# Budget Minimization with Precedence Constraints

## TL;DR

This paper investigates the complexity of minimizing the budget in precedence-constrained scheduling, proving NP-hardness of approximation and providing polynomial algorithms for specific order types.

## Contribution

It extends previous NP-hardness results, introduces structural insights for simpler algorithms, and offers polynomial solutions for series-parallel and convex bipartite orders.

## Key findings

- NP-hard to approximate within any factor on bipartite partial orders
- Existence of optimal solutions that partition jobs for independent scheduling
- Polynomial algorithms for series-parallel and convex bipartite partial orders

## Abstract

Budget Minimization is a scheduling problem with precedence constraints, i.e., a scheduling problem on a partially ordered set of jobs $(N, \unlhd)$. A job $j \in N$ is available for scheduling, if all jobs $i \in N$ with $i \unlhd j$ are completed. Further, each job $j \in N$ is assigned real valued costs $c_{j}$, which can be negative or positive. A schedule is an ordering $j_{1}, \dots, j_{\vert N \vert}$ of all jobs in $N$. The budget of a schedule is the external investment needed to complete all jobs, i.e., it is $\max_{l \in \{0, \dots, \vert N \vert \} } \sum_{1 \le k \le l} c_{j_{k}}$. The goal is to find a schedule with minimum budget. Rafiey et al. (2015) showed that Budget Minimization is NP-hard following from a reduction from a molecular folding problem. We extend this result and prove that it is NP-hard to $\alpha(N)$-approximate the minimum budget even on bipartite partial orders. We present structural insights that lead to arguably simpler algorithms and extensions of the results by Rafiey et al. (2015). In particular, we show that there always exists an optimal solution that partitions the set of jobs and schedules each subset independently of the other jobs. We use this structural insight to derive polynomial-time algorithms that solve the problem to optimality on series-parallel and convex bipartite partial orders.