The counting complexity of a simple scheduling problem

Abstract

Let $T$ be a set of tasks. Each task has a non-negative processing time and a deadline. The problem of determining whether or not there is a schedule of the tasks in $T$ such that a single machine can finish processing each of them before its deadline is polynomially solvable. We prove that counting the number of schedules satisfying this condition is #P-complete.

Introduction

We consider the single machine scheduling problem with processing times and deadlines which can be described as follows. Let $T=\{1,\dots ,n\}=\left[n\right]$ be a set of tasks. With each task are associated two non-negative integers, a processing time $p_i$ and a deadline $d_i$. Let $S_n$ be the symmetric group on $\left[n\right]$. A schedule of the tasks consists of a permutation $\sigma \in S_n$. A schedule is said to be feasible if ${\sum }_{i=1}^j{p}_{\sigma \left(i\right)}\le d_{\sigma \left(j\right)}$ for all $1\le j\le n$. In other words, $\sigma$ is feasible if a single machine can process the tasks following the order of $\sigma$ and is able to finish each of them before its deadline. Determining whether or not there exists a feasible schedule can be done by ordering tasks according to their respective deadline in increasing order. If such a schedule, generally called “earliest deadline first”, is not feasible then no feasible schedule exists. Several extensions to this problem are also polynomial time solvable. Two examples are the problem of finding a schedule that minimizes the number of tardy tasks [1] and the problem of determining whether or not there exists a feasible schedule which respects a given set of precedence constraints between the tasks [2]. In this article we prove that the problem of counting the number of feasible schedules for this problem belongs to the class #P-complete, a class of hard counting problems introduced by Valiant [3]. This result shows that even for a very simple scheduling problem, heuristic or approximate algorithms for counting the number of solutions may be the only useful ones. The complexity of counting solutions in scheduling was also studied by [4] but in more complex multi-criteria problems.

In Section 2 we define the counting version of the problem studied, as well as other relevant definitions. The hardness result is presented in Section 3. An open question is given in Section 4.