Neural Networks for Vertex Covering Vertex Covering Problem For a given graph G = (V, E), find a minimum subset such that every edge is covered by some vertex in A combinatorial optimization problem (NP-hard) Constraints: Covering (hard); Minimality (soft) Competition but not exclusion

Neural net Update rules: Each node corresponds to one vertex in V with activation Update rules: no explicit weights provides support from

Energy function Experiments It can be shown that Random graphs with different densities 40 with |V| = 20, 40 with |V| = 50 Compare with true minimum covers and greedy alg A = 3.0, initially and gradually increasing True min +1 +2 +3 |V| = 20 35 (24) 5 (13) 0 (3) 0 (0) |V| = 50 18 (8) 15 (18) 5 (9) 2 (5) Numbers in parentheses are results using sequential greedy algorithm

Generate weight by gradient descent The results were not as good Hopfield approach Energy function (reflecting the constraints) Generate weight by gradient descent The results were not as good Penalty for covering constraint Penalty for minimality constraint