In Exaclty One Triple of T. from a Steiner Triple System We Derive a Graph the Cardinality of a Minimum Cover for a Steiner Triple System Deened On

Test case generators and computational results for the maximum clique problem. A polynomial algorithm for maximum weighted vertex packings on graphs without long odd cycles. A strong cutting plane/branch-and-bound algorithm for node packing. A probabilistic heuristic for a computa-tionally diicult set covering problem. Two computation-ally diicult set covering problems that arise in computing the 1-width of incidence matrices of steiner triple systems. Mathematical Programming Study, 2:72{81, 1974. 14] F. Gavril. Algorithms for a maximum clique and a maximum independent set of a circle graph. Details about this model can be found in 8]. These graphs can have a huge number of nodes and are very dense. To our knowledge these are the largest`real life' instances of MSS problems solved to optimality. In this case the value of the parameter has been set equal to one for all the experiments, and node xing has been disabled. The results are shown in table 6. Observe that all of these graphs are very dense. For this reason they are not diicult instances of MSS and they can be solved in a reasonable amount of time also by a quasi-enumerative method such as the one described in 9]. References 1] D. Avis. A note on some computationally diicult set covering problems. f(w j ; u ij) : w j 2 W and u ij 2 Ug. In other words, the nodes in W represent the elements of A while the nodes in U represent the triples with jWj = jAj and jUj = 3jT j. Moreover to each triple t i corresponds a triangle K i with nodes in U and each node w j 2 W is connected to the corresponding nodes u ij whenever a j 2 t i. It is well known that the problem of nding a subset C (called cover) of A of minimum cardinality such that C \ t i 6 = ;, for all i, is NP-hard. This problem can be stated as a Set Covering Problem; and, in this form, it was proposed by Fulkerson et al. 12] as a class of test problems. Computational experience has shown that they are hard to solve in practice 10,20]. In 1], Avis suggests an explanation to the computational hardness of this class of problems, by proving that any implementation of branch-and-bound that employs linear or dynamic programming as a fathoming device requires at least the examination of 2 …