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The focus of this paper is to increase our understanding of the Longest Processing Time First (LPT) heuristic. LPT is a classical heuristic for the fundamental problem of uniform machine scheduling. For different machine speeds, LPT was first considered by Gonzalez et al (SIAM J. Computing, 1977). Since then, extensive work has been done to improve the approximation factor of the LPT heuristic. However, all known implementations of the LPT heuristic take $O(mn)$ time, where $m$ is the number of machines and $n$ is the number of jobs. In this work, we come up with the first near-linear time implementation for LPT. Specifically, the running time is $O((n+m)(\log^2{m}+\log{n}))$. Somewhat surprisingly, the result is obtained by mapping the problem to dynamic maintenance of lower envelope of lines, which has been well studied in the computational geometry community. Our second contribution is to analyze the performance of LPT for the Drones Warehouse Problem (DWP), which is a natural generalization of the uniform machine scheduling problem motivated by drone-based parcel delivery from a warehouse. In this problem, a warehouse has multiple drones and wants to deliver parcels to several customers. Each drone picks a parcel from the warehouse, delivers it, and returns to the warehouse (where it can also get charged). The speeds and battery lives of the drones could be different, and due to the limited battery life, each drone has a bounded range in which it can deliver parcels. The goal is to assign parcels to the drones so that the time taken to deliver all the parcels is minimized. We prove that the natural approach of solving this problem via the LPT heuristic has an approximation factor of $\phi$, where $\phi \approx 1.62$ is the golden ratio., Comment: To appear in FSTTCS 2024

Greenberger-Horne-Zeilinger (GHZ) states are quantum states involving at least three entangled particles. They are of fundamental interest in quantum information theory, and the construction of such states of high dimension has various applications in quantum communication and cryptography. They are of fundamental interest in quantum information theory, and the construction of such states of high dimension has various applications in quantum communication and cryptography. Krenn, Gu and Zeilinger discovered a correspondence between a large class of quantum optical experiments which produce GHZ states and edge-weighted edge-coloured multi-graphs with some special properties called the \emph{GHZ graphs}. On such GHZ graphs, a graph parameter called \emph{dimension} can be defined, which is the same as the dimension of the GHZ state produced by the corresponding experiment. Krenn and Gu conjectured that the dimension of any GHZ graph with more than $4$ vertices is at most $2$. An affirmative resolution of the Krenn-Gu conjecture has implications for quantum resource theory. On the other hand, the construction of a GHZ graph on a large number of vertices with a high dimension would lead to breakthrough results. In this paper, we study the existence of GHZ graphs from the perspective of the Krenn-Gu conjecture and show that the conjecture is true for graphs of vertex connectivity at most 2 and for cubic graphs. We also show that the minimal counterexample to the conjecture should be $4$-connected. Such information could be of great help in the search for GHZ graphs using existing tools like PyTheus. While the impact of the work is in quantum physics, the techniques in this paper are purely combinatorial, and no background in quantum physics is required to understand them., Comment: To appear in MFCS 2024