Most interesting real-world optimization problems are very challenging from a computational point of view. In fact, quite often, finding an optimal or even a near-optimal solution to a large-scale optimization problem may require computational resources far beyond what is practically available. There is a substantial body of literature exploring the computational properties of optimization prob...

Many problems of great practical importance are hard to solve computationally, at least if exact solutions are required. We survey a number of (NPor P-complete) problems for which fast parallel approximation algorithms are known: The O-l knapsack problem, binpacking, the minimal makeshift problem, the list scheduling problem, greedy scheduling, and the high density subgraph problem. Algorithms ...

We have seen several randomized approximation algorithms before. Recall that for the MAXCUT and MAX-3SAT problems we proved that choosing any solution uniformly at random gives a constant factor approximation in the expectation. For MAX-CUT the approximation factor was 2. For MAX-3SAT, a random assignment of truth values to a variable satisfies a given clause with probability 78 . In the case o...

Let G = (V,E) be an undirected graph with arc weights w : V → R+. Define xv for each vertex v as follows: xv = 1, if v is in the vertex cover; xv = 0, if v is not chosen. Our goal is to find min ( ∑ v∈V wvxv), such that xu + xv ≥ 1, ∀e = (u, v) ∈ E, xv ∈ {0, 1}. However, we can’t solve Integer Linear Programming (ILP) problems in polynomial time. So we have to use Linear Programming (LP) to app...

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The Peace of Olivia. How sweat and peaceful it sounds! There the great powers noticed for the first time that the land of the Poles lends itself admirably to partition. – The tin drum, Gunter Grass In this chapter, we are going to discuss two basic geometric algorithms. The first one, computes the closest pair among a set of n points in linear time. This is a beautiful and surprising result tha...

Approximation Algorithms for Clustering Clustering items into groups is a fundamental problem in the information sciences. Many typical clustering optimization problems are NP-hard and so cannot be expected to be solved optimally in a reasonable amount of time. Although the use of heuristics is common, in this dissertation we seek approximation algorithms, whose performance ratio in relation to...

Consider the following problem: given a metric space, some of whose points are “clients”, open a set of at most k facilities to minimize the average distance from the clients to these facilities. This is just the well-studied k-median problem, for which many approximation algorithms and hardness results are known. Note that the objective function encourages opening facilities in areas where the...

We discuss two closely related NP Optimization problems, namely Set Cover and Maximum Coverage in this lecture. Set Cover was among the first problems for which approximation algorithms were analyzed [1]. This problem is also significant from a practical point of view, since the problem itself and several of its generalizations arise quite frequently in a number of application areas. We will co...