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We consider a variant of a knapsack problem with a fixed cardinality constraint. There are three objective functions to be optimized: one real-valued and two integer-valued objectives. We show that this problem can be solved efficiently by a local search. The algorithm utilizes connectedness of a subset of feasible solutions and has optimal run-time.

The hypervolume subset selection problem consists of finding a subset, with a given cardinality, of a nondominated set of points that maximizes the hypervolume indicator. This problem arises in selection procedures of population-based heuristics for multiobjective optimization, and for which practically efficient algorithms are strongly required. In this article, we provide two new formulations for the two-dimensional variant of this problem.
The first is an integer programming formulation that can be solved by solving its linear relaxation. The second formulation is a \(k\)-link shortest path formulation on a special digraph with Monge property that can be solved by dynamic programming in \(\mathcal{O}(n^2)\) time complexity. This improves upon the existing result of \(O(n^3)\) in Bader.

The hypervolume subset selection problem consists of finding a subset, with a given cardinality \(k\), of a set of nondominated points that maximizes the hypervolume indicator. This problem arises in selection procedures of evolutionary algorithms for multiobjective optimization, for which practically efficient algorithms are required. In this article, two new formulations are provided for the two-dimensional variant of this problem.
The first is a (linear) integer programming formulation that can be solved by solving its linear programming relaxation. The second formulation is a \(k\)-link shortest path formulation on a special digraph with the Monge property that can be solved by dynamic programming in \(\mathcal{O}(n(k + \log n))\) time. This improves upon the \(\mathcal{O}(n^2k)\) result of Bader (2009), and matches the recent result of Bringmann et al. (2014), which was developed independently from this work using different techniques. Moreover, it is shown that these bounds may be further improved under mild conditions on \(k\).