Adding Isolated Vertices Makes some Online Algorithms Optimal

An unexpected difference between online and offline algorithms is observed. The natural greedy algorithms are shown to be worst case online optimal for Online Independent Set and Online Vertex Cover on graphs with 'enough' isolated vertices, Freckle Graphs. For Online Dominating Set, the greedy algorithm is shown to be worst case online optimal on graphs with at least one isolated vertex. These algorithms are not online optimal in general. The online optimality results for these greedy algorithms imply optimality according to various worst case performance measures, such as the competitive ratio. It is also shown that, despite this worst case optimality, there are Freckle graphs where the greedy independent set algorithm is objectively less good than another algorithm. It is shown that it is NP-hard to determine any of the following for a given graph: the online independence number, the online vertex cover number, and the online domination number.Comment: A footnote in the .tex file didn't show up in the last version. This was fixe

Approximation Algorithms for Stochastic Boolean Function Evaluation and Stochastic Submodular Set Cover

Stochastic Boolean Function Evaluation is the problem of determining the value of a given Boolean function f on an unknown input x, when each bit of x_i of x can only be determined by paying an associated cost c_i. The assumption is that x is drawn from a given product distribution, and the goal is to minimize the expected cost. This problem has been studied in Operations Research, where it is known as "sequential testing" of Boolean functions. It has also been studied in learning theory in the context of learning with attribute costs. We consider the general problem of developing approximation algorithms for Stochastic Boolean Function Evaluation. We give a 3-approximation algorithm for evaluating Boolean linear threshold formulas. We also present an approximation algorithm for evaluating CDNF formulas (and decision trees) achieving a factor of O(log kd), where k is the number of terms in the DNF formula, and d is the number of clauses in the CNF formula. In addition, we present approximation algorithms for simultaneous evaluation of linear threshold functions, and for ranking of linear functions. Our function evaluation algorithms are based on reductions to the Stochastic Submodular Set Cover (SSSC) problem. This problem was introduced by Golovin and Krause. They presented an approximation algorithm for the problem, called Adaptive Greedy. Our main technical contribution is a new approximation algorithm for the SSSC problem, which we call Adaptive Dual Greedy. It is an extension of the Dual Greedy algorithm for Submodular Set Cover due to Fujito, which is a generalization of Hochbaum's algorithm for the classical Set Cover Problem. We also give a new bound on the approximation achieved by the Adaptive Greedy algorithm of Golovin and Krause

Greedy Algorithm for Set Cover in Context of Knowledge Discovery Problems

AbstractIn the paper some problems connected with a process of knowledge discovery are considered. These problems are reduced to the set cover problem. It is known that under a plausible assumption on the class N P the greedy algorithm is close to best approximate polynomial algorithms for the set cover problem solving. Unfortunately, the performance ratio of this algorithm grows almost as natural logarithm on the cardinality of covered set. Instead of usual greedy algorithm we consider greedy algorithm with threshold. This algorithm constructs a partial cover, which covers at least a fixed part (for example, 90%) of the set. We prove that the cardinality of constructed partial cover is bounded from above by a linear function on the minimal cardinality of exact cover Cmin. In the case of 90% -cover, for example, in the capacity of such function we can take the function 2.31,·,Cmin+1. This bound is independent of the cardinality of covered set. Notice that the concept of partial cover in context of knowledge discovery problems is very close to the concept of approximate reduct

Independent sets in bounded-degree hypergraphs

AbstractIn this paper we analyze several approaches to the Maximum Independent Set (MIS) problem in hypergraphs with degree bounded by a parameter Δ. Since independent sets in hypergraphs can be strong and weak, we denote by MIS (MSIS) the problem of finding a maximum weak (strong) independent set in hypergraphs, respectively. We propose a general technique that reduces the worst case analysis of certain algorithms on hypergraphs to their analysis on ordinary graphs. This technique allows us to show that the greedy algorithm for MIS that corresponds to the classical greedy set cover algorithm has a performance ratio of (Δ+1)/2. It also allows us to apply results on local search algorithms on graphs to obtain a (Δ+1)/2 approximation for the weighted MIS and (Δ+3)/5−ϵ approximation for the unweighted case. We improve the bound in the weighted case to ⌈(Δ+1)/3⌉ using a simple partitioning algorithm. We also consider another natural greedy algorithm for MIS that adds vertices of minimum degree and achieves only a ratio of Δ−1, significantly worse than on ordinary graphs. For MSIS, we give two variations of the basic greedy algorithm and describe a family of hypergraphs where both algorithms approach the bound of Δ

Survey of Approximation Algorithms for Set Cover Problem

In this thesis, I survey 11 approximation algorithms for unweighted set cover problem. I have also implemented the three algorithms and created a software library that stores the code I have written. The algorithms I survey are: 1. Johnson's standard greedy; 2. f-frequency greedy; 3. Goldsmidt, Hochbaum and Yu's modified greedy; 4. Halldorsson's local optimization; 5. Dur and Furer semi local optimization; 6. Asaf Levin's improvement to Dur and Furer; 7. Simple rounding; 8. Randomized rounding; 9. LP duality; 10. Primal-dual schema; and 11. Network flow technique. Most of the algorithms surveyed are refinements of standard greedy algorithm