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1. Minimum L_∞ Hausdorff distance of point sets under translation: Generalizing Klee's measure problem

   https://doi.org/10.4230/LIPIcs.SoCG.2023.24  

   Chan, Timothy M. (June 2023, Proc. 39th Sympos. Computational Geometry (SoCG))

   Chambers, Erin W.; Gudmundsson, Joachim (Ed.)

   We present a (combinatorial) algorithm with running time close to O(n^d) for computing the minimum directed L_∞ Hausdorff distance between two sets of n points under translations in any constant dimension d. This substantially improves the best previous time bound near O(n^{5d/4}) by Chew, Dor, Efrat, and Kedem from more than twenty years ago. Our solution is obtained by a new generalization of Chan’s algorithm [FOCS'13] for Klee’s measure problem. To complement this algorithmic result, we also prove a nearly matching conditional lower bound close to Ω(n^d) for combinatorial algorithms, under the Combinatorial k-Clique Hypothesis. 

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2. Finding a planted clique by adaptive probing

   https://doi.org/10.30757/ALEA.v17-30  

   Racz, Miklos Z; Schiffer, Benjamin (August 2020, Alea)

   null (Ed.)

   We consider a variant of the planted clique problem where we are allowed unbounded computational time but can only investigate a small part of the graph by adaptive edge queries. We determine (up to logarithmic factors) the number of queries necessary both for detecting the presence of a planted clique and for finding the planted clique. Specifically, let $$G \sim G(n,1/2,k)$$ be a random graph on $$n$$ vertices with a planted clique of size $$k$$. We show that no algorithm that makes at most $q = o(n^2 / k^2 + n)$ adaptive queries to the adjacency matrix of $$G$$ is likely to find the planted clique. On the other hand, when $$k \geq (2+\epsilon) \log_2 n$$ there exists a simple algorithm (with unbounded computational power) that finds the planted clique with high probability by making $$q = O( (n^2 / k^2) \log^2 n + n \log n)$$ adaptive queries. For detection, the additive $$n$$ term is not necessary: the number of queries needed to detect the presence of a planted clique is $n^2 / k^2$ (up to logarithmic factors). 

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3. A Clique-Based Separator for Intersection Graphs of Geodesic Disks in ℝ²

   https://doi.org/10.4230/LIPIcs.SoCG.2024.9  

   Aronov, Boris; de_Berg, Mark; Theocharous, Leonidas (June 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Mulzer, Wolfgang; Phillips, Jeff M (Ed.)

   Let d be a (well-behaved) shortest-path metric defined on a path-connected subset of ℝ² and let 𝒟 = {D_1,…,D_n} be a set of geodesic disks with respect to the metric d. We prove that 𝒢^×(𝒟), the intersection graph of the disks in 𝒟, has a clique-based separator consisting of O(n^{3/4+ε}) cliques. This significantly extends the class of objects whose intersection graphs have small clique-based separators. Our clique-based separator yields an algorithm for q-Coloring that runs in time 2^O(n^{3/4+ε}), assuming the boundaries of the disks D_i can be computed in polynomial time. We also use our clique-based separator to obtain a simple, efficient, and almost exact distance oracle for intersection graphs of geodesic disks. Our distance oracle uses O(n^{7/4+ε}) storage and can report the hop distance between any two nodes in 𝒢^×(𝒟) in O(n^{3/4+ε}) time, up to an additive error of one. So far, distance oracles with an additive error of one that use subquadratic storage and sublinear query time were not known for such general graph classes. 

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4. On Single-Distance Graphs on the Rational Points in Euclidean Spaces

   https://doi.org/10.4153/S0008439520000181  

   Bau, Sheng; Johnson, Peter; Noble, Matt (March 2021, Canadian Mathematical Bulletin)

   null (Ed.)

   Abstract For positive integers n and d > 0, let $$G(\mathbb {Q}^n,\; d)$$ denote the graph whose vertices are the set of rational points $$\mathbb {Q}^n$$ , with $$u,v \in \mathbb {Q}^n$$ being adjacent if and only if the Euclidean distance between u and v is equal to d . Such a graph is deemed “non-trivial” if d is actually realized as a distance between points of $$\mathbb {Q}^n$$ . In this paper, we show that a space $$\mathbb {Q}^n$$ has the property that all pairs of non-trivial distance graphs $$G(\mathbb {Q}^n,\; d_1)$$ and $$G(\mathbb {Q}^n,\; d_2)$$ are isomorphic if and only if n is equal to 1, 2, or a multiple of 4. Along the way, we make a number of observations concerning the clique number of $$G(\mathbb {Q}^n,\; d)$$ . 

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5. Why Is Maximum Clique Often Easy in Practice?

   https://doi.org/10.1287/opre.2019.1970  

   Walteros, Jose L.; Buchanan, Austin (November 2020, Operations Research)

   null (Ed.)

   To this day, the maximum clique problem remains a computationally challenging problem. Indeed, despite researchers’ best efforts, there exist unsolved benchmark instances with 1,000 vertices. However, relatively simple algorithms solve real-life instances with millions of vertices in a few seconds. Why is this the case? Why is the problem apparently so easy in many naturally occurring networks? In this paper, we provide an explanation. First, we observe that the graph’s clique number ω is very near to the graph’s degeneracy d in most real-life instances. This observation motivates a main contribution of this paper, which is an algorithm for the maximum clique problem that runs in time polynomial in the size of the graph, but exponential in the gap [Formula: see text] between the clique number ω and its degeneracy-based upper bound d+1. When this gap [Formula: see text] can be treated as a constant, as is often the case for real-life graphs, the proposed algorithm runs in time [Formula: see text]. This provides a rigorous explanation for the apparent easiness of these instances despite the intractability of the problem in the worst case. Further, our implementation of the proposed algorithm is actually practical—competitive with the best approaches from the literature. 

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