algorithm,graph,graph-algorithm,linear-programming,np-complete , Heuristic to find the maximum weight independent set in an arbritary graph

## Question:

Tag: algorithm,graph,graph-algorithm,linear-programming,np-complete

The MWIS (Maximum weight independent set) is a NP-complete problem, so if P!=NP we cannot find a solution in a good enough time complexity.

I am looking for an algorithm that can find an approximation of the MWIS in an arbitrary graph within a good time complexity. I am currently working on a connected graph with 128 nodes and 3051 edges.

I have found this paper, but it seems that it is only working for bipartite graph with an unique MWIS.

I will be glad if anyone can help me with some references or even better with a pseudo-code of a working algorithm.

It's possible to formulate this as the following problem. Suppose each vertex v in the graph has weight w(v). You define a variable x(v), and use some out-of-the-box linear programming solver to solve

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

subject to

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

and

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

This is a combinatorical problem (the last constraint is exponential in the number of vertices). There are two ways to continue from here:

• Switch the last constraint to

x(v) \in [0, 1] (extent to which you choose a vertex)

solve it with an LP solver, and continue along this paper, 4.3.

• In the comment below, David Eisenstat claims that for the sizes of your graph, an integer solver will do just fine (and yield better results)

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