We study a generalization of the Set Cover problem called the Partial Set Cover in the context of geometric set systems. The input to this problem is a set system (X ,S), where X is a set of elements and S is a collection of subsets of X , and an integer k ≤ |X |. The goal is to cover at least k elements of X by using a minimum-weight collection of sets from S. In the geometric version, X typic...

We consider the replica placement problem: given a graph with clients and nodes, place replicas on a minimum set of nodes to serve all the clients; each client is associated with a request and maximum distance that it can travel to get served and there is a maximum limit (capacity) on the amount of request a replica can serve. The problem falls under the general framework of capacitated set cov...

Minimally nonideal matrices are a key to understanding when the set covering problem can be solved using linear programming. The complete classification of minimally nonideal matrices is an open problem. One of the most important results on these matrices comes from a theorem of Lehman, which gives a property of the core of a minimally nonideal matrix. Cornuéjols and Novick gave a conjecture on...

This paper proposes a greedy algorithm named as Big step greedy set cover algorithm to compute approximate minimum set cover. The Big step greedy algorithm, in each step selects p sets such that the union of selected p sets contains greatest number of uncovered elements and adds the selected p sets to partial set cover. The process of adding p sets is repeated until all the elements are covered...

We design a new learning algorithm for the Set Covering Machine from a PAC-Bayes perspective and propose a PAC-Bayes risk bound which is minimized for classifiers achieving a non trivial margin-sparsity trade-off.

Interactive submodular set cover is an interactive variant of submodular set cover over a hypothesis class of submodular functions, where the goal is to satisfy all sufficiently plausible submodular functions to a target threshold using as few (cost-weighted) actions as possible. It models settings where there is uncertainty regarding which submodular function to optimize. In this paper, we pro...

A variety of methods exist for inductive learning of classification rules using crisp sets. In this paper we illustrate an inductive learner that uses fuzzy sets, where the membership functions of the linguistic terms are given in advance. We also show how the induction of conjunctive rules fit into a fuzzy set covering framework (FuzzyBexa) that we introduced before.

Abstract Consider the following variant of the set cover problem. We are given a universe U = {1, ..., n} and a collection of subsets C = {S1, ..., Sm} where Si ⊆ U . For every element u ∈ U we need to find a set φ (u) ∈ C such that u ∈ φ (u). Once we construct and fix the mapping φ : U 7→ C a subset X ⊆ U of the universe is revealed, and we need to cover all elements from X with exactly φ(X) :...

We use a new approximation measure, the differential approximation ratio, to derive polynomial-time approximation algorithms for minimum set covering (for both weighted and unweighted cases), minimum graph coloring and bin-packing. We also propose differentialapproximation-ratio preserving reductions linking minimum coloring, minimum vertex covering by cliques, minimum edge covering by cliques ...