scholarly journals On the Complexity of Chore Division

2018 ◽  
Author(s):  
Alireza Farhadi ◽  
MohammadTaghi Hajiaghayi
Keyword(s):  
Lower Bound ◽  
Division Problem ◽  
Cake Cutting ◽  
Cutting Problem ◽  
Cutting Problems

We study the proportional chore division problem where a protocol wants to divide an undesirable object, called chore, among n different players. This problem is the dual variant of the cake cutting problem in which we want to allocate a desirable object. In this paper, we show that chore division and cake cutting problems are closely related to each other and provide a tight lower bound for proportional chore division.

scholarly journals Deterministic, Strategyproof, and Fair Cake Cutting

2017 ◽  
Author(s):  
Vijay Menon ◽  
Kate Larson
Keyword(s):  
Mechanism Design ◽  
Impossibility Result ◽  
Restricted Class ◽  
Piecewise Constant ◽  
Incentive Compatible ◽  
Negative Results ◽  
Cake Cutting ◽  
Design Perspective ◽  
A Minor ◽  
Cutting Problem

We study the classic cake cutting problem from a mechanism design perspective, in particular focusing on deterministic mechanisms that are strategyproof and fair. We begin by looking at mechanisms that are non-wasteful and primarily show that for even the restricted class of piecewise constant valuations there exists no direct-revelation mechanism that is strategyproof and even approximately proportional. Subsequently, we remove the non-wasteful constraint and show another impossibility result stating that there is no strategyproof and approximately proportional direct-revelation mechanism that outputs contiguous allocations, again, for even the restricted class of piecewise constant valuations. In addition to the above results, we also present some negative results when considering an approximate notion of strategyproofness, show a connection between direct-revelation mechanisms and mechanisms in the Robertson-Webb model when agents have piecewise constant valuations, and finally also present a (minor) modification to the well-known Even-Paz algorithm that has better incentive-compatible properties for the cases when there are two or three agents.

A Lower Bound for Cake Cutting

2003 ◽  
pp. 459-469 ◽  
Author(s):  
Jiří Sgall ◽  
Gerhard J. Woeginger
Keyword(s):  
Lower Bound ◽  
Cake Cutting

scholarly journals Almost Tight Lower Bounds for Hard Cutting Problems in Embedded Graphs

2021 ◽  
Vol 68 (4) ◽  
pp. 1-26
Author(s):  
Vincent Cohen-Addad ◽  
Éric Colin De Verdière ◽  
Dániel Marx ◽  
Arnaud De Mesmay
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Undirected Graph ◽  
Time Algorithm ◽  
Planar Case ◽  
Embedded Graphs ◽  
Multiway Cut ◽  
Hard Cutting ◽  
Cutting Problems ◽  
Unweighted Graph

We prove essentially tight lower bounds, conditionally to the Exponential Time Hypothesis, for two fundamental but seemingly very different cutting problems on surface-embedded graphs: the Shortest Cut Graph problem and the Multiway Cut problem. A cut graph of a graph  G embedded on a surface S is a subgraph of  G whose removal from S leaves a disk. We consider the problem of deciding whether an unweighted graph embedded on a surface of genus  G has a cut graph of length at most a given value. We prove a time lower bound for this problem of n Ω( g log g ) conditionally to the ETH. In other words, the first n O(g) -time algorithm by Erickson and Har-Peled [SoCG 2002, Discr. Comput. Geom. 2004] is essentially optimal. We also prove that the problem is W[1]-hard when parameterized by the genus, answering a 17-year-old question of these authors. A multiway cut of an undirected graph  G with t distinguished vertices, called terminals , is a set of edges whose removal disconnects all pairs of terminals. We consider the problem of deciding whether an unweighted graph  G has a multiway cut of weight at most a given value. We prove a time lower bound for this problem of n Ω( gt + g 2 + t log ( g + t )) , conditionally to the ETH, for any choice of the genus  g ≥ 0 of the graph and the number of terminals  t ≥ 4. In other words, the algorithm by the second author [Algorithmica 2017] (for the more general multicut problem) is essentially optimal; this extends the lower bound by the third author [ICALP 2012] (for the planar case). Reductions to planar problems usually involve a gridlike structure. The main novel idea for our results is to understand what structures instead of grids are needed if we want to exploit optimally a certain value  G of the genus.

scholarly journals Fair Cake Division Under Monotone Likelihood Ratios

2021 ◽  
Author(s):  
Siddharth Barman ◽  
Nidhi Rathi
Keyword(s):  
Lipschitz Constant ◽  
Concave Function ◽  
Likelihood Ratios ◽  
Exponential Distributions ◽  
Monotone Likelihood Ratio ◽  
Cake Cutting ◽  
Bit Complexity ◽  
Arbitrary Precision ◽  
Cutting Problem ◽  
Positive Results

This work develops algorithmic results for the classic cake-cutting problem in which a divisible, heterogeneous resource (modeled as a cake) needs to be partitioned among agents with distinct preferences. We focus on a standard formulation of cake cutting wherein each agent must receive a contiguous piece of the cake. Although multiple hardness results exist in this setup for finding fair/efficient cake divisions, we show that, if the value densities of the agents satisfy the monotone likelihood ratio property (MLRP), then strong algorithmic results hold for various notions of fairness and economic efficiency. Addressing cake-cutting instances with MLRP, first we develop an algorithm that finds cake divisions (with connected pieces) that are envy free, up to an arbitrary precision. The time complexity of our algorithm is polynomial in the number of agents and the bit complexity of an underlying Lipschitz constant. We obtain similar positive results for maximizing social, egalitarian, and Nash social welfare. Many distribution families bear MLRP. In particular, this property holds if all the value densities belong to any one of the following families: Gaussian (with the same variance), linear, Poisson, and exponential distributions, linear translations of any log-concave function. Hence, through MLRP, the current work obtains novel cake-cutting algorithms for multiple distribution families.

scholarly journals Dynamic Fair Division Problem with General Valuations

2018 ◽  
Author(s):  
Bo Li ◽  
Wenyang Li ◽  
Yingkai Li
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Fair Division ◽  
Division Problem ◽  
Logarithmic Factor ◽  
Dynamic Algorithms ◽  
Dynamic Setting ◽  
Over Time

In this paper, we focus on how to dynamically allocate a divisible resource fairly among n players who arrive and depart over time. The players may have general heterogeneous valuations over the resource. It is known that the exact envy-free and proportional allocations may not exist in the dynamic setting [Walsh, 2011]. Thus, we will study to what extent we can guarantee the fairness in the dynamic setting. We first design two algorithms which are O(log n)-proportional and O(n)-envy-free for the setting with general valuations, and by constructing the adversary instances such that all dynamic algorithms must be at least Omega(1)-proportional and Omega(n/log n)-envy-free, we show that the bounds are tight up to a logarithmic factor. Moreover, we introduce the setting where the players' valuations are uniform on the resource but with different demands, which generalize the setting of [Friedman et al., 2015]. We prove an O(log n) upper bound and a tight lower bound for this case. 

Two-Dimensional Stock Cutting Problems and Solution Methodologies

1983 ◽  
Vol 105 (3) ◽  
pp. 155-160 ◽  
Author(s):  
S. C. Sarin
Keyword(s):  
Optimization Procedure ◽  
Two Dimensional ◽  
Interactive Optimization ◽  
Stock Cutting ◽  
Cutting Problem ◽  
Cutting Problems

In this paper methodologies available in the literature to solve two-dimensional stock cutting problems are reviewed. Several variations of the problem are discussed. An interactive optimization procedure for a general two-dimensional stock cutting problem is introduced.

The least distance between extremums and minimal period of solutions to autonomous vector differential equations

2019 ◽  
Vol 485 (2) ◽  
pp. 142-144
Author(s):  
A. A. Zevin
Keyword(s):  
Lower Bound ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Lipschitz Constant ◽  
Minimal Period ◽  
Arbitrary Vector ◽  
Vector Norm ◽  
Sharp Lower Bound

Solutions x(t) of the Lipschitz equation x = f(x) with an arbitrary vector norm are considered. It is proved that the sharp lower bound for the distances between successive extremums of xk(t) equals π/L where L is the Lipschitz constant. For non-constant periodic solutions, the lower bound for the periods is 2π/L. These estimates are achieved for norms that are invariant with respect to permutation of the indices.

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