CS 6310 Advanced Design and Analysis of Algorithms Rajani Pingili Set Cover Problem CS 6310 Advanced Design and Analysis of Algorithms Rajani Pingili

The set cover problem is well-known and classical question of in computer science, operations research, combinatorics, and complexity theory. It is one of Karp's 21 popular NP- complete problems. It is a problem "whose study has led to the development of fundamental techniques for the entire field" of approximation algorithms Set Cover Problem

Set Cover Problem The Set Cover problem: We have a universe of elements 𝑼= 𝒙 𝟏 ,…, 𝒙 𝑵 We have a collection of subsets of 𝑼, 𝑺= 𝑺 𝟏 ,…, 𝑺 𝒏 Given a universe 𝑼 and a family 𝑺 of subsets of 𝑼,a cover is a subfamily 𝑪⊆𝑺 of sets whose union is 𝑼. (𝑼, S) is the input pair and k is an integer in the set covering decision problem; whether there is a set covering of size k or less. This problem is NP-Complete (𝑼, S) is the input pair in the set covering optimization problem and the task is to find a set covering that utilizes the fewest sets. This problem is NP-Hard.

Set Cover Problem Example Given a set of elements {1,2,….,n} (called the universe) and a collection of S of m sets whose union equals the universe. The set cover problem is to identify the smallest sub- collection of S whose union equals the universe. U = {1,2,3,4,5} and the collection of sets S = {{1,2,3},{2,4},{3,4},{4,5}}. U = Union of S Smaller number of sets: {{1,2,3}, {4,5}}

Greedy Algorithm Rule of greedy algorithm for polynomial time is “at each stage, choose the set that contains the largest number of uncovered elements.” Approximation ratio H(s), s is the size of the sets to be covered. H(n) is the 𝑛 𝑡ℎ harmonic number, H(n) = 1≤𝑘≤𝑛 1 𝑘 ≤ ln n + 1

Standard Example Universe U: 2 (𝑘+1) - 2 elements The set system consists of k pairwise disjoint sets 𝑆 1 ,…, 𝑆 𝑘 with sizes 2, 4, 8,……., 2 𝑘 respectively, as well as two disjoint sets 𝑇 0 , 𝑇 1 , each of which contains half of the elements of U. Approx. ratio = k/2 = O(log n), since the obtained solution contains the k sets 𝑆 1 ,…, 𝑆 𝑘 , while the optimal solution consists only of 𝑇 0 and 𝑇 1 . Approximation ratio: ln 𝑛 − ln ln 𝑛+ 𝛩(1) with a tighter analysis of the greedy algorithm [2].

References Content http://en.wikipedia.org/wiki/Set_cov er_problem. Slavík Petr A tight analysis of the greedy algorithm for set cover. STOC'96, Pages 435-441, doi:10.1145/237814.237991 References