K1,3-covering red and blue points in the plane

We say that a finite set of red and blue points in the plane in general position can be K1, 3-covered if the set can be partitioned into subsets of size 4, with 3 points of one color and 1 point of the other color, in such a way that, if at each subset the fourth point is connected by straight-line segments to the same-colored points, then the resulting set of all segments has no crossings. We consider the following problem: Given a set R of r red points and a set B of b blue points in the plane in general position, how many points of R ¿ B can be K1, 3-covered? and we prove the following results: (1) If r = 3g + h and b = 3h + g, for some non-negative integers g and h, then there are point sets R ¿ B, like {1, 3}-equitable sets (i.e., r = 3b or b = 3r) and linearly separable sets, that can be K1, 3-covered. (2) If r = 3g + h, b = 3h + g and the points in R ¿ B are in convex position, then at least r + b - 4 points can be K1, 3-covered, and this bound is tight. (3) There are arbitrarily large point sets R ¿ B in general position, with r = b + 1, such that at most r + b - 5 points can be K1, 3-covered. (4) If b = r = 3b, then at least 9 8 (r + b- 8) points of R ¿ B can be K1, 3-covered. For r > 3b, there are too many red points and at least r - 3b of them will remain uncovered in any K1, 3-covering. Furthermore, in all the cases we provide efficient algorithms to compute the corresponding coverings

Maximum rectilinear convex subsets

Let P be a set of n points in the plane. We consider a variation of the classical Erdos-Szekeres problem, presenting efficient algorithms with (formula presented) running time and (formula presented) space complexity that compute: (1) A subset S of P such that the boundary of the rectilinear convex hull of S has the maximum number of points from P, (2) a subset S of P such that the boundary of the rectilinear convex hull of S has the maximum number of points from P and its interior contains no element of P, (3) a subset S of P such that the rectilinear convex hull of S has maximum area and its interior contains no element of P, and (4) when each point of P is assigned a weight, positive or negative, a subset S of P that maximizes the total weight of the points in the rectilinear convex hull of S

On the verge of below-ground speciation: A new species complex of microendemic endogean carabid beetles, Typhlocharis Dieck, 1869 (Coleoptera: Carabidae: Anillini), from south-west Iberian Peninsula

A new species complex of genus Typhlocharis Dieck, 1869 (Coleoptera: Carabidae: Trechinae: Anillini: Typhlocharina) is described. Six populations from southern Badajoz (Spain), referred as the >coenobita species complex>, are the first documented case of an expected situation within Typhlocharina and potentially other lineages of endogean ground beetles: the presence of closely related allopatric populations within a reduced geographical range that, despite certain genetic isolation, show a gradient of morphological differences that challenge taxonomic assignment. Previous phylogenies of Typhlocharina recovered these populations as a monophyletic lineage, represented by three potential new species in need of further examination to validate their status. Here, we test the congruence of this taxonomic hypothesis through direct observation, statistical analyses applied to morphological characters and analysis of COI sequences. Such integrative approach, revealed as a powerful tool to solve situations where phenotypic differences are very subtle, is used for the first time to discriminate Anillini species. The results are coherent with the three species hypothesis, formally described as T. coenobita sp.n., T. eremita sp.n. and T. anachoreta sp.n. The implications of the internal variability within this species complex to the systematics of Typhlocharina and their affinities to other Typhlocharis species are discussed. The entity of T. eremita sp.n. as new species is well established within the standards of the genus. However, the populations of T. coenobita sp.n. show high variability and their relationship with T. anachoreta sp.n. is in the verge of what can be considered species-level differentiation, suggestive of an incipient speciation process. The proposed species boundaries maximize the consistence among the different sources of evidence. The intraspecific variability within T. coenobita sp.n. is properly described, contributing to elucidate the ongoing differentiation processes within this endogean lineage. Finally, an identification key for the coenobita species complex is provided.This research has been partially funded by projects CGL2010-16944 (Spanish Ministry of Education) and GR35/10-A Research Group 921632 (Santander-UCM). S.P.G. received support from a postdoctoral Juvenile Employment grant (CT53/16-CT54/16) at the Universidad Complutense de Madrid and C.A. from a Marie Curie postdoctoral grant (Project SOILBIODIV-705639; Program H2020-MSCA-IF-2015).Peer Reviewe

Matching colored points with rectangles

Let S be a point set in the plane such that each of its elements is colored either red or blue. A matching of S with rectangles is any set of pairwise-disjoint axis-aligned closed rectangles such that each rectangle contains exactly two points of S. Such a matching is monochromatic if every rectangle contains points of the same color, and is bichromatic if every rectangle contains points of different colors. We study the following two problems: (1) Find a maximum monochromatic matching of S with rectangles. (2) Find a maximum bichromatic matching of S with rectangles. For each problem we provide a polynomial-time approximation algorithm that constructs a matching with at least 1 / 4 of the number of rectangles of an optimal matching. We show that the first problem is -hard even if either the matching rectangles are restricted to axis-aligned segments or S is in general position, that is, no two points of S share the same x or y coordinate. We further show that the second problem is also -hard, even if S is in general position. These -hardness results follow by showing that deciding the existence of a matching that covers all points is -complete in each case. Additionally, we prove that it is -complete to decide the existence of a matching with rectangles that cover all points in the case where all the points have the same color, solving an open problem of Bereg et al

Computing the Coarseness with Strips or Boxes

Recently, the concept of coarseness was introduced as a measure of how blended a 2-colored point set S role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3eS is. In the definition of this measure, a convex partition Π role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3eΠ, that is, a partition of S role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3eS into sets {S1,…,Sk} role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3e{S1,…,Sk} of S role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3eS whose convex hulls are pairwise disjoint, is considered. The discrepancy of Π role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3eΠ, denoted by d(S,Π) role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3ed(S,Π), is the smallest (bichromatic) discrepancy of the elements of Π role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3eΠ. The coarseness of S role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3eS, denoted by C(S) role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3eC(S), is then defined as the maximum of d(S,Π) role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3ed(S,Π) over all convex partitions Π role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3eΠ of S role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3eS. Roughly speaking, the value of the coarseness is high when we can split S role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3eS into blocks, each with large discrepancy. It has been conjectured that computing the coarseness is NP-hard. In this paper, we study how to compute the coarseness for two constrained cases: (1) when the k role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3ek elements of Π role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3eΠ are separated by k−1 role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3ek−1 pairwise parallel lines (strips) and, (2) the case in which the cardinality of the partition is fixed and the elements of Π role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3eΠ are covered by pairwise disjoint axis-aligned rectangles (boxes). For the first case we present an O(n2log2n) role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3eO(n2log2n)-time algorithm, and show that such a computation problem is 3SUM-hard; for the second, we show that computing the coarseness with k role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3ek boxes is NP-hard, when k role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3ek is part of the input. For k role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3ek fixed, we show that the coarseness can be computed in O(n2k−1) role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3eO(n2k−1) time and propose more efficient algorithms for k=2,3,4 role= presentation style= box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative; \u3ek=2,3,4