Classical and Quantum Distributed Algorithms for the Survivable Network Design Problem
Phillip Kerger, David E. Bernal Neira, Zoe Gonzalez Izquierdo, Eleanor G. Rieffel
TL;DR
The paper develops distributed classical and quantum algorithms for the survivable network design problem (SNDP) in the CONGEST-CLIQUE setting, framing the solution as a distributed realization of the GoemansBertsimas tree-heuristic. It hinges on distributed APSP with routing tables, Shortest Path Forests, and a modified MST step to build a near-optimal SNDP subgraph, preserving known approximation guarantees for Steiner, TSP, and k-connected variants. The authors demonstrate asymptotic quantum speedups by replacing classical APSP with quantum APSP (and routing computations) in the qCCM, achieving rounds $ ilde{O}(n^{1/4})$ versus the classical $ ilde{O}(n^{1/3})$, under reasonable assumptions on the number of connectivity types. The work highlights a potential separation between classical and quantum distributed models for large-scale SNDP instances and points to future enhancements via local improvements and distributed MWPM methods.
Abstract
We investigate distributed classical and quantum approaches for the survivable network design problem (SNDP), sometimes called the generalized Steiner problem. These problems generalize many complex graph problems of interest, such as the traveling salesperson problem, the Steiner tree problem, and the k-connected network problem. To our knowledge, no classical or quantum algorithms for the SNDP have been formulated in the distributed settings we consider. We describe algorithms that are heuristics for the general problem but give concrete approximation bounds under specific parameterizations of the SNDP, which in particular hold for the three aforementioned problems that SNDP generalizes. We use a classical, centralized algorithmic framework first studied in (Goemans & Bertsimas 1993) and provide a distributed implementation thereof. Notably, we obtain asymptotic quantum speedups by leveraging quantum shortest path computations in this framework, generalizing recent work of (Kerger et al. 2023). These results raise the question of whether there is a separation between the classical and quantum models for application-scale instances of the problems considered.