Minconvex Factors of Prescribed Size in Graphs

We provide a polynomial algorithm that determines for any given undirected graph $G=(V,E)$, positive integer $k$, and convex functions $f_v:\mathbb{N}\rightarrow\mathbb{R}$ ($v\in V$) a subgraph $H=(V,F)$ of $k$ edges that minimizes $\sum_{v\in V}f_v(d_H(v))$, where $d_H(v)$ is the degree of $v$ in $H$. The motivation and at the same time the main application of the results is the problem of finding a subset of $k$ vertices in a line graph that covers as many edges as possible. The latter problem generalizes the vertex cover problem for line graphs, which is in turn equivalent to the maximum matching problem in graphs. Improving paths or walks for factorization problems have to be completed by pairs of such walks for this problem. We provide several solutions leading to different variants of the problem and also show the limits of the methods by proving the NP-completeness of some direct extensions, in particular to all convex functions.