performance ratio, we call for a new type of approximative kernelization race. Our α-shrinking transformations can be used to obtain kernels which are smaller than the best known for a given problem. For the Vertex Cover problem we obtain a kernel size of 2(2 − α)k. The smaller "α-fidelity" kernels allow us to solve exactly problem instances more efficiently, while obtaining an approximate solution for the original instance. We show that such transformations exist for several fundamental problems, including Vertex Cover, d-Hitting Set, Connected Vertex Cover and Steiner Tree. We note that most of our algorithms are easy to implement and are therefore practical in use.
In the d-dimensional (vector) knapsack problem given is a set of items, each having a d-dimensional size vector and a profit, and a d-dimensional bin. The goal is to select a subset of the items of maximum total profit such that the sum of all vectors is bounded by the bin capacity in each dimension. It is well known that, unless P = N P , there is no fully polynomial time approximation scheme for d-dimensional knapsack, already for d = 2. The best known result is a polynomial time approximation scheme (PTAS) due to Frieze and Clarke (European J. of Operational Research, 100-109, 1984 ) for the case where d ≥ 2 is some fixed constant. A fundamental open question is whether the problem admits an efficient PTAS (EPTAS).In this paper we resolve this question by showing that there is no EPTAS for ddimensional knapsack, already for d = 2, unless W [1] = F P T . Furthermore, we show that unless all problems in SNP are solvable in sub-exponential time, there is no approximation scheme for two-dimensional knapsack whose running time is f (1/ε)|I| o( √ 1/ε) , for any function f . Together, the two results suggest that a significant improvement over the running time of the scheme of Frieze and Clarke is unlikely to exist.
We prove a general result demonstrating the power of Lagrangian relaxation in solving constrained maximization problems with arbitrary objective functions. This yields a unified approach for solving a wide class of subset selection problems with linear constraints. Given a problem in this class and some small ε ∈ (0, 1), we show that if there exists a ρ-approximation algorithm for the Lagrangian relaxation of the problem, for some ρ ∈ (0, 1), then our technique achieves a ratio of ρ ρ+1 −ε to the optimal, and this ratio is tight. The number of calls to the ρ-approximation algorithm, used by our algorithms, is linear in the input size and in log(1/ε) for inputs with cardinality constraint, and polynomial in the input size and in log(1/ε) for inputs with arbitrary linear constraint. Using the technique we obtain approximation algorithms for natural variants of classic subset selection problems, including real-time scheduling, the maximum generalized assignment problem (GAP) and maximum weight independent set.
We give a comprehensive study of bin packing with conflicts (BPC). The input is a set I of items, sizes s : I → [0, 1], and a conflict graph G = (I, E). The goal is to find a partition of I into a minimum number of independent sets, each of total size at most 1. Being a generalization of the notoriously hard graph coloring problem, BPC has been studied mostly on polynomially colorable conflict graphs. An intriguing open question is whether BPC on such graphs admits the same best known approximation guarantees as classic bin packing. We answer this question negatively, by showing that (in contrast to bin packing) there is no asymptotic polynomial-time approximation scheme (APTAS) for BPC already on seemingly easy graph classes, such as bipartite and split graphs. We complement this result with improved approximation guarantees for BPC on several prominent graph classes. Most notably, we derive an asymptotic 1.391-approximation for bipartite graphs, a 2.445-approximation for perfect graphs, and a 1 + 2 e -approximation for split graphs. To this end, we introduce a generic framework relying on a novel interpretation of BPC allowing us to solve the problem via maximization techniques. Our framework may find use in tackling BPC on other graph classes arising in applications.
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