Balanced partitions of 3-colored geometric sets in the plane

Bereg, Sergey; Hurtado, Ferrán; Kanō, Mikio; Korman, Matias; Lara, Dolores; Seara, Carlos; Silveira, Rodrigo I.; Urrutia, Jorge; Verbeek, Kevin

doi:10.1016/j.dam.2014.10.015

Cited by 17 publications

(15 citation statements)

References 11 publications

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“…Another relaxation of Ham-Sandwich cuts was introduced by Bereg et al [8] and has received some attention lately (see [6,10,13,17]): instead of cutting with a single hyperplane, how many masses can we bisect if we cut with several hyperplanes? In this setting, the masses are distributed into two parts according to a natural 2-coloring of the induced arrangement.…”

Section: Introductionmentioning

confidence: 99%

Equipartitions with Wedges and Cones

Schnider¹

2019

Preprint

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A famous result about mass partitions is the so called Ham-Sandwich theorem. It states that any d mass distributions in R d can be simultaneously bisected by a single hyperplane. In this work, we study two related questions.The first one is, whether we can bisect more than d masses, if we allow for bisections with more general objects such as cones, wedges or double wedges. We answer this question in the affirmative by showing that with all of these objects, we can simultaneously bisect d + 1 masses. For double wedges, we prove a stronger statement, namely that d families of d + 1 masses each can each by simultaneously bisected by some double wedge such that all double wedges have one hyperplane in common.The second question is, how many masses we can simultaneously equipartition with a k-fan, that is, k half-hyperplanes in R d , emanating from a common (d − 2)-dimensional apex. This question was already studied in the plane, our contribution is to extend the planar results to higher dimensions.All of our results are proved using topological methods. We use some well-established techniques, but also some newer methods. In particular, we introduce a Borsuk-Ulam theorem for flag manifolds, which we believe to be of independent interest.

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Section: Introductionmentioning

confidence: 99%

Equipartitions with Wedges and Cones

Schnider¹

2019

Preprint

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“…Now we are ready to describe the algorithm to solve Problem 1. In Algorithm 1, we describe how to get maximum balanced subtree with root t for a tree T rooted at t. P v = {(0, 0), (0, 1)}; 5 if v be a vertex with red color and v has k children u 1 , u 2 , ..., u k in T with root at r, then…”

Section: Treesmentioning

confidence: 99%

The Balanced Connected Subgraph Problem

Bhore

Chakraborty

Jana

et al. 2019

Algorithms and Discrete Applied Mathematics

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The problem of computing induced subgraphs that satisfy some specified restrictions arises in various applications of graph algorithms and has been well studied. In this paper, we consider the following Balanced Connected Subgraph (shortly, BCS) problem. The input is a graph G = (V, E), with each vertex in the set V having an assigned color, "red" or "blue". We seek a maximum-cardinality subset V ′ ⊆ V of vertices that is color-balanced (having exactly |V ′ |/2 red nodes and |V ′ |/2 blue nodes), such that the subgraph induced by the vertex set V ′ in G is connected. We show that the BCS problem is NP-hard, even for bipartite graphs G (with red/blue color assignment not necessarily being a proper 2-coloring). Further, we consider this problem for various classes of the input graph G, including, e.g., planar graphs, chordal graphs, trees, split graphs, bipartite graphs with a proper red/blue 2coloring, and graphs with diameter 2. For each of these classes either we prove NP-hardness or design a polynomial time algorithm.

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“…One way to bisect more than d masses is to use more complicated cuts, such as algebraic surfaces of fixed degree [44] or piece-wise linear cuts with a fixed number of turns [28,40]. Another option is to use several straight cuts, as introduced by Bereg et al [8]: Consider some arrangement of n hyperplanes in R d . The cells of this arrangement allow a natural 2-coloring, where two cells get a different color whenever they share a (d − 1)-dimensional face.…”

Section: Introduction 1mass Partitionsmentioning

confidence: 99%

The complexity of sharing a pizza

Schnider

2021

Preprint

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Assume you have a 2-dimensional pizza with 2n ingredients that you want to share with your friend. For this you are allowed to cut the pizza using several straight cuts, and then give every second piece to your friend. You want to do this fairly, that is, your friend and you should each get exactly half of each ingredient. How many cuts do you need?It was recently shown using topological methods that n cuts always suffice. In this work, we study the computational complexity of finding such n cuts. Our main result is that this problem is PPA-complete when the ingredients are represented as point sets. For this, we give a new proof that for point sets n cuts suffice, which does not use any topological methods.We further prove several hardness results as well as a higher-dimensional variant for the case where the ingredients are well-separated.

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Balanced partitions of 3-colored geometric sets in the plane

Cited by 17 publications

References 11 publications

Equipartitions with Wedges and Cones

Equipartitions with Wedges and Cones

The Balanced Connected Subgraph Problem

The complexity of sharing a pizza

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