SDP Approach to Quadratic Vertex-Disjoint Paths Problem

Abstract: We study the quadratic $k$-vertex-disjoint paths problem (Q-$k$-VDP), which seeks $k$ vertex-disjoint paths in a directed graph that minimize a nonconvex quadratic objective function. We formulate the problem as a binary quadratic program and apply a systematic graph reduction to manage its dimensionality. To obtain a tractable bounding model, we drop the subtour-elimination constraints and derive a semidefinite programming (SDP) relaxation. We then solve this relaxed model within a branch-and-bound framework, where the bounds are computed from the SDP relaxation using a tailored alternating direction method of multipliers. Computational results show that our proposed method consistently outperforms Gurobi by solving more instances to optimality, especially on challenging large-scale instances.

• The paper introduces an innovative graph reduction that fixes over 95% of arcs, dramatically reducing the binary quadratic program's dimensionality.
• It develops a semidefinite programming relaxation with facial reduction to capture quadratic interactions and tighten the feasible region.
• A custom ADMM algorithm within a branch-and-bound framework efficiently tackles large-scale NP-hard quadratic VDP instances, outperforming traditional solvers.

SDP Relaxation for the Quadratic $k$-Vertex-Disjoint Paths Problem

Introduction and Problem Formulation

The paper studies a quadratic generalization of the $k$-vertex-disjoint paths problem (Q-$k$-VDP) in directed graphs, focused on determining $k$ mutually vertex-disjoint paths connecting given source–target pairs, with a nonconvex quadratic objective. The quadratic term captures interactions between paths—a critical modeling aspect for applications in VLSI design and robust network routing, where congestion or mutual influence among simultaneous routes is relevant.

The problem is formulated as a Binary Quadratic Program (BQP), with variables representing arc selection in a disjoint union of $k$ isomorphic graph copies, equipped with flow conservation, vertex-disjointness, and subtour elimination constraints. The nonconvex quadratic cost makes the problem NP-hard even for moderate $k$, especially as the SECs grow exponentially with graph size. The matrix $\tilde{Q}$ encodes both linear and interaction (quadratic) costs, and the feasible region is highly structured by the combinatorics of flow and disjointness.

Graph Reduction and Model Tightening

A key contribution is a systematic reduction procedure that dramatically decreases problem dimensionality. The reduction pivots on the observation that, due to vertex-disjointness and source–target structure, many arcs in the union graph must be fixed to zero or one prior to optimization. Fixed-zero arcs cannot appear in any feasible solution, determined by infeasibility in derived augmented $k'$-VDP subproblems. Fixed-one arcs, conversely, must necessarily be traversed in all feasible solutions; their identification involves infeasibility checks upon their removal and further precludes conflicts due to capacity constraints, recursively fixing incident arcs to zero.

For planar graphs, this reduction is provably polynomial time, leveraging tractability results for $k$-vertex-disjoint paths on planar instances. After reduction, the original BQP is faithfully projected onto a smaller reduced BQP (R-BQP) on a new graph $G=(V,E)$, preserving all optimal solutions but with a drastically compacted decision space. The theoretical development rigorously establishes the correctness and order structure induced by mandatory arcs, and the partitioning of $k$0 into blocks corresponding to these fixed arcs. This not only yields computational benefits but strictly tightens subsequent relaxations as the set of feasible (fractional) solutions shrinks post-reduction.

Semidefinite Programming Relaxations

Due to the nonconvex quadratic objective and residual combinatorial constraints, direct optimization remains intractable. The authors develop a semidefinite programming (SDP) relaxation by "lifting" the binary variables to a matrix space. The relaxation encodes the quadratic form linearly in a lifted matrix variable $k$1, with additional linear constraints capturing arc exclusions (from the conflict set $k$2), binary idempotency (arrow constraints), and lifted flow conservation. This relaxation is provably stronger when applied after reduction, since the reduced feasible polytope is strictly smaller.

Facial reduction is performed analytically: the feasible set is projected onto the minimal face of the positive semidefinite cone containing the feasible region, further reducing the problem to a lower-dimensional matrix $k$3, with $k$4, greatly mitigating computational cost and addressing degeneracy when equality constraints place all feasible solutions on a boundary of the cone.

The authors provide necessary and sufficient conditions—via an alternative theorem and a tailored auxiliary system—for the satisfaction of Slater's condition post-reduction. The analysis is illustrated with explicit examples, including exact construction of exposing vectors when facial reduction is incomplete.

Alternating Direction Method of Multipliers (ADMM) Algorithm

For scalability and robustness on large instances, the SDP relaxations are solved using a custom ADMM algorithm rather than interior-point methods, which are often numerically unstable or excessively memory-demanding for large or non-strictly feasible problems. ADMM alternates between updating $k$5 and $k$6 (the primal variables) and the dual matrix, each subproblem possessing a closed-form or efficiently computable solution due to problem structure. The authors further improve numerical behavior by explicitly imposing bounds $k$7, and efficiently manage the projection operations required for the ADMM updates.

The SDP relaxation is embedded as the bounding operation in a branch-and-bound (B&B) global optimization framework ("SABB"), with upper bounds extracted from fractional solutions via rounding heuristics and lower bounds from the dual. This guarantees valid certificates for optimality or tightness up to the prescribed numerical gap.

Numerical Experiments and Comparative Analysis

Experiments are conducted on planar instances, with benchmarks generated from random grid graphs subject to realistic path selection and pruning, then processed by the reduction framework. The reduction eliminates a dominant fraction of arcs—over 95% for dense instances—leading to smaller, more tractable models.

Empirical comparisons between SABB (SDP+ADMM-based B&B) and Gurobi (commercial MIP/MIQP solver) are performed across a comprehensive suite of instance sizes, binned by arc count. On small instances ($k$8), both methods are competitive, but SABB achieves full optimality on all, matching or slightly exceeding Gurobi's speed.

For medium to large-scale problems ($k$9), SABB’s advantage is strongly pronounced: it solves notably more instances to optimality, achieves significantly lower average optimality gaps, and does so with better runtime profiles as dimensionality increases. On the largest, most challenging test cases ($k$0), Gurobi yields extremely loose gaps and nearly always times out, whereas SABB still produces meaningful bounds and occasionally achieves optimality.

The key numerical result is that SABB's tight relaxations and robust bounding permit effective cutting planes and pruning in the B&B tree, while general solvers suffer from weak lower bounds and combinatorial explosion. This is supported by direct evidence: after reduction, Gurobi solves all R-BQP subproblems easily, whereas many raw BQP instances remain fundamentally out of reach.

Implications and Future Directions

The paper demonstrates that for nonconvex combinatorial problems involving quadratic interactions under routing/disjointness constraints, tight SDP relaxations—especially when combined with aggressive combinatorial pre-processing—yield dramatic practical advantages over conventional solvers. The branch-and-bound algorithm, equipped with the facially reduced SDP formulation and custom ADMM solver, provides numerically stable, scalable performance, enabling solution of instances beyond the capability of vanilla MIP/MIQP techniques.

This framework sets a strong precedent for leveraging structure-preserving reduction, advanced convex relaxations, and specialized first-order algorithms in hard combinatorial optimization with quadratic terms. In the broader context, potential directions include integrating cutting plane methods to strengthen relaxations iteratively, extension to other interaction-structured path packing, and systematic study of approximation properties versus practical optimality for even larger graphs. Application-specific enhancements, informed by physical characteristics (e.g., in VLSI or robust networking), could also yield further performance gains.

Conclusion

Through a rigorous combination of graph-theoretic reduction, semidefinite relaxation, facial reduction, and scalable first-order optimization, the paper delivers a powerful toolset for tackling the Q-$k$1-VDP in directed planar graphs. Substantial empirical evidence supports the superiority of the SDP-ADMM approach ("SABB") for both bounding and solving large-scale, nonconvex quadratic disjoint path problems. This methodology is broadly applicable to other combinatorial optimization problems with complex interaction costs and imposes a high standard for future research on similarly structured discrete optimization challenges.

Reference: "SDP Approach to Quadratic Vertex-Disjoint Paths Problem" (2604.03452)