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Problem: Given a graph G with positive integral weights on nodes as well as edges, find an induced subgraph H with maximum total weight per node, i.e., the goal is to maximize $$ \frac{ \sum_{e \in H}w(e) + \sum_{v \in H} w(v) }{|V(H)|}.$$

If you allow negative weights, then it is NP-hard (easy reduction from Clique). I have a strong feeling that the above is not NP-Hard.

I would love to have some first-thoughts on the above, if not a full-fledged optimal algorithm. Does it seem to be NP-Hard? What is the closest known problem -- my thoughts were Prize Collection, Clique -- but both seem much harder.

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  • $\begingroup$ Isn't G the induced subgraph of G with maximal weight? $\endgroup$
    – qz-
    Apr 19 at 19:16
  • $\begingroup$ @qz no, you want the subgraph that maximizes average weight (... so to speak) $\endgroup$
    – Pål GD
    Apr 19 at 19:21
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    $\begingroup$ Have a look at the Dense Subgraph Problem. $\endgroup$
    – Pål GD
    Apr 19 at 19:27
  • $\begingroup$ @PålGD ah, thanks for editing the question to be more clear $\endgroup$
    – qz-
    Apr 19 at 19:30
  • $\begingroup$ Yes, I see that "dense subgraph problem" is very close. Thanks -- that should work. $\endgroup$
    – Brian
    Apr 19 at 20:17
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The Dense Subgraph Problem (thanks @PalGD) is a special case of the above problem--with no node weights and unit edge weights. See the solutions for the dense subgraph problem here. These solutions (LP and 2-approximation Greedy) easily generalize to the above problem (generalizing LP requires a bit of understanding/effort, but generalizing Greedy is trivial).

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