Abstract We demonstrate how a generic automated theorem prover can be applied to establish the non-orderability of groups. Our approach incorporates various tools such as reasoning from the first principles, positive cones, torsions, generalised torsions and cofinal elements.

We study the unit-demand capacitated vehicle routing problem in the random setting of the Euclidean plane. The objective is to visit $n$ random terminals in a square using a set of tours of minimum total length, such that each tour visits the depot and at most $k$ terminals. We design an elegant algorithm combining the classical sweep heuristic and Arora's framework for the Euclidean traveling salesman problem [Journal of the ACM 1998]. We show that our algorithm is a polynomial-time approximation of ratio at most $1.55$ asymptotically almost surely. This improves on previous approximation ratios of $1.995$ due to Bompadre, Dror, and Orlin [Journal of Applied Probability 2007] and $1.915$ due to Mathieu and Zhou [Random Structures and Algorithms 2022]. In addition, we conjecture that, for any $\varepsilon>0$, our algorithm is a $(1+\varepsilon)$-approximation asymptotically almost surely.

We study m by m random matrices M with jointly Gaussian entries. Assuming a global small-ball probability bound

This paper studies online correlated selection (OCS). Suppose that we receive a pair of elements in each round and select one of them. Can we select with negative correlation to be more effective than independent random selections? Our contributions are threefold. For semi-OCS, which considers the probability that an element remains unselected after appearing in$k$rounds, we give an optimal algorithm that minimizes this probability for all k. It leads to 0.536-competitive unweighted and vertex-weighted on-line bipartite matching algorithms that randomize over only two options in each round, improving the previous 0.508-competitive ratio by Fahrbach et al. (2020). Further, we develop the first multi-way semi-OCS that allows an arbitrary number of elements with arbitrary masses in each round. As an application, it rounds the Balance algorithm in unweighted and vertex-weighted online bi-partite matching to get a 0.593-competitive ratio. Finally, we study OCS, which further considers the probability that an element is unselected in any subset of rounds. We prove that the optimal “level of negative correlation” is between 0.167 and 0.25, improving the previous bounds of 0.109 and 1 by Fahrbach et al. (2020). Our OCS gives a 0.519-competitive edge-weighted online bipartite matching algorithm, improving the previous 0.508-competitive ratio by Fahrbach et al. (2020).