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Classes of set functions along with a choice of ground set are a bedrock todetermine and develop corresponding variants of greedy algorithms to obtainefficient solutions for combinatorial optimization problems. The class ofapproximate constrained submodular optimization has seen huge advances at theintersection of good computational efficiency, versatility and approximationguarantees while exact solutions for unconstrained submodular optimization areNP-hard. What is an alternative to situations when submodularity does not hold?Can efficient and globally exact solutions be obtained? We introduce one suchnew frontier: The class of quasi-concave set functions induced as a dual classto monotone linkage functions. We provide a parallel algorithm with a timecomplexity over $n$ processors of $\mathcal{O}(n^2g)+\mathcal{O}(\log{\log{n}})$ where $n$ is the cardinality of the ground set and$g$ is the complexity to compute the monotone linkage function that induces acorresponding quasi-concave set function via a duality. The complexity reducesto $\mathcal{O}(gn\log(n))$ on $n^2$ processors and to $\mathcal{O}(gn)$ on$n^3$ processors. Our algorithm provides a globally optimal solution to amaxi-min problem as opposed to submodular optimization which is approximate. Weshow a potential for widespread applications via an example of diverse featuresubset selection with exact global maxi-min guarantees upon showing that astatistical dependency measure called distance correlation can be used toinduce a quasi-concave set function.

Parallel Quasi-concave set optimization: A new frontier that scales without needing submodularity