Logic-based Benders Decomposition

• Logic-based Benders Decomposition is a method that partitions decision variables into master and subproblems, allowing arbitrary combinatorial structures instead of relying solely on linear relaxations.
• It leverages specialized algorithms and logic inference to derive strong, application-specific cuts that accelerate convergence even for NP-hard subproblems.
• The approach has broad applications including scheduling, robust optimization, and Bayesian inference, enabling exact or near-optimal solutions in otherwise intractable problems.

Logic-based Benders Decomposition (LBBD) generalizes classical Benders decomposition by enabling the subproblem to be an arbitrary combinatorial or inference optimization problem, as opposed to requiring a specific linear or convex structure. LBBD extends the decomposition paradigm to domains where specialized algorithms (such as constraint programming, dynamic programming, simulation, or logic inference) yield strong, non-linearizable subproblem relaxations and combinatorial Benders cuts, thus facilitating tractability in scheduling, planning, stochastic, robust, and hierarchical design problems that are otherwise intractable via direct mixed-integer programming or classical Benders approaches.

1. Formal Principles and Mathematical Framework

The primary abstraction in LBBD partitions the decision variables into a “master” set $x\in D_x$ and a “subproblem” set $y\in D_y$, subject to a global objective $f(x,y)$ and constraints $C(x,y), C'(x)$. When the master variables $x$ are fixed, the subproblem

$$\min_y\ f(x, y) \quad \text{s.t.}\ C(x, y),\ y \in D_y$$

is solved (optimally or via specialized inference). The Benders cut is then extracted as a function $B_{x^k}(x)$ of the master variables only, providing a valid lower bound on the subproblem’s contribution for all $x \in D_x$. The master problem at iteration $k$ accumulates these cuts: $$\min\ z\quad \text{s.t.}\ z \geq B_{x^i}(x)\ \forall i \leq k,\ x \in D_x,\ C'(x)$$ and is re-solved until convergence is achieved, typically when the lower and upper bounds coincide (Hooker, 2019).

LBBD does not assume the subproblem is LP/MILP or even convex; the only requirement is that inference from the fixed-$x$ subproblem must produce a valid, efficiently checkable master cut, possibly exploiting logic, combinatorics, or dual optimality of specialized algorithms (Hooker, 2019).

2. Master-Subproblem Decomposition and Cut Derivation

A distinguishing feature of LBBD is its capacity to exploit problem structure by formulating the subproblem in a form best suited to the domain—for instance, scheduling via constraint programming, or complex recourse decisions via simulation, as in

$$(\mathrm{MP})\quad \min_{x,\theta}\ C(x) + \theta\quad\text{s.t.}\ \theta \geq \alpha^k + \sum_i \beta^k_i (x_i - \bar x_i^k),\ x \in X,\ \theta \in \mathbb{R} \tag{1}$$

Each $\bar x^k$ triggers a subproblem: an evaluation of the true (possibly stochastic or simulated) performance $F(\bar x^k)$. LBBD then derives “combinatorial” Benders cuts by monotonicity properties or by other logic-based reasoning (Forbes et al., 2021).

For monotone subproblems, a fundamental cut is: $$\theta \geq F(\bar x) - \sum_{i: \bar x_i \geq 1} [F(\bar x) - F(\bar x - \mathbf e_i)] (x_i - \bar x_i)$$ General subproblem structures (such as those in answer set programming (Cappanera et al., 2023) or factor-graph MAP inference (Dubey et al., 24 Oct 2024)) allow for arbitrary optimality and feasibility cuts, augmenting the master with

$z \leq p(\bar y) + u(\bar y)^\top (y - \bar y)$

if the subproblem is feasible, or

$0 \leq w(\lambda) + \lambda^\top (y - \bar y)$

otherwise. In discrete and combinatorial settings, so-called “combinatorial” or “logic-based” cuts can be far tighter and more informative than generic dual-based constraints (Hooker, 2019, Forbes et al., 2021).

3. Algorithmic Loop and Convergence Properties

The generic LBBD algorithm alternates between solving the master for an incumbent $x^k$, then solving the subproblem for $x^k$:

1. Solve Master: $x^k \leftarrow \arg\min_{x} C(x) + \theta$ with all previously generated cuts.
2. Solve Subproblem (e.g., simulation, CP, scheduling): Estimate $F(x^k)$ or resolve logic constraints.
3. Cut Generation: If solution is suboptimal/infeasible, derive and add logic-based Benders cut $B_{x^k}(x)$ to the master.
4. Repeat until no improving $x$ is available (Forbes et al., 2021, Hooker, 2019).

When the space of master variables is finite (e.g., in 0-1 integer problems), and each cut eliminates at least one previously feasible solution or improves the lower bound, LBBD provably converges in finitely many iterations to a globally optimal solution (Assunção et al., 2020, Hooker, 2019, Forbes et al., 2021). This holds even when the subproblem is NP-hard and cannot be modeled or solved via LP duality (Assunção et al., 2020).

In practice, convergence can often be accelerated by embedding subproblem relaxations within the master, using strengthened cuts, or deploying branch-and-check single-tree strategies (Hooker, 2019, Elci et al., 2020).

4. Strength of Logic-Based Benders Cuts: Combinatorial and Monotonicity Classes

The ability of LBBD to derive strong, application-specific cuts is critical to its effectiveness. In resource-monotonic stochastic systems, monotonicity cuts exploit realized, finite-difference sensitivity of the subproblem to changes in $x$: $$\theta \geq F(\bar x) - \sum_{i} [F(\bar x) - F(\bar x - \mathbf e_i)] (x_i - \bar x_i)$$ Such cuts dominate basic no-good or feasibility cuts, and under mild assumptions can be facet-defining for the convex hull of feasible $(x,\theta)$ (Forbes et al., 2021).

LBBD also supports “supercut” strategies, where a single Benders cut may exclude all assignments in a $k$-OPT neighborhood, as demonstrated in resource-constrained scheduling: $$\sum_{(i,j,m): \bar x_{ijm}=1}(1 - x_{ijm}) + \sum_{(j,m): \bar y_{jm}=1}(1 - y_{jm}) \geq \delta$$ for all schedules within Hamming distance $k$ of a reference (Avgerinos et al., 2023). This mechanism enables exponential-size neighborhoods to be eliminated in a single iteration.

In simulated or stochastic settings, sample average approximation (SAA) preserves the monotonicity structure, permitting cuts that are valid under high-probability stochastic realizations (Forbes et al., 2021, Guo et al., 2019).

5. Applications Across Domains

LBBD has been applied to a wide array of domains, including:

• Stochastic resource allocation and scheduling: Incorporating simulation-based performance evaluation directly in the optimization loop via monotonically valid cuts allows exact solutions where earlier methods only yielded approximations or heuristics. In experiments, LBBD achieved 0% optimality gap with runtimes of 10–30 minutes, and required only 15–40 simulation calls (Forbes et al., 2021).
• Robust and min-max regret optimization: For interval uncertainty in 0–1 models, logic-based Benders cuts—derived from optimal solutions to induced subproblems—guarantee finite, optimal convergence even for NP-hard separation instances (Assunção et al., 2020).
• Intermodal transport, health care operations, and facility layout: By partitioning complicating assignments and leveraging combinatorial cuts or feasibility “nogoods,” substantial reductions in master problem size, solution time, and optimality gap are achieved (Avgerinos et al., 2022, Cappanera et al., 2023, Grus et al., 23 Dec 2025).
• Bayesian inference under logic constraints: Incorporating logical must-link/cannot-link/minimum-cluster-size constraints in master MAP inference while the subproblem exploits structured factor graphs enables proof of finite-time convergence, outperforming Gibbs and variational Bayes (Dubey et al., 24 Oct 2024).
• Stochastic and robust scheduling: CP or custom subproblems for scheduling, married with analytical or combinatorial cuts, gains orders-of-magnitude speedup over integer L-shaped approaches (Elci et al., 2020, Guo et al., 2019).

6. Algorithmic Extensions and Practical Innovations

Enhancements commonly observed in the recent literature include adaptive control of simulation sample size, hierarchical and recursive LBBD (as in hierarchical rectangle packing (Grus et al., 23 Dec 2025)), integration with partial relaxations and heuristics, and hybridization with branch-and-check or multi-shot approaches (notably in ASP and stochastic settings (Cappanera et al., 2023, Elci et al., 2020)).

LBBD variants systematically balance cut strength, subproblem scalability, and master problem tractability; cut management, warm starts, and analytic lower bounds embedded in the master further expedite convergence (Hooker, 2019, Forbes et al., 2021, Guo et al., 2019). In some settings, a single logic-based optimality cut may simultaneously penalize node removal and reward node addition to optimize algorithmic progress (Roohnavazfar et al., 2021).

7. Impact, Generality, and Future Directions

LBBD’s scope has expanded from scheduling and resource allocation to encompass stochastic, robust, and logical modeling domains, higher-order combinatorial systems, and hierarchical design. Extensions may include multi-stage settings, generalized stochastic recourse, and more expressive logic constraints. Its capacity to deploy subproblem-specific algorithms (discrete event simulation, CP, BDD-based submodels, factor-graph inference) makes it a major tool for large-scale, intractable systems across operations research, AI planning, robust control, and logistics.

Across these domains, the use of application-tailored, logic-based combinatorial Benders cuts—often exploiting monotonicity, finite difference, or strong dual inference—enables exact or near-optimal solutions to problems well beyond the reach of classical mixed-integer programming or convex decomposition, with strong iteration and runtime scaling across tested benchmarks (Forbes et al., 2021, Assunção et al., 2020, Grus et al., 23 Dec 2025, Guo et al., 2019).