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