I need to minimize a supermodular function and I am well aware of the fact that minimizing supermodular functions is equivalent to maximize submodular functions and that there are many good approximation algorithms for that problem. All of those algorithms are for non-negative submodular functions. My function is a non-negative supermodular function and therefore its negative counterpart will not be a NON NEGATIVE submodular function. I could add a constant to make the function positive but then the approximation ratio will not be guaranteed anymore. Did anyone incur into this problem? I am aware of the data correcting-algorithm by Goldering but this is a branch-and-bound algorithm and it doesn't run in polynomial time. I'm looking for a constant ratio approximation algorithm that runs in polynomial time if it's out there (my supermodular function is not monotone).
2 Answers
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It is unlikely that your problem has a constant approximaiton. Consider the case where the optimal solution of your problem is zero. Then a constant approximation means you have to find the optimal solution exactly, which is NP-hard in general.
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$\begingroup$ Thank you for your reply. Forgive me if I'm being naive but would not this be a problem for any approx algorithm for a minimization problem? Are there any particular conditions under which we can find a guaranteed approximation for minimization algorithms? Any literature you could point me to would also be great. Thank you for your help. $\endgroup$Rebecca– Rebecca2014-02-12 15:58:43 +00:00Commented Feb 12, 2014 at 15:58
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1$\begingroup$ In fact, for many combinatorial minimization problems which admits a constant approximation, the optimal solution (for nontrivial instances) is non-zero, e.g., vertex cover problem, TSP, steiner tree and so on. There is in general no such particular condition. But in order to get a bounded approximation ratio, the following should be true: "deciding whether the optimal solution is zero should be in P". $\endgroup$jian– jian2014-02-23 02:43:15 +00:00Commented Feb 23, 2014 at 2:43
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The paper "An FPTAS for optimizing a class of low-rank functions over a polytope" by Mittal and Schulz has this theorem:
Theorem 8 Let $f : 2^S → \mathbb{Z}_+$ be a supermodular function defined over the subsets of $S$. Then it is not possible to approximate the minimum of $f$ to within any factor, unless P = NP.
Therefore, unfortunately, there is no such algorithm.
