data scientists in combinatorial optimization

Abstract: L-extendable functions on the integer lattice are functions that admit a $\frac12$UJ-convex relaxation; that is, values can be assigned to the additional half-integral points, while keeping the original values at integer points, so that the resulting function is $\frac12$UJ-convex. Various examples of L-extendable functions and their applications are known. A general characterization of L-extendability for functions on the integer lattice, however, remains open. In this paper, we study the L-extendability of functions on the integer lattice. We first obtain a necessary condition: L-extendable functions are integrally convex. We then consider a special subclass of L-extendable functions, namely, those whose local convex extensions give L-convex relaxations. This subclass has a useful optimization property: the optimal value of the original function coincides with that of the $\frac12$UJ-convex relaxation given by its local convex extension, a property not guaranteed for general $\frac12$UJ-convex relaxations. We characterize this subclass by showing that a function is L-extendable through its local convex extension if and only if it is integrally convex and its restriction to every unit hypercube is D-convex, where D-convexity is a discrete convexity associated with reflected Freudenthal simplicial divisions. This subclass strictly contains the class of D-convex functions. We also give a combinatorial characterization of D-convex subsets of the vertices of a unit hypercube: a set $X\subseteq\{0,1\}^n$ is D-convex if and only if it is majority-closed and contains a relative opposite pair.

Abstract: Generative Marginalization Models (MaMs) have been recently introduced as efficient neural sampling models for any-order autoregressive modelling of discrete distributions. By learning both the marginal and conditional probabilities of a persistent-block Gibbs sampler, MaMs enable fast posterior evaluation with a single neural network forward pass. While prior work has considered MaMs to be distinct from Generative Flow Networks (GFlowNets), a well-established paradigm for inference in discrete stochastic models, we show that they are equivalent. Then, we also extend MaMs' sampling strategy to non-autoregressive generative processes. In particular, we describe an automatic criterion for full-state rejuvenation of the Gibbs sampler, derived from the Gelman-Rubin statistic, which plays a key role in speeding up learning convergence. Our experiments show that our method, called Particle GFlowNets, markedly accelerates training in large combinatorial spaces.