algorithm - Heuristic to find the maximum weight independent set in an arbritary graph -

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the mwis (maximum weight independent set) np-complete problem, if p!=np cannot find solution in enough time complexity.

i looking algorithm can find approximation of mwis in arbitrary graph within time complexity. working on connected graph 128 nodes , 3051 edges.

i have found this paper, seems working bipartite graph unique mwis.

i glad if can me references or better pseudo-code of working algorithm.

it's possible formulate following problem. suppose each vertex v in graph has weight w(v). define variable x(v), , use out-of-the-box linear programming solver solve

max \sum_v w(v) x(v) (maximize weight of chosen vertices)

subject

x(u) + x(v) <= 1, (u, v) \in e (don't take neighbors)

and

x(v) \in {0, 1} (can choose take or not take vertex)

this combinatorical problem (the last constraint exponential in number of vertices). there 2 ways continue here:

  • switch last constraint

    x(v) \in [0, 1] (extent choose vertex)

    solve lp solver, , continue along this paper, 4.3.

  • in comment below, david eisenstat claims sizes of graph, integer solver fine (and yield better results)


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