The paper introduces the concept of supermodular rank, defined as the minimal number of supermodular components required to express a function on a lattice. It characterizes the maximum possible rank and the structure of functions with a fixed rank, and analogously defines submodular rank. Leveraging submodular decompositions, the authors propose an algorithm that partitions a set‑function optimization problem into submodular subproblems, thereby improving approximation guarantees for monotone maximization and ratio minimization while incurring a computational cost tied to the submodular rank.
arXiv:2606. 04946v1 Announce Type: cross Abstract: Consistency is an important property in dynamic submodular maximization and entails maintaining a near-optimal solution at all times, making only a small number of adjustments to the solution in each step.
arXiv:2607. 05759v1 Announce Type: cross Abstract: Submodular maximization is an important building block for developing algorithms in many areas such as machine learning and data mining.
arXiv:2510. 23634v4 Announce Type: replace-cross Abstract: Motivated by applications for set containment problems, we consider the following fundamental problem: can we design set-to-vector functions so that the natural partial order on sets is preserved, namely $S\subseteq T \text{ if and only if } F(S)\leq F(T) $.
arXiv:2602. 21312v4 Announce Type: replace-cross Abstract: This work considers a number of optimization problems and reductive relations between them.
arXiv:2606. 05919v1 Announce Type: cross Abstract: Identifying most influential sets (MIS) - size-$k$ subsets whose removal maximally changes a target estimand - is typically infeasible because it requires searching over $\binom{n}{k}$ subsets.
arXiv:2607. 09739v1 Announce Type: new Abstract: We study LLM benchmark coreset selection: selecting a small subset of prompts over multiple benchmarks whose induced model scores and rankings approximate those obtained from the full benchmark suite.
This paper extends the Jaccard distance to arbitrary lattices and real valuations, proving that the triangle inequality holds when the valuation is strictly positive, monotone, and modular. It further shows that in relatively complemented distributive lattices the inequality remains valid under positive, monotone, supermodular, and log‑submodular valuations, and adapts the symmetric‑difference Jaccard formulation for submodular valuations in sectionally complemented distributive lattices. The authors also establish that supermodularity is a necessary condition for the generalized Jaccard distance to be a metric and discuss the implications of these relaxed structural constraints for fields such as quantum information theory, formal concept analysis, and machine learning.
arXiv:2606. 07403v1 Announce Type: cross Abstract: Benders decomposition is a fundamental framework for solving large-scale mixed-integer optimization problems with complicating variables that, when fixed, yield significantly easier subproblems.
arXiv:2604. 23628v2 Announce Type: replace-cross Abstract: Hierarchical clustering is a fundamental task in data analysis, but classical methods have long lacked a principled objective function.
arXiv:2602. 20578v2 Announce Type: replace Abstract: We study online maximization of non-monotone Diminishing-Return(DR)-submodular functions over down-closed convex sets, a regime where existing projection-free online methods suffer from suboptimal regret and limited feedback guarantees.
arXiv:2608. 25221v1 Announce Type: new Abstract: We study exact representations of $\mathrm{MAX}_N(x)=\max{x_1,\ldots,x_N}$ using two-hidden-layer ReLU neural networks.
arXiv:2604. 12036v3 Announce Type: replace-cross Abstract: We study a well-known task of constructing a decision tree identifying an unknown hypothesis from a given ground set of hypotheses under both the average- and worst-case cost.