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There are many open algorithmic problems. All problems below (other than the last bullet) are NP-hard, so we are interested in the best approximation ratio we can achieve in polynomial time. The following are just a sample: Given a non-negative submodular function on a universe $U$, find a set $A$ of size at most $k$ maximizing $f(A)$. The best known ...

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A FOCS'15 paper by Lee, Sidford, and Wong [LSW15] can be leveraged to obtain such minimization guarantees -- cf. Section 5 (specifically, Corollary 5.4) in our recent paper ([BCELR16]). Corollary 5.4. Let $g\colon 2^{[m]} \to \mathbb{R}$ be a submodular function. There exists an algorithm that, when given access to an approximate oracle $\mathcal{O}^{\pm}... 4 Unless I'm mistaken, you can solve your problem in$O(n\log n)$time using a greedy algorithm. Minimizing$f(S)$is equivalent to maximizing $$\textstyle g(S') = \sum_{i \in S'} b_i + \big(\sum_{i\in S'} \alpha_i\big) \log \sum_{j \in S'} \alpha_j$$ for$b_i=w_i + \alpha_i\log(\alpha_i)$. Here$S'$is the complement of your$S$. Assume WLOG that$\alpha_i&...

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The notion of sub-modularity you use is non-standard. Usually you consider set functions with domain $\lbrace 0, 1\rbrace^n$. But to answer your questions, the Lovász extension establishes the following relationship between sub-modularity and convexity: A set-function $F$ is sub-modular if and only if its Lovász extension $f$ is convex. For a proof see ...

4

I think this is an example showing no kind of approximation is possible except with exponential$(k)$ value queries. Let $f(S) = 0$ if $|S| \leq k$, otherwise $f(S) = |S| - k$. Now pick a special set $S^*$ uniformly at random from all sets of size $k$, and let $f(S^*) = 0.5$. I'm claiming that this function is supermodular because every element initially ...

3

I think both of your problems are in P. Here's an algorithm that works in P without any restriction on the $(a_i,b_i)$ pairs. Formulate your problem as the problem of choosing $x\in\{0,1\}^n$ subject to $\sum_i x_i = k$ so as to maximize $$\textstyle f(x) = \pi(x) + \sum_i x_i a_i$$ where $\pi(x)=\prod_i \exp( x_i \beta_i)$ and $\beta_i = \ln b_i$. Relax ...

3

Sasho already gave you a yes/no answer, but here's an actual convex combination for you: If $B_\ell$ is the $k \times k$ matrix which is $1$ when $|S_i \cap S_j| \ge \ell$ and zero otherwise, then $M = \sum_{\ell=1}^{n'} \frac{1}{n'} B_\ell$.

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It is NP-hard. Svitkina and Fleischer showed that there is no $o(\sqrt{n/\log n})$ approximation using only a polynomial number of queries.

2

Consider a graph $G=(V,E)$. Define the set function $f$ over $V$ where $f(A)$ is the number of edges with both end points in $A$. Then $f$ is supermodular and monotone. Suppose $G$ has an independent set of size $k$. Then the minimum of $f(S)$ under the constraints $|S| = k$ (the uniform matroid constraint) will be $0$. If $G$ does not have an independent ...

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If the only thing that you know about $f$ is that it is non-increasing, then there is a simple adversary argument to show that you need to examine the value of $f$ on at least ${n \choose k}$ inputs (e.g., all ${n \choose k}$ different sets of cardinality $k$) in the worst case. For instance, consider functions $f$ that are $+\infty$ on all sets of ...

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Definition. For a given finite set $A$, a set function $f:2^A \rightarrow \mathbb{R}$ is submodular if for any $X, Y \subseteq A$ it holds that: $$f(X) + f(Y) \geq f(X \cup Y) + f(X \cap Y).$$ Lemma Given a bipartite graph $G = (A \cup B, E)$ with positive edge weights, let $f: 2^A \rightarrow \mathbb{R}^+$ be the function that maps $S\subseteq A$ to ...

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There is a paper by Gotovos, Hassani, and Krause (NIPS, 2015) titled "Sampling from Probabilistic Submodular Models" which describes a MCMC method based on Gibbs sampling. They show that the mixing time $t_{\text{mix}}(\epsilon)$ is bounded above by $$t_{\text{mix}}(\epsilon) \leq 2 n^2 \exp(\zeta_f) \log \left( \frac{1}{\epsilon p_\min} \right)$$ where $... 1 To simplify life, let$\mathcal V = [n] := \{1,2,\ldots,n\}$. For$A \subseteq [n]$, define$h(A):=\sum_{i \in A}c_i$. Note that$h$defines a modular (i.e additive) set function. Now, suppose there exists$\gamma \in [0, 1]$such that$f$is weakly$\gamma$-submodular, i.e such that$\$ \sum_{i \in B\setminus A}f(A \cup \{i\}) - f(A) \ge \gamma (f(A \cup B)...

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