The Set Covering and Other Problems: An Empiric Complexity Analysis Using the Minimum Ellipsoidal Width

Abstract

This research aims to explain the intrinsic difficulty of Karp’s list of twenty-one problems through the use of empirical complexity measures based on the ellipsoidal width of the polyhedron generated by the constraints of the relaxed linear programming problem. The variables used as complexity measures are the number of nodes visited by the B&B and the CPU time spent solving the problems. The measurements used as explanatory variables correspond to the Dikin ellipse eigenvalues within the polyhedron. Other variables correspond to the constraint clearance with respect to the analytical center used as the center of the ellipse. The results of these variables in terms of the number of nodes and CPU time are particularly satisfactory. They show strong correlations, above 60%, in most cases.

Keywords

• integer programming
• branch and bound
• combinatorial optimization
• set covering problem