Upload
mildred-park
View
216
Download
0
Tags:
Embed Size (px)
Citation preview
The Districting Problem
Esther Arkin, Irina Kostitsyna, Joseph Mitchell, Valentin Polishchuk,
Girishkumar Sabhnani
The Districting Problem
Problem statement
• Districting problem (DP) Given a polygon partitioned into sub-districts with weights, join them into minimum number of simple districts with total weight less than M
• Conjugate problem (CDP)Given a limit of k districts, minimize maximum weight
Motivation
Political districting• Voting
Objective:• Find a partition into districts
Requirements:• Bounded weights • Population equality• Contiguity• “Nice shape”• …
Our districting problem• Air traffic management
Objective:• Find a partition into districts
Requirements:• Bounded weights
• Contiguity• “Nice shape”
Results
• In 1D case: optimal solution• In 2D case: DP, CDP are NP-hard• DP is weakly hard to approximate with 3/2
factor• CDP is hard to approximate with 5/4 factor• Approximations– Hamiltonian: 4-approximation– Non-Hamiltonian: 2Δ-approximation
Weak hardness
• Reduction from PARTITION• Hard to approximate with factor better than 3/2
Strong hardness
• Proof similar to rectangle tiling hardness (Khanna et al., ‘98)
• CDP is strongly hard to approximate with factor better than 5/4
1D case
• Greedy algorithm is optimal
Hamiltonian path in dual graph
Algorithm creates holes
After breaking districts with holes we get 4-approximation
Spanning tree of max degree Δ
• Degree Δ*+1 (Fürer, Raghavachari, ‘92)
•
After breaking districts with holes we get 2Δ-appx
Future work and open problems
• Dynamic districting problem (hardness)• Introduce “nice shape” requirement• Better approximation algorithms