Sperner’s Lemma and Fair Division Problems

نویسنده

  • SUTHEE RUANGWISES
چکیده

This paper is a seminar paper that summarizes Fran­ cis Edward Su’s paper “Rental harmony: Sperner’s lemma in fair division” [1] in 1999. In this paper we introduce and prove Sperner’s Lemma. Then, we introduce the fair division problem and an al­ gorithm based on Sperner’s lemma that can solve some variants of the fair division problem.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Rental Harmony: Sperner's Lemma in Fair Division

My friend’s dilemma was a practical question that mathematics could answer, both elegantly and constructively. He and his housemates were moving to a house with rooms of various sizes and features, and were having trouble deciding who should get which room and for what part of the total rent. He asked, “Do you think there’s always a way to partition the rent so that each person will prefer a di...

متن کامل

Multilabeled versions of Sperner's and Fan's lemmas and applications

We propose a general technique related to the polytopal Sperner lemma for proving old and new multilabeled versions of Sperner’s lemma. A notable application of this technique yields a cake-cutting theorem where the number of players and the number of pieces can be independently chosen. We also prove multilabeled versions of Fan’s lemma, a combinatorial analogue of the Borsuk-Ulam theorem, and ...

متن کامل

Sperner's Colorings, Hypergraph Labeling Problems and Fair Division

We prove three results about colorings of the simplex reminiscent of Sperner’s Lemma, with applications in hardness of approximation and fair division. First, we prove a coloring lemma conjectured by [5]: Let Vk,q = {v ∈ Z+ : ∑k i=1 vi = q} and Ek,q = {{a + e1,a + e2, . . . ,a + ek} : a ∈ Z+, ∑k i=1 ai = q − 1}. Then for every Sperner-admissible labeling (` : Vk,q → [k] such that v`(v) > 0 for ...

متن کامل

Sperner’s Lemma Implies Kakutani’s Fixed Point Theorem

Kakutani’s fixed point theorem has many applications in economics and game theory. One of its most well-known applications is in John Nash’s paper [8], where the theorem is used to prove the existence of an equilibrium strategy in n-person games. Sperner’s lemma, on the other hand, is a combinatorial result concerning the labelling of the vertices of simplices and their triangulations. It is kn...

متن کامل

Tree-like resolution complexity of two planar problems

We consider two CSP problems: the first CSP encodes 2D Sperner’s lemma for the standard triangulation of the right triangle on n2 small triangles; the second CSP encodes the fact that it is impossible to match cells of n× n square to arrows (two horizontal, two vertical and four diagonal) such that arrows in two cells with a common edge differ by at most 45◦, and all arrows on the boundary of t...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2015