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1. Locally Fair Partitioning

   https://doi.org/10.1609/aaai.v36i5.20401  

   Agarwal, Pankaj K.; Ko, Shao-Heng; Munagala, Kamesh; Taylor, Erin (June 2022, Proceedings of the AAAI Conference on Artificial Intelligence)

   We model the societal task of redistricting political districts as a partitioning problem: Given a set of n points in the plane, each belonging to one of two parties, and a parameter k, our goal is to compute a partition P of the plane into regions so that each region contains roughly s = n/k points. P should satisfy a notion of "local" fairness, which is related to the notion of core, a well-studied concept in cooperative game theory. A region is associated with the majority party in that region, and a point is unhappy in P if it belongs to the minority party. A group D of roughly s contiguous points is called a deviating group with respect to P if majority of points in D are unhappy in P. The partition P is locally fair if there is no deviating group with respect to P.This paper focuses on a restricted case when points lie in 1D. The problem is non-trivial even in this case. We consider both adversarial and "beyond worst-case" settings for this problem. For the former, we characterize the input parameters for which a locally fair partition always exists; we also show that a locally fair partition may not exist for certain parameters. We then consider input models where there are "runs" of red and blue points. For such clustered inputs, we show that a locally fair partition may not exist for certain values of s, but an approximate locally fair partition exists if we allow some regions to have smaller sizes. We finally present a polynomial-time algorithm for computing a locally fair partition if one exists. 

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2. Minimum Plane Bichromatic Spanning Trees

   https://doi.org/10.4230/LIPIcs.ISAAC.2024.4  

   A_Akitaya, Hugo; Biniaz, Ahmad; Demaine, Erik D; Kleist, Linda; Stock, Frederick; Tóth, Csaba D (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Mestre, Julián; Wirth, Anthony (Ed.)

   For a set of red and blue points in the plane, a minimum bichromatic spanning tree (MinBST) is a shortest spanning tree of the points such that every edge has a red and a blue endpoint. A MinBST can be computed in O(n log n) time where n is the number of points. In contrast to the standard Euclidean MST, which is always plane (noncrossing), a MinBST may have edges that cross each other. However, we prove that a MinBST is quasi-plane, that is, it does not contain three pairwise crossing edges, and we determine the maximum number of crossings. Moreover, we study the problem of finding a minimum plane bichromatic spanning tree (MinPBST) which is a shortest bichromatic spanning tree with pairwise noncrossing edges. This problem is known to be NP-hard. The previous best approximation algorithm, due to Borgelt et al. (2009), has a ratio of O(√n). It is also known that the optimum solution can be computed in polynomial time in some special cases, for instance, when the points are in convex position, collinear, semi-collinear, or when one color class has constant size. We present an O(log n)-factor approximation algorithm for the general case. 

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3. Optimal point sets determining few distinct angles

   Henry L. Fleischmann*, Steven J. (January 2023, AUSTRALASIAN JOURNAL OF COMBINATORICS)

   Let P(k) denote the largest size of a non-collinear point set in the plane admitting at most k distinct angles. We prove P(2) = P(3) = 5, and we characterize the optimal sets. We also leverage results from Fleischmann et al. [Disc. Comput. Geom. (2023)] to provide the general bounds k+2 ≤ P(k) ≤ 6k, although the upper bound may be improved pending progress toward the Strong Dirac Conjecture. We conjecture that the lower bound is tight, providing infinite families of configurations meeting the bound and ruling out several classes of potential counterexamples. 

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4. Zarankiewicz’s problem for semilinear hypergraphs

   https://doi.org/10.1017/fms.2021.52  

   Basit, Abdul; Chernikov, Artem; Starchenko, Sergei; Tao, Terence; Tran, Chieu-Minh (January 2021, Forum of Mathematics, Sigma)

   Abstract A bipartite graph $$H = \left (V_1, V_2; E \right )$$ with $$\lvert V_1\rvert + \lvert V_2\rvert = n$$ is semilinear if $$V_i \subseteq \mathbb {R}^{d_i}$$ for some $$d_i$$ and the edge relation E consists of the pairs of points $$(x_1, x_2) \in V_1 \times V_2$$ satisfying a fixed Boolean combination of s linear equalities and inequalities in $$d_1 + d_2$$ variables for some s . We show that for a fixed k , the number of edges in a $$K_{k,k}$$ -free semilinear H is almost linear in n , namely $$\lvert E\rvert = O_{s,k,\varepsilon }\left (n^{1+\varepsilon }\right )$$ for any $$\varepsilon> 0$$ ; and more generally, $$\lvert E\rvert = O_{s,k,r,\varepsilon }\left (n^{r-1 + \varepsilon }\right )$$ for a $$K_{k, \dotsc ,k}$$ -free semilinear r -partite r -uniform hypergraph. As an application, we obtain the following incidence bound: given $$n_1$$ points and $$n_2$$ open boxes with axis-parallel sides in $$\mathbb {R}^d$$ such that their incidence graph is $$K_{k,k}$$ -free, there can be at most $$O_{k,\varepsilon }\left (n^{1+\varepsilon }\right )$$ incidences. The same bound holds if instead of boxes, one takes polytopes cut out by the translates of an arbitrary fixed finite set of half-spaces. We also obtain matching upper and (superlinear) lower bounds in the case of dyadic boxes on the plane, and point out some connections to the model-theoretic trichotomy in o -minimal structures (showing that the failure of an almost-linear bound for some definable graph allows one to recover the field operations from that graph in a definable manner). 

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5. Subdivided Claws and the Clique-Stable Set Separation Property

   https://doi.org/10.1007/978-3-030-62497-2  

   Chudnovsky, Maria and (February 2021, MATRIX book series)

   null (Ed.)

   Let C be a class of graphs closed under taking induced subgraphs. We say that C has the clique-stable set separation property if there exists c ∈ N such that for every graph G ∈ C there is a collection P of partitions (X, Y ) of the vertex set of G with |P| ≤ |V (G)| c and with the following property: if K is a clique of G, and S is a stable set of G, and K ∩ S = ∅, then there is (X, Y ) ∈ P with K ⊆ X and S ⊆ Y . In 1991 M. Yannakakis conjectured that the class of all graphs has the clique-stable set separation property, but this conjecture was disproved by M. G¨o¨os in 2014. Therefore it is now of interest to understand for which classes of graphs such a constant c exists. In this paper we define two infinite families S, K of graphs and show that for every S ∈ S and K ∈ K, the class of graphs with no induced subgraph isomorphic to S or K has the clique-stable set separation property. 

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