Scheduling with Obligatory Tests

TL;DR

The paper studies scheduling with obligatory tests on a single machine, where each job $j$ has a known test time $t_j$ revealing the unknown processing time $p_j$, with the goal to minimize $\sum_j C_j$. It introduces 1-SORT for arbitrary test times and proves a tight $\approx1.861$-competitive bound using a novel delay-graph decomposition; for uniform test times it presents a threshold-based algorithm achieving about $1.585$-competitiveness and establishes a $\sqrt{2}$ lower bound for determinism. The work also provides a rigorous lower bound for arbitrary test times showing no deterministic algorithm can beat $\approx1.618$ (at $\beta=1$) and extends the uniform-case analysis with a separate SIDLE approach. Overall, the results advance the understanding of scheduling with obligatory testing by offering improved competitive ratios and a new analytical framework based on edge-delay decompositions, with practical implications for medical, maintenance, and diagnostic workflows where tests precede processing.

Abstract

Motivated by settings such as medical treatments or aircraft maintenance, we consider a scheduling problem with jobs that consist of two operations, a test and a processing part. The time required to execute the test is known in advance while the time required to execute the processing part becomes known only upon completion of the test. We use competitive analysis to study algorithms for minimizing the sum of completion times for $n$ given jobs on a single machine. As our main result, we prove using a novel analysis technique that the natural $1$-SORT algorithm has competitive ratio at most 1.861. For the special case of uniform test times, we show that a simple threshold-based algorithm has competitive ratio at most 1.585. We also prove a lower bound that shows that no deterministic algorithm can be better than $\sqrt{2}$-competitive even in the case of uniform test times.

Figures (6)

• Figure 1: Instance with two jobs where the delay ratio on the arc $(1,2)$ is arbitrarily close to $2$. A job with test time $t_j$ and processing time $p_j$ is written as a pair $(t_j,p_j)$.
• Figure 2: Instance with $2k$ jobs to illustrate red (drawn solid), blue (drawn dashed) and green (drawn dotted) arcs. A job with test time $t_j$ and processing time $p_j$ is written as a pair $(t_j,p_j)$.
• Figure 3: Illustration of the idea underlying the analysis of a red vertex $j$: If the blue arcs have not yet been used in the analysis of a previous red vertex, they can be used in combination with the incoming red arcs of $j$ (left). If blue arcs have already been used in the analysis of a previous red vertex $j'$, there must be a green arc between $j'$ and $j$ that is also available to be used in the analysis of the incoming red arcs of $j$.
• Figure 4: Tree of possibilities for $D(j,k)$ and $D^*(j,k)$ and $\rho_{jk}$ -- the top label at each leaf is the order of execution by $1$-SORT, the bottom label is $D(j,k)$. If we assume that $t_j+p_j\le t_k+p_k$, then Leaf $5$ is impossible to reach and $D^*(j,k)=t_j+p_j$ for all other leaves.
• Figure 5: Illustration of Invariant \ref{['inv:full']}. Only blue arcs between vertices in $N^-_{\ge j}(k)$ are shown.

Theorems & Definitions (16)

• Theorem 1
• Corollary 2
• Definition 3
• Definition 4
• Lemma 5
• Lemma 6
• Lemma 7
• Lemma 8
• Lemma 9
• Lemma 10