# Fair and Efficient Cake Division with Connected Pieces

## TL;DR

This paper develops algorithms and hardness results for fair and efficient cake division with connected pieces, focusing on envy-freeness, Nash social welfare, and mean welfare, balancing fairness and efficiency.

## Contribution

It introduces new approximation algorithms for envy-freeness and Nash social welfare in connected cake division and proves their computational hardness.

## Key findings

- Efficient algorithm achieves 2+o(1) envy-freeness approximation.
- Algorithm attains 3+o(1) approximation for Nash social welfare.
- Maximizing Nash social welfare is APX-hard in this setting.

## Abstract

The classic cake-cutting problem provides a model for addressing fair and efficient allocation of a divisible, heterogeneous resource (metaphorically, the cake) among agents with distinct preferences. Focusing on a standard formulation of cake cutting, in which each agent must receive a contiguous piece of the cake, this work establishes algorithmic and hardness results for multiple fairness/efficiency measures. First, we consider the well-studied notion of envy-freeness and develop an efficient algorithm that finds a cake division (with connected pieces) wherein the envy is multiplicatively within a factor of 2+o(1). The same algorithm in fact achieves an approximation ratio of 3+o(1) for the problem of finding cake divisions with as large a Nash social welfare (NSW) as possible. NSW is another standard measure of fairness and this work also establishes a connection between envy-freeness and NSW: approximately envy-free cake divisions (with connected pieces) always have near-optimal Nash social welfare. Furthermore, we develop an approximation algorithm for maximizing the $\rho$-mean welfare--this unifying objective, with different values of $\rho$, interpolates between notions of fairness (NSW) and efficiency (average social welfare). Finally, we complement these algorithmic results by proving that maximizing NSW (and, in general, the $\rho$-mean welfare) is APX-hard in the cake-division context.