SDP-based bounds for graph partition via extended ADMM

We study two NP-complete graph partition problems, k-equipartition problems and graph partition problems with knapsack constraints (GPKC). We introduce tight SDP relaxations with nonnegativity constraints to get lower bounds, the SDP relaxations are solved by an extended alternating direction method of multipliers (ADMM). In this way, we obtain high quality lower bounds for k-equipartition on large instances up to n = 1000 vertices within as few as 5 min and for GPKC problems up to n = 500 vertices within as little as 1 h. On the other hand, interior point methods fail to solve instances from n = 300 due to memory requirements. We also design heuristics to generate upper bounds from the SDP solutions, giving us tighter upper bounds than other methods proposed in the literature with low computational expense.