(2023) An EPTAS for Budgeted Matroid Independent Set.
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Abstract
We consider the {\em budgeted matroid independent set} problem. The input is a ground set, where each element has a cost and a non-negative profit, along with a matroid over the elements and a budget. The goal is to select a subset of elements which maximizes the total profit subject to the matroid and budget constraints. Several well known special cases, where we have, e.g., a uniform matroid and a budget, or no matroid constraint (i.e., the classic knapsack problem), admit a fully polynomial-time approximation scheme (FPTAS). In contrast, already a slight generalization to the {\em multi-budgetedmatroid independent set} problem has a PTAS but does not admit an efficient polynomial-time approximation scheme (EPTAS). This implies a PTAS for our problem, which is the best known result prior to this work. Our main contribution is an EPTAS for the budgeted matroid independent set problem. A key idea of the scheme is to find a {\em representative set} for the instance, whose cardinality depends solely on $1/\eps$, where $\eps >0$ is the accuracy parameter of the scheme. The representative set is identified via matroid basis minimization, which can be solved by a simple greedy algorithm. Our scheme enumerates over subsets of the representative set and extends each subset using a linear program. The notion of representative sets may be useful in solving other variants of the budgeted matroid independent set problem.
| Item Type: | Conference or Workshop Item (A Paper) (Paper) |
|---|---|
| Divisions: | Dániel Marx (DM) |
| Conference: | UNSPECIFIED |
| Depositing User: | Ariel Kulik |
| Date Deposited: | 05 May 2023 11:23 |
| Last Modified: | 11 May 2023 06:46 |
| Primary Research Area: | NRA1: Trustworthy Information Processing |
| URI: | https://publications.cispa.saarland/id/eprint/3939 |
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- An EPTAS for Budgeted Matroid Independent Set. (deposited 05 May 2023 11:23) [Currently Displayed]
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