The Entropy Rounding Method in Approximation Algorithms

TL;DR

This paper introduces a novel randomized rounding method based on entropy properties of matrices, improving approximation algorithms for problems like Bin Packing and Train Delivery by leveraging discrepancy theory and semidefinite programming.

Contribution

It applies Beck's entropy method to approximation algorithms, providing a new randomized rounding procedure for matrices with bounded entropy, and achieves improved approximation ratios.

Findings

Achieves polynomial-time OPT + O(log^2 OPT) approximation for Bin Packing With Rejection.

Provides the first AFPTAS for the Train Delivery problem.

Demonstrates the versatility of entropy-based rounding in various optimization problems.

Abstract

Let A be a matrix, c be any linear objective function and x be a fractional vector, say an LP solution to some discrete optimization problem. Then a recurring task in theoretical computer science (and in approximation algorithms in particular) is to obtain an integral vector y such that Ax is roughly Ay and c*y exceeds c*x by only a moderate factor. We give a new randomized rounding procedure for this task, provided that A has bounded Delta-approximate entropy. This property means that for uniformly chosen random signs chi(j) in {-1,+1} on any subset of the columns, the outcome A*chi can be approximately described using a sub-linear number of bits in expectation. To achieve this result, we modify well-known techniques from the field of discrepancy theory, especially we rely on Beck's entropy method, which to the best of our knowledge has never been used before in the context of approximation algorithms. Our result can be made constructive using the Bansal framework based on semidefinite programming. We demonstrate the versatility of our procedure by rounding fractional solutions to column-based linear programs for some generalizations of Bin Packing. For example we obtain a polynomial time OPT + O(log^2 OPT) approximation for Bin Packing With Rejection and the first AFPTAS for the Train Delivery problem.