Generalized Colorful Linear Programming and Further Applications

Colorful linear programming (CLP) is a generalization of linear programming that was introduced by Bárány and Onn. Given k point sets C1, . . . , Ck ⊂ R that each contain a point b ∈ R in their positive span, the problem is to compute a set C ⊆ C1 ∪ · · · ∪ Ck that contains at most one point from each set Ci and that also contains b in its positive span, or to state that no such set exists. CLP is known to be NP-hard. We consider a generalization of CLP in which we are given additionally for each set Ci a number li ∈ N, i = 1, . . . , k, and we want to find a set that contains at most li points from Ci. We call this problem generalized colorful linear programming (GCLP). While we show that even seemingly simple cases of GCLP remain NP-hard, we present a weakly-polynomial algorithm for the special case that there are only two colors and that the vectors of each set Ci contain b in their positive span. This case is particularly interesting due to its connection with the colorful Carathéodory theorem. Furthermore, we consider additional applications of CLP to problems on colored graphs.