Post-Processing Heuristics in Network Design

• Post-processing heuristics are algorithmic strategies applied after LP or ILP rounding to iteratively refine network design solutions, reducing over-provisioning.
• They employ methods like 1-DWN and 1-UP to fix fractional variables one-by-one, maintaining feasibility for any scaled demand matrix.
• Empirical evaluations show these heuristics yield near-optimal solutions with a 1–3% gap to ILP optima and significantly lower computational times compared to full ILP solvers.

Post-processing heuristics are algorithmic strategies employed after an initial algorithmic solution—typically an LP or ILP relaxation and rounding step—to further improve the quality or feasibility of computed network designs, flows, or allocations. In the context of traffic-oblivious multi-commodity flow network design and related green traffic engineering tasks, post-processing heuristics specifically refer to methods that iteratively adjust or refine a solution obtained from continuous relaxations/rounding steps, with the goal of reducing resource usage (such as the number of active links or connections) while still guaranteeing specified universal multi-commodity flow properties. These heuristics are crucial for bridging the gap between the theoretical approximation ratios and practical near-optimality.

1. Formal Context and Definition

Post-processing heuristics arise most prominently in computational settings where a combinatorial network design problem is NP-hard and the best available polynomial-time algorithms proceed by (a) formulating a continuous (LP or convex) relaxation, (b) solving this relaxation to obtain a fractional solution, and (c) rounding the fractional solution to an integer one. A canonical example is the Minimum Multi-Commodity Flow Subgraph (MMCFS) problem, where the goal is to find a sparse subgraph supporting all scaled routable traffic matrices in a traffic-oblivious fashion (Chimani et al., 23 Apr 2025), or, more generally, activating a minimal set of network resources such that all downscaled demand matrices remain feasible (Ilsen et al., 19 Jan 2026).

Post-processing in this context refers to an additional algorithmic layer applied after initial rounding—such as simple ceiling rounding of LP solutions—to tighten the solution, potentially reducing over-provisioning introduced by the rounding and yielding solutions much closer to the true integral optimum without sacrificing universal feasibility guarantees. These techniques are called "heuristics" because, while they do not offer better worst-case theoretical approximation guarantees than the rounding step they refine, in practice they regularly deliver more efficient solutions.

2. Motivations and Implications in Traffic-Oblivious Green TE

The main motivation for applying post-processing heuristics in traffic-oblivious green traffic engineering is to minimize energy consumption by reducing the number of active network connections, subject to the constraint that any scaled admissible demand matrix remains routable. LP rounding, while fast and theoretically robust (e.g., achieving a max $\max((\varrho\lambda_{\min,2})^{-1}, 2)$-approximation in the link-deactivation problem (Ilsen et al., 19 Jan 2026)), can be pessimistic: it may result in activating unnecessary links due to integrality gaps, local relaxation artifacts, or worst-case-oriented conservative rounding tails.

The role of post-processing heuristics is to explore the feasible region more granularly, fixing variables iteratively and re-optimizing conditionally, thereby tightening the solution and potentially eliminating excess active links. The practical implication is energy savings (fewer active links) without the need for frequent network reconfigurations, reduced network management complexity, and increased robustness to unforeseen traffic patterns or diurnal cycles (Ilsen et al., 19 Jan 2026).

3. Methodological Details: 1-DWN and 1-UP Heuristics

For the traffic-oblivious link deactivation problem, the key post-processing heuristics proposed are the "1-DWN" (one-by-one downward rounding) and "1-UP" (one-by-one upward rounding) methods (Ilsen et al., 19 Jan 2026). Both begin with a basic feasible solution $x^*$ to the LP relaxation, but differ in their subsequent variable-fixing strategies.

• 1-DWN heuristic: Iteratively selects the edge $e$ with the smallest fractional part $x^*_e-\lfloor x^*_e \rfloor$, attempts to fix $x_e$ permanently to $\lfloor x^*_e \rfloor$, and then re-optimizes the reduced LP. If this yields an infeasible solution, $x_e$ is instead fixed to $\lceil x^*_e \rceil$.
• 1-UP heuristic: Symmetrically, always selects the edge $e$ with the smallest upward gap $\lceil x^*_e \rceil-x^*_e$, rounds $x^*_e$ up to $\lceil x^*_e \rceil$, and resolves the restricted LP.

Both heuristics involve at most $|E|$ LP re-solves (as each iteration fixes one variable integrally) and guarantee feasibility for the traffic-oblivious scaling constraint at each step. They maintain the overall worst-case approximation guarantee of the original rounding but empirically achieve significantly lower numbers of active connections (Ilsen et al., 19 Jan 2026).

4. Theoretical Properties and Approximation Guarantees

Post-processing heuristics as described above are constructed to maintain the worst-case approximation ratio derived from the LP rounding analysis. In particular, for the traffic-oblivious link deactivation problem, the approximation ratio remains $r = \max((\varrho\,\lambda_{\min,2})^{-1}, 2)$, where $\lambda_{\min,2}$ is the minimum available link bundle size. This is ensured by arguments (see Lemma 3.3 and Theorem 3.4 in (Ilsen et al., 19 Jan 2026)) bounding the possible increase in the total number of active connections due to iterative rounding steps.

At the core, the feasibility of the heuristically refined solution is protected by the maintainance of flow-conservation and per-link capacity constraints for the "super-arc" matrix, which, via reduction [Theorem 3.2 in (Ilsen et al., 19 Jan 2026)], is sufficient to guarantee universal traffic-oblivious routability for all valid scaled demands.

5. Empirical Results and Practical Impact

Empirical evaluation within large ISP topologies shows that both 1-DWN and 1-UP heuristics yield solutions that are within 1–3% of the ILP optimum (computed via commercial solvers), compared with up to 10% gap for simple rounding (RND) and sometimes much larger gaps for traffic-aware (snapshot-specific) designs (Ilsen et al., 19 Jan 2026). They also substantially reduce computational times: for topologies with ≥60 nodes, RND and post-processing heuristics are 10x–150x faster than ILP, especially as ILP runtimes become prohibitive beyond ~10 minutes.

Furthermore, in terms of maximum link utilization under 50% downscaling (MLU), traffic-oblivious solutions obtained via these heuristics stay below or near 1.0, demonstrating robustness, while traffic-aware (snapshot-optimized) topologies can violate feasibility on unseen demands—highlighting the practical robustness of traffic-oblivious, post-processed solutions.

6. Limitations, Extensions, and Open Directions

Post-processing heuristics retain all standing modeling assumptions of the underlying LP relaxation: static physical topologies, uniform link-bundle models, and idealized MCF-routability oracles (as opposed to segment or shortest-path routing constraints). Their effectiveness diminishes primarily in cases where link granularity ($\lambda_{\min,2}$) is small, as the rounding ratio then dominates.

Extensions identified include incorporation of heterogeneous connection costs, adaptation to $k$-segment or unsplittable routing models (where the super-matrix trick fails), and dynamic/online scenarios where nested solutions for different downscaling ratios are needed (Ilsen et al., 19 Jan 2026). Theoretical improvements beyond the rounding barrier—either in the worst-case or by exploiting structural properties of real-world topologies—remain open.

7. Summary Table: Heuristics Properties

Heuristic	Rounding Direction	LP Resolves	Approximation Guarantee	Typical Gap to ILP
RND	Up	1	$\max((\varrho\lambda_{\min,2})^{-1},2)$	≤10% (oblivious)
1-DWN	Downward first	≤	E
1-UP	Upward first	≤	E

The table summarizes the key technical properties of the post-processing heuristics evaluated for oblivious link deactivation (Ilsen et al., 19 Jan 2026).

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In sum, post-processing heuristics such as 1-DWN and 1-UP are essential for translating the provable but conservative guarantees of LP-based algorithms into near-optimal practical solutions in universal network design and green traffic engineering. They systematically explore the rounding polytope to eliminate superfluous decisions, all while maintaining the strong universal feasibility and robustness central to traffic-oblivious design (Ilsen et al., 19 Jan 2026).