Stefan Boettcher

Stefan Boettcher

In Nature Machine Intelligence 4, 367 (2022), Schuetz et al provide a scheme to employ graph neural networks (GNN) as a heuristic to solve a variety of classical, NP-hard combinatorial optimization problems. It describes how the network is trained on sample instances and the resulting GNN heuristic is evaluated applying widely used techniques to determine its ability to succeed. Clearly, the idea of harnessing the powerful abilities of such networks to ``learn'' the intricacies of complex, multimodal energy landscapes in such a hands-off approach seems enticing. And based on the observed performance, the heuristic promises to be highly scalable, with a computational cost linear in the input size $n$, although there is likely a significant overhead in the pre-factor due to the GNN itself. However, closer inspection shows that the reported results for this GNN are only minutely better than those for gradient descent and get outperformed by a greedy algorithm, for example, for Max-Cut. The discussion also highlights what I believe are some common misconceptions in the evaluations of heuristics.

Stefan Boettcher, Stephan Mertens

The Karmarkar-Karp differencing algorithm is the best known polynomial time heuristic for the number partitioning problem, fundamental in both theoretical computer science and statistical physics. We analyze the performance of the differencing algorithm on random instances by mapping it to a nonlinear rate equation. Our analysis reveals strong finite size effects that explain why the precise asymptotics of the differencing solution is hard to establish by simulations. The asymptotic series emerging from the rate equation satisfies all known bounds on the Karmarkar-Karp algorithm and projects a scaling $n^{-c\ln n}$, where $c=1/(2\ln2)=0.7213...$. Our calculations reveal subtle relations between the algorithm and Fibonacci-like sequences, and we establish an explicit identity to that effect.

Stefan Boettcher

In Changjun Fan et al. [Nature Communications https://doi.org/10.1038/s41467-023-36363-w (2023)], the authors present a deep reinforced learning approach to augment combinatorial optimization heuristics. In particular, they present results for several spin glass ground state problems, for which instances on non-planar networks are generally NP-hard, in comparison with several Monte Carlo based methods, such as simulated annealing (SA) or parallel tempering (PT). Indeed, those results demonstrate that the reinforced learning improves the results over those obtained with SA or PT, or at least allows for reduced runtimes for the heuristics before results of comparable quality have been obtained relative to those other methods. To facilitate the conclusion that their method is ''superior'', the authors pursue two basic strategies: (1) A commercial GUROBI solver is called on to procure a sample of exact ground states as a testbed to compare with, and (2) a head-to-head comparison between the heuristics is given for a sample of larger instances where exact ground states are hard to ascertain. Here, we put these studies into a larger context, showing that the claimed superiority is at best marginal for smaller samples and becomes essentially irrelevant with respect to any sensible approximation of true ground states in the larger samples. For example, this method becomes irrelevant as a means to determine stiffness exponents $θ$ in $d>2$, as mentioned by the authors, where the problem is not only NP-hard but requires the subtraction of two almost equal ground-state energies and systemic errors in each of $\approx 1\%$ found here are unacceptable. This larger picture on the method arises from a straightforward finite-size corrections study over the spin glass ensembles the authors employ, using data that has been available for decades.

Stefan Boettcher

We analyze the transformation of QUBO from its conventional Boolean presentation into an equivalent spin glass problem with coupled $\pm1$ spin variables exposed to a site-dependent external field. We find that in a widely used testbed for QUBO these fields tend to be rather large compared to the typical coupling and many spins in each optimal configurations simply align with the fields irrespective of their constraints. Thereby, the testbed instances tend to exhibit large redundancies - seemingly independent variables which contribute little to the hardness of the problem, however. We demonstrate various consequences of this insight, for QUBO solvers as well as for heuristics developed for finding spin glass ground states. To this end, we implement the Extremal Optimization (EO) heuristic, in a new adaptation for the QUBO problem. We also propose a novel way to assess the quality of heuristics for increasing problem sizes based on asymptotic scaling.