An application of the greedy heuristic of set cover to traffic checks

Let us suppose that X is a given, finite, not empty set and is a given collection of subsets of X such that their union equals X , in other words covers X . Set cover is the problem of selecting as few as possible subsets from such that their union covers X . Max k -cover is the problem of selecting k subsets from such that their union has maximum cardinality. Both problems are NP-hard. There is a polynomial time greedy heuristic that iteratively selects the subset from that covers the largest number of yet uncovered elements. We implemented this greedy algorithm to support the planning of a checking system that is aimed to check the vehicles in a road network. We would like to answer such questions: How many and which links are sufficient to check a given percentage of all traffic flow? What percentage of traffic can be checked with given links? This paper defines the necessary data and basic knowledge, gives algorithms to answer the previous questions and also shows the results of an implementation in a road network that contains about 11,000 junctions, 3,000 origin–destination junctions and 26,000 links.