Introduction

Rmaxcut finds an approximate solution to a weighted max-cut problem via random perturbation. Each line in an input file consists of the first nodeID, the second nodeID and an integer weight. The output gives

N  nodeID   spin     w++  w+-  w-+  w--
E  nodeID1  nodeID2  weight  spin1  spin2

where spin is either 1 or -1, indicating the partition of nodeID. On an N-line, w++ is the sum of positive weights of neighbors with positive spins; other w** numbers are similar. These numbers are mostly for debugging purposes.

To try rmaxcut, you may acquire x-all.txt.gz from the download page and run

rmaxcut -r20000 x-all.txt.gz > test.out 2> test.err

It will take a couple of minutes on a single thread. Increasing option -r, the number of iterations, often leads to a better solution for the largest few connected components. Rmaxcut emits good enough solutions to problems at my hand. I haven't compared it to other more sophisticated max-cut solvers. Use with caution.

Algorithms

The weighted max-cut problem is to find a bipartition in an undirected graph such that the sum of edge weights between the two partitions are maximized. Max-cut is NP-complete. In physics, solving max-cut is equivalent to finding optimal states in an Ising model without external fields. We often take ideas in physics to find approximate solutions to large-scale max-cut problems. This repo provides such an implementation. It uses a fairly naive algorithm:

1. Find connected component. Do the following in each component.
2. Randomly partition nodes, or equivalently, randomly assign a spin {-1,1} to each node.
3. For each node, flip its spin if doing that increases the weight. Repeat until the total weight can't be improved this way. This is a local maximum.
4. Randomly flip 10% of spins, or flip the spins of nodes within a certain distance from a random node (done by BFS). Do step 3 again. If the new local maximum is better, do step 4 on the new state; otherwise, do step 4 on the old state.
5. Repeat 3-4 for many times. Then move to the next component.

The algorithm is akin to simulated annealing but without proper annealing. When I have time, I need to try more sophisticated procedures in vein of Monte Carlo methods. This may speed up heuristic search.