Pinwheel Scheduling is a fundamental scheduling problem, in which each task \(i\) is associated with a positive integer deadline \(d_i\), and the objective is to schedule one task per time slot, ensuring each task perpetually appears at least once in every \(d_i\) time slots. Although conjectured to be PSPACE-complete, it remains open whether Pinwheel Scheduling is NP-hard (unless a compact input encoding is used) or even contained in NP.

The Subset Sum Ratio problem (SSR) asks, given a multiset $A$ of positive integers, to find two disjoint subsets of $A$ such that the largest-to-smallest ratio of their sums is minimized. In this paper we study the $k$-version of SSR, namely $k$-Subset Sum Ratio ($k$-SSR), which asks to minimize the largest-to-smallest ratio of sums of $k$ disjoint subsets of $A$. We develop an approximation scheme for $k$-SSR running in $O({n^{2k}}/{\varepsilon^{k-1}})$ time, where $n=|A|$ and $\varepsilon$ is the error parameter. To the best of our knowledge, this is the first FPTAS for $k$-SSR for fixed $k>2$. We also study the $k$-way Number Partitioning Ratio ($k$-PART) problem, which differs from $k$-SSR in that the $k$ subsets must constitute a partition of $A$; this problem in fact corresponds to the objective of minimizing the largest-to-smallest sum ratio in the family of Multiway Number Partitioning problems. We present a more involved FPTAS for $k$-PART, also achieving $O({n^{2k}}/{\varepsilon^{k-1}})$ time complexity. Notably, $k$-PART is also equivalent to the Minimum Envy-Ratio problem with identical valuation functions, which has been studied in the context of fair division of indivisible goods. Thus, for the case of identical valuations, our FPTAS represents a significant improvement over the $O(n^{4k^2+1}/\varepsilon^{2k^2})$ bound obtained by Nguyen and Rothe's FPTAS for Minimum Envy-Ratio with general additive valuations. Lastly, we propose a second FPTAS for $k$-SSR, which employs carefully designed calls to the first one; the new scheme has a time complexity of $\widetilde{O}(n/{\varepsilon^{3k-1}})$, thus being much faster when $n\gg 1/ \varepsilon$.

Non-convex minimization problems are universally considered hard, and even guaranteeing that a computed solution is locally minimizing is known to be NP-hard. In this general context, our paper focuses on the problem of finding stationary points that satisfy an approximate second-order optimality condition, which serves to exclude strict saddles and other non-minimizing stationary points. Our main result is that the problem of finding approximate second-order stationary points (SOSPs) is PLS-complete, i.e., of the same complexity as the problem of finding first-order stationary points (FOSPs), thus resolving an open question in the field. In particular, our results imply that, under the widely believed complexity conjecture that PLS $\neq$ FNP, finding approximate SOSPs in unconstrained domains is *easier* than in constrained domains, which is known to be NP-hard. This comes in stark contrast with earlier results which implied that, unless PLS = CLS, finding approximate FOSPs in unconstrained domains is *harder* than in constrained domains.