8 research outputs found
Knapsack Cover Subject to a Matroid Constraint
We consider the Knapsack Covering problem subject to a matroid constraint. In this problem, we are given an universe U of n items where item i has attributes: a cost c(i) and a size s(i). We also have a demand D. We are also given a matroid M = (U, I) on the set U. A feasible solution S to the problem is one such that (i) the cumulative size of the items chosen is at least D, and (ii) the set S is independent in the matroid M (i.e. S is in I). The objective is to minimize the total cost of the items selected, sum_{i in S}c(i).
Our main result proves a 2-factor approximation for this problem.
The problem described above falls in the realm of mixed packing covering problems. We also consider packing extensions of certain other covering problems and prove that in such cases it is not possible to derive any constant factor pproximations
Density Functions subject to a Co-Matroid Constraint
In this paper we consider the problem of finding the {\em densest} subset
subject to {\em co-matroid constraints}. We are given a {\em monotone
supermodular} set function defined over a universe , and the density of
a subset is defined to be f(S)/\crd{S}. This generalizes the concept of
graph density. Co-matroid constraints are the following: given matroid \calM
a set is feasible, iff the complement of is {\em independent} in the
matroid. Under such constraints, the problem becomes \np-hard. The specific
case of graph density has been considered in literature under specific
co-matroid constraints, for example, the cardinality matroid and the partition
matroid. We show a 2-approximation for finding the densest subset subject to
co-matroid constraints. Thus, for instance, we improve the approximation
guarantees for the result for partition matroids in the literature