Computing the sequence of $k$-cardinality assignments

TL;DR

This paper extends an existing $O(n^3)$ algorithm to compute the entire sequence of $k$-cardinality assignments in bipartite graphs, optimizing the process by adding a second stage for semi-essential terms.

Contribution

It introduces a two-stage algorithm that efficiently computes all $k$-cardinality assignments with potential speed improvements in certain cases.

Findings

The extended algorithm computes all assignments in $O(n^3)$ time.

The second stage accelerates computation when many semi-essential terms remain.

No overall speed benefit over existing algorithms in general cases.

Abstract

The $k$-cardinality assignment problem asks for finding a maximal (minimal) weight of a matching of cardinality $k$ in a weighted bipartite graph $K_{n,n}$, $k \leq n$. The algorithm of Gassner and Klinz from 2010 for the parametric assignment problem computes in time $O(n^3)$ the set of $k$-cardinality assignments for those integers $k \leq n$ which refer to "essential" terms of a corresponding maxpolynomial. We show here that one can extend this algorithm and compute in a second stage the other "semi-essential" terms in time $O(n^2)$, which results in a time complexity of $O(n^3)$ for the whole sequence of $k=1,...,n$-cardinality assignments. The more there are assignments left to be computed at the second stage the faster the two-stage algorithm runs. In general, however, there is no benefit for this two-stage algorithm on the existing algorithms, e.g. the simpler network flow algorithm based on the successive shortest path algorithm which also computes all the $k$-cardinality assignments in time $O(n^3)$.