The objective of this paper is to characterize classes of problems for which a greedy algorithm finds solutions provably close to optimum. To that end, we introduce the notion of k-extendible systems, a natural generalization of matroids, and show that a greedy algorithm is a 1 k -factor approximation for these systems. Many seemly unrelated problems fit in our framework, e.g.: b-matching, maxi...

We describe approximation algorithms with bounded performance guarantees for the following problem: A graph is given with edge weights satisfying the triangle inequality, together with two numbers k and p. Find k disjoint subsets of p vertices each, so that the total weight of edges within subsets is maximized.

Motivation Want to learn a combinatorial parameter of a graph: the maximum matching size the independence number α(G), the minimum vertex cover size, the minimum dominating set size Krzysztof Onak – Sublinear Graph Approximation Algorithms – p. 2/32 Motivation Want to learn a combinatorial parameter of a graph: the maximum matching size the independence number α(G), the minimum vertex cover siz...

In this survey, we offer an overview of approximation algorithms for constraint satisfaction problems (CSPs) – we describe main results and discuss various techniques used for solving CSPs. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems

Sparse tensor best rank-1 approximation (BR1Approx), which is a sparsity generalization of the dense BR1Approx, and higher-order extension sparse matrix one most important problems in decomposition related arising from statistics machine learning. By exploiting multilinearity as well structure problem, four polynomial-time algorithms are proposed, easily implemented, low computational complexit...

In an online linear optimization problem, on each period t, an online algorithm chooses st ∈ S from a fixed (possibly infinite) set S of feasible decisions. Nature (who may be adversarial) chooses a weight vector wt ∈ R, and the algorithm incurs cost c(st, wt), where c is a fixed cost function that is linear in the weight vector. In the full-information setting, the vector wt is then revealed t...

The main aim of NP-completeness theory is the analysis of intractabil-ity. Many optimization problems were rst proved to be NP-hard. Since the complete solution of these problems requires exponential time, polynomial time algorithms to nd \near-optimal" solutions, i.e., approximation algorithms, appear to be viable. In this paper we show the basic principles of Approximation Theory for NP-compl...

To date, thousands of natural optimization problems have been shown to be NP-hard [8, 18]. To deal with these problems, two approaches are commonly adopted: (a) approximation algorithms, (b) randomized algorithms. Roughly speaking, approximation algorithms aim to find solutions whose costs are as close to be optimal as possible in polynomial time. Randomized algorithms can be looked at from dif...