Arc-Set Relaxations in Optimization

• Arc-set relaxations are techniques that reformulate optimization problems by redefining feasible regions on aggregated network arcs for improved computational efficiency.
• They leverage disjunctive programming, dynamic programming, and cutting planes to manage complementarity constraints and time-indexed formulations effectively.
• Applications include vehicle routing, cutting stock, and resource scheduling, with empirical results showing near-complete closure of integrality gaps.

Arc-set relaxations are a class of techniques for reformulating and tightening mathematical programming relaxations—especially mixed-integer and pseudo-polynomial-sized models—by replacing difficult constraints or state-spaces with relaxations defined on carefully chosen sets of arcs. Such relaxations have enabled strong linear bounds and efficient formulations for a range of challenging optimization problems, notably those with complementarity constraints (Pia et al., 2022), dynamic resource constraints (Lima et al., 2020), or time-dependent network structure (Dyk et al., 2023). They are grounded in network flow theory and dynamic programming, often leveraging aggregation, disjunctive programming, extended formulations, and polyhedral cutting-plane techniques.

1. Foundations of Arc-Set Relaxations

Arc-set relaxation refers to constructing a surrogate or relaxed model for an optimization problem by redefining its feasible region based on flows (or transitions) along selected subsets of arcs in a (possibly aggregated) network representation. The central idea is to exploit problem structure—such as the disjunctive logic of complementarity relations, dynamic programming state transitions, or time expansion in network design—by either aggregating large state spaces, relaxing combinatorial constraints, or enabling tighter convexifications in extended variable spaces.

The relaxation operates by reformulating the original (often intractable) constraints into tractable subproblems defined on arc sets, where the “arc” may represent:

• a conflicting variable pair (as in complementarity constraints),
• a DP transition (in arc-flow models),
• a time-indexed activity in dynamic networks.

The arc-set is then relaxed through aggregation, disjunction, or projection to yield a model that is computationally efficient and yet produces high-quality bounds.

2. Arc-Set Relaxations in Complementarity-Constrained Linear Programs

In linear programs with complementarity constraints (LPCCs), arc-set relaxations exploit the structure of variable pairs that cannot simultaneously take positive values. The canonical LPCC introduces a conflict graph $G=(V,E)$, where each edge $\{i,j\}\in E$ encodes $y_i \cdot y_j = 0$.

Edge-by-edge arc-set relaxation constructs, for each edge $e = \{i,j\}$, a disjunctive Balas-style extended formulation introducing selector variables $q_e \in [0,1]$ and auxiliary copies of variables. The convex hull of the union of feasible sets under $y_i = 0$ or $y_j = 0$ is explicitly described. Projection yields relaxations such as $y_i + y_j \le 1$. By applying such a block for every $e \in E$ and merging back variables, the global relaxation $\mathcal{M}$ is obtained.

Vertex cover-based relaxation generalizes this idea: select a covering of $E$ by disjoint vertex subsets, treating each set as a disjunction. The extended system involves variables and selector scalars $q_T$ per set $T$, and constraints are imposed to partition, project, and prohibit simultaneous activation across neighboring nodes. This construction generalizes the ERLT of Nguyen–Richard–Tawarmalani to general graphs, with the property $$P^{\perp} \subseteq_\mathrm{proj} P_{\mathrm{VC}} \subseteq P$$, and exactness upon full covering refinement (Pia et al., 2022).

3. Arc-Set Relaxations in Network Flow and Dynamic Programming

Arc-flow formulations establish a direct mapping from dynamic programming (DP) models to network flow problems, with each DP state or transition represented as a node or arc. For large or exponential DP state spaces, classical arc-flow models become too large; arc-set (state-space) relaxation aggregates multiple DP states via a surjective $g: S \to S'$, forming a relaxed network $N' = (S',A')$.

The relaxed flow model operates on $S'$ and $A'$, where each $A'$ arc aggregates all original transitions between its endpoints. Arc costs are set to the maximal DP reward/cost among pre-images. This relaxation yields a valid upper bound satisfying

$$\max\,\text{path-cost in } N' \geq \max\,\text{path-cost in } N_{\mathrm{DP}}$$

and substantially reduces the problem size. For applications such as cutting stock and vehicle routing, the arc-set relaxations are known to provide nearly tight LP bounds (e.g., $\leq 1$ unit in cutting stock, $<0.1\%$ gap in CVRP benchmarks) and are amenable to efficient MILP solution (Lima et al., 2020).

4. Arc-Dependent Discretization in Time-Indexed Network Design

Time-dependent network design and routing problems often utilize time-indexed formulations, which suffer from scalability due to the number of time-expanded nodes/arcs. The dynamic discretization discovery (DDD) method introduces a node-level decomposition of time grids. The arc-set relaxation framework extends this by associating a distinct departure time set $T_a$ to each arc $a$, enabling adaptive sparsification.

Under this scheme, the time-expanded network only contains copies of arcs for each $t \in T_a$ (subject to time feasibility), and the integer program is constructed over this "arc-sparse" structure. The auxiliary network splits nodes per arc group and maintains only a necessary subset of timed nodes. During iterative DDD, the feasible region is successively expanded by adding timed nodes/arcs where required, guided by path extraction and failure diagnostics.

Computationally, arc-level discretization yields order-of-magnitude reductions in variable and constraint counts and up to 57% reductions in runtime for certain service network design instances compared to node-grid approaches, as confirmed across benchmark families (fixed-route, hub-and-spoke) (Dyk et al., 2023).

5. Polyhedral Strengthening and Cutting Planes

Embedding arc-set relaxations into an extended formulation facilitates the addition of powerful linear cutting planes:

• Stable set (clique/odd-cycle) inequalities: Any valid inequality for the stable set polytope of the conflict graph ($\alpha^{\top} y \leq \beta$) can be applied, and is linearizable in the extended $(q,v,\bar{v})$ space.
• Bipartite Boolean Quadric Polytope (BQP) inequalities: Odd-cycle cuts in the bipartite BQP defined over the copies of selection and activation variables further tighten the feasible region, removing fractional "bilinear" artifacts prevalent in relaxed extended spaces (Pia et al., 2022).

Such cuts dramatically close the LP optimality gap, typically reducing residual gaps from several percent to below 1% or tighter, with cases observed where the gap falls to 0.00% (e.g., on TPESC with $|S|=10$, $|D|=10$, conflict density 40%).

6. Guidelines and Theoretical Insights

Designing effective arc-set relaxations typically follows these principles (Lima et al., 2020):

• Identify underlying DP/network structure: Formulate the full network, even if exponential.
• Select aggregation dimension: Choose state components (e.g., resource, time, memory) to aggregate based on tractability and bound strength trade-offs.
• Aggregate transitions and cost functions: For each arc in the relaxed network, assign an aggregated cost (maximizing over projections) to guarantee bound validity.
• Balance model size and relaxation strength: Aggressive aggregation yields compact relaxations but may require branch-and-cut to recover tightness. Keeping more DP dimensions yields larger but stronger models, often nearly integral at the LP level.
• Apply post-processing: “Fake” arcs may introduce infeasibilities; these can sometimes be resolved via branching or solution augmentation.
• Cutting and fixing: Apply dominance, graph reduction, and reduced-cost fixing to further shrink the search space and accelerate solution.

7. Practical Impact and Applications

Arc-set relaxations have delivered state-of-the-art codes for mixed-integer and pseudo-polynomial network problems, including:

• Linear programs with general complementarity constraints (via arc- and vertex-cover relaxations and polyhedral cuts) (Pia et al., 2022)
• Time-expanded network design with arc-based dynamic discretization (Dyk et al., 2023)
• Resource-constrained scheduling, cutting stock, bin-packing, and vehicle routing, leveraging state-space relaxations aggregated via arc-flow structure (Lima et al., 2020)

Empirical results consistently demonstrate dramatically reduced problem sizes and integrality gaps—often closing nearly 100% of gaps that remain open under classical formulations—making arc-set relaxations a foundational tool in modern combinatorial and mixed-integer optimization.