1. 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

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

3. 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
|V| = 20
35 (24) (13) (3) (0)
|V| = 50
(8) (18) (9) (5)
Numbers in parentheses are results using sequential greedy algorithm

4. 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