Boosting Sparsity in Graph Decompositions with QAOA Sampling

TL;DR

This work targets the NP-hard problem of decomposing a weighted graph into a small number of matchings to enable efficient scheduling. It introduces E-FCFW, a hybrid quantum–classical extension of the Fully-Corrective Frank–Wolfe algorithm that samples multiple matchings per iteration and optimises their weights to produce sparse decompositions. A QAOA-based matching sampler is developed via a QUBO formulation with a penalty to enforce feasibility, enabling diverse, high-weight matchings to feed the decomposition process. Empirical results across complete, bipartite, and heavy-hex graphs show that E-FCFW with QAOA sampling often yields sparser decompositions and better approximation on smaller/topologies, while hardware-scale experiments reveal noise as a limiting factor, suggesting avenues for deeper QAOA, improved parameter tuning, and post-processing improvements for practical quantum-assisted graph scheduling.

Abstract

We study the problem of decomposing a graph into a weighted sum of a small number of matchings, a task that arises in network resource allocation problems such as peer-to-peer energy exchange. Computing such decompositions is challenging for classical algorithms, even for small instances. To address this problem, we propose E-FCFW, a hybrid quantum-classical algorithm based on the Fully-Corrective Frank-Wolfe (FCFW) algorithm that incorporates a matching-sampling subroutine. We design a QAOA version of this subroutine and benchmark it against classical approaches (random sampling and simulated annealing) on demand graphs derived from complete, bipartite, and heavy-hex topologies. The quantum subroutine is executed using the Qiskit Aer state-vector and MPS simulators and on IBM Kingston hardware (7-111 qubits). On complete and bipartite graphs with 6-10 nodes, E-FCFW with QAOA yields consistently sparser decompositions than the classical baselines, and even beats the best-known solution for one instance. On heavy-hex graphs with 50, 70 and 100 nodes, E-FCFW with QAOA outperforms the other methods in terms of approximation error, demonstrating performance on utility-scale quantum hardware. For the largest graphs (100 nodes) E-FCFW with QAOA performs much better when using MPS circuit simulation, compared to using quantum hardware. This indicates that at this scale, the performance is severely impacted by hardware noise.

Figures (11)

• Figure 1: Demand graph (thick edges) overlaid on a complete graph (dashed edges) with 6 nodes. Nodes represent devices (e.g., EVs), and each dashed edge indicates a feasible exchange of resources such as energy. The weight on each thick edge shows the fraction of time the corresponding devices wish to be connected. Fig. \ref{['fig:matchings_example']} shows how the graph can be decomposed as the sum of four different matchings.
• Figure 2: Example of how the graph in Fig. \ref{['fig:graph_example']} can be decomposed as the sum of graph matchings. A matching captures how nodes can exchange resources in practice. The number below each matching indicates the fraction of time each matching should be used to satisfy the demand graph in Fig. \ref{['fig:graph_example']}
• Figure 3: Illustrating the decomposition length of the Birkhoff+ algorithm for bipartite graphs of size $n \in \{3,4,\dots,30\}$. There are 10 different graph instances for each graph size. Graphs are generated by sampling $n$ matchings and weights uniformly at random. The red line (best-known solution) indicates the number of matchings used to create the graph (i.e. $n$). The light green area indicates the maximum and minimum decomposition length obtained for a graph size. The thick green line is the average decomposition length for all instances.
• Figure 4: Illustration showing one step of the Frank-Wolfe algorithm. The grey dots represent discrete objects to choose from (e.g. matchings), and the polytope is the convex hull of the grey dots. The white dot ($X_k$) represents the current decomposition, and the black dot ($M$) is the maximum-weight matching selected by Frank-Wolfe with the update in Eq. \ref{['eq:matching_selection']}. The dashed arrow indicates the convex combination between $X_k$ and $M$, for which there exists a new graph decomposition ($X_{k+1})$ that is closer to the target demand matrix $D^*$ (black star).
• Figure 5: Schematic illustration showing how QAOA may help to find better matchings explained in Sec. \ref{['sec:qaoa_motivation']}. The grey dots with a cross (outside the polytope) indicate invalid matchings (i.e. bitstrings) that may be selected during the sampling process. We note the figure is schematic as matchings are extreme points and therefore cannot be represented as the convex combination of two or more other matchings.