Single-Machine Coupled Task Scheduling

• Single-Machine Coupled Task Scheduling is a scheduling problem where each coupled task has two sub-tasks separated by a fixed idle period, and compatibility allows nesting of tasks.
• The problem is strongly NP-hard and exhibits complex behavior even under homogeneous or stretched task conditions, with various graph structures influencing its tractability.
• Algorithmic approaches such as matching methods, constraint programming, and hybrid heuristics provide near-optimal approximations in practical applications.

A single-machine coupled task scheduling problem concerns the non-preemptive scheduling of a set of "coupled-tasks," where each task comprises two sub-tasks separated by an exact (mandatory) idle interval. The defining feature is that the single processor can execute at most one sub-task at a time, but compatibility between tasks permits certain sub-tasks to be scheduled within the idle interval of another, drastically affecting the achievable makespan or total completion time. The problem, motivated by applications such as data acquisition in submarine torpedoes or industrial processes, is structurally complex and displays a rich interplay between scheduling, graph theory, and combinatorial optimization.

1. Formal Problem Definition

Each coupled-task $T_i$ consists of two sub-tasks ($a_i$, $b_i$) with processing times $p(a_i)=a_i$ and $p(b_i)=b_i$. These must be separated by a prescribed idle time $L_i$ (for many studied variants, $L_i=a_i+b_i$ or $L_i = a_i = b_i$). The central constraints are:

• Precedence: $b_i$ can only start after $a_i$ completes and the idle period $L_i$ has elapsed.
• Non-preemption: No task may be interrupted once started.
• Single-processor: No overlap between any two sub-tasks.
• Compatibility: An undirected graph $G_c=(V,E)$ encodes admissible task "nestings": if $(T_i,T_j)\in E$, then some sub-task of $T_i$ can be scheduled during the idle period of $T_j$ (and vice versa).

A schedule $\sigma$ assigns start times to all sub-tasks, observing all constraints. The main objectives are:

• Makespan minimization: $$C_{\max}=\max_{1\leq i\leq n}\bigl(\sigma(b_i)+b_i\bigr)$$.
• Sum of completion times: $\sum_{i=1}^{n} C_i$, where $C_i$ is the completion time of the second sub-task $b_i$.

Variants differ in whether task parameters are common or arbitrary, the structure of $G_c$, or are subject to specific application-motivated symmetries (Simonin et al., 2017, Giroudeau et al., 2017, Barbosa et al., 29 Dec 2025, Fischer et al., 2022).

2. Complexity and Structural Hardness

The coupled-task scheduling problem is strongly NP-hard in its general form, both for makespan and sum of completion times objectives (Simonin et al., 2017, Fischer et al., 2022, Barbosa et al., 29 Dec 2025). The NP-hardness persists under severe restrictions:

• Homogeneous tasks: Even when all $a_i=a$, $b_i=b$, $L_i=a+b$, and $b<a$, the decision problem is NP-complete via reduction from Hamiltonian Path (Simonin et al., 2017).
• Stretched tasks: In the "stretched" case ($a_i = b_i = L_i = p$), the problem remains NP-hard (Darties et al., 2017, Giroudeau et al., 2017).
• Restricted compatibility graphs: Inapproximability results hold for $1$-stage bipartite graphs, quasi split-graphs, and even for star-like compatibility structures (Giroudeau et al., 2017, Darties et al., 2017).
• Sum of completion times: Strong NP-hardness is proved for cases such as $1\,(p_j,L,p_j)\,\sum C_j$ and for fixed orderings of sub-tasks (Fischer et al., 2022).

ETH-based lower bounds exclude sub-exponential exact algorithms for a range of topologies (Giroudeau et al., 2017).

3. Graph-Theoretic Reformulations

Several formulations leverage graph-theoretic models to encapsulate the nested structure imposed by compatibility:

• Minimum Schedule-Linked Disjoint-Path Cover (Min-SLDPC): Each schedule corresponds to a vertex-disjoint path cover of $G_c$. The cost links directly to the number and structure of these paths. For homogeneous tasks, $C_{\max}(\sigma)=n(a+b)+\sum_{p\in\mathcal{P}}w(p)$, where $w(p) = a$ for paths of length one and $w(p) = a+b$ otherwise (Simonin et al., 2017).
• Packing interpretations: For stretched tasks, the problem can be mapped to bin packing, subset-sum, and (multiple-)knapsack variants, especially for bipartite and star-structured compatibility graphs (Darties et al., 2017). The interplay between the compatibility graph structure and the combinatorics of packing determines tractability.

4. Algorithmic Approaches and Approximability

Despite pervasive intractability, several constant-factor algorithms are known for specific situations and graph topologies:

Graph/Parameters	Complexity	Best Known Approximation Ratio
Edgeless (no compatibility)	Trivial	1
Path/Chain	Polynomial	O($n^3$) (Darties et al., 2017)
Star (center idle hosts satellites)	Polynomial	O($n$) (Darties et al., 2017)
Star (center only nested inside)	NP-complete	FPTAS (Darties et al., 2017)
1-stage bipartite, deg$(Y)\leq2$	Polynomial	O($n^3$) (Darties et al., 2017)
1-stg bipartite (general)	NP-complete	7/6 (Darties et al., 2017), $5/4$ ((Giroudeau et al., 2017), quasi split-graph $|Y|=1$)
1-stage complete bipartite	NP-complete	PTAS, $13/12$ if $Y$ uniform (Darties et al., 2017)
General $G_c$	NP-hard	$3/2 \leq (3a+2b)/(2a+2b) < 5/4$ (Simonin et al., 2017), $\approx 1.37$ for special classes

Algorithmic paradigms include:

• Maximum matching for general $G_c$: Matching-based approximation achieves $(3a+2b)/(2a+2b)$ (Simonin et al., 2017).
• Min-DPC method: If Min-DPC is solvable in polynomial time on $G_c$, a $\frac{a+b}{a}$-approximation is achievable. On interval graphs and other "easy" classes, this improves the ratio to $\approx 1.37$ (Simonin et al., 2017).
• Flow plus matching for quasi split-graphs: A $5/4$-approximation is achieved by reducing to a flow and matching problem (Giroudeau et al., 2017).
• FPTAS via subset-sum: For some star graphs, the problem reduces to subset-sum, admitting an efficient FPTAS (Darties et al., 2017).
• Knapsack-based reductions: Bipartite models yield $7/6$ or $13/9$ approximations by recasting as (restricted) multi-knapsack problems (Darties et al., 2017).

For sum-of-completion-time objectives, greedy, block-oriented, and chain-based scheduling play key roles, with ratios such as $2$ for fixed symmetric tasks and $1.5$ in special cases (Fischer et al., 2022).

5. Practical Optimization and Heuristics

Recent experimental research has produced effective practical heuristics and exact models for large instances:

• Constraint Programming (CP): Encodes precedence, exact delays, and non-overlap via global disjunctive constraints. Achieves proven optimality or very tight bounds for moderate-scale instances ($n \leq 25$) when allowed sufficient run-time and parallelization (Barbosa et al., 29 Dec 2025).
• Biased Random-Key Genetic Algorithm (BRKGA): Decodes chromosomes as orderings, then inserts jobs optimally via a first-fit approach that respects schedule constraints. Augmented with local search and periodic "shake" perturbations, BRKGA efficiently explores the solution space, regularly obtaining near-optimal makespans ($<$1% gap for $n \leq 100$) in seconds (Barbosa et al., 29 Dec 2025).
• Hybrid/bi-objective frameworks: Approximate solutions for makespan and sum-of-completion-times objectives simultaneously using the Stein-Wein composition bound, achieving $(\rho,\rho)$-approximations that synthesize the best available single-objective algorithms (Fischer et al., 2022).

A summary of observed practical performance:

Approach	Instance Size	Optimality/BK Ratio	Typical Runtime	Applicability
CP (multi-core)	$n\leq 25$	90.6% BK found	$~1$h	Exact or tight bounds
BRKGA	$n\leq 100$	$<1\%$ gap	Seconds	Anytime, large-scale

*BK: Best-known solution (Barbosa et al., 29 Dec 2025). The combination of shake and local search within BRKGA is critical to closing the optimality gap.

6. Impact of Compatibility Constraints and Topological Structure

Allowing compatibility fundamentally transforms the single-machine coupled-task problem from a trivial sum-of-lengths scheduling task into a combinatorial packing problem in which the ability to nest sub-tasks within the idle intervals of others introduces a new dimension of complexity. The structure of $G_c$ plays a decisive role in tractability and approximability:

• Edgeless $G_c$ produces trivial sequential schedules.
• Dense or well-structured $G_c$ (e.g., interval graphs, cographs, cacti) enable improved approximations.
• Even restricted topologies such as quasi split-graphs or 1-stage bipartite graphs may already induce NP-hardness and inapproximability.
• In the most favorable structured cases (trees, chains), efficient polynomial algorithms are achievable (Darties et al., 2017, Simonin et al., 2017).

Implications include the necessity for problem-specific algorithm selection and topological analysis before selecting or designing solution methods (Giroudeau et al., 2017, Simonin et al., 2017).

7. Open Problems and Research Directions

Key challenges and open questions persist in the study of single-machine coupled-task scheduling:

• Complexity characterization for tree-shaped or bounded-degree compatibility graphs is unresolved (Darties et al., 2017).
• Extension of polynomial-time or FPT algorithms to broader classes of $k$-stage bipartite graphs or other natural topologies remains an open avenue (Darties et al., 2017).
• Tightening of approximation ratios, closing the worst-case gap between $1.37$ (Min-DPC classes) and $1.5$ (general $G_c$), and parameterized complexity results.
• For sum-of-completion-time objectives, the existence of constant-factor approximations for the most general parameter setting is still open (Fischer et al., 2022).
• The integration of advanced metaheuristics with hybrid exact-approximate models promises further practical advances, particularly for very large-scale or online settings (Barbosa et al., 29 Dec 2025).

Single-machine coupled task scheduling thus remains a technically rich, highly structured area at the intersection of scheduling theory, graph algorithms, and combinatorial optimization, with significant open theoretical and engineering challenges.