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On the correlation gap of matroids.

Edin Husić1, Zhuan Khye Koh2, Georg Loho3

  • 1IDSIA, USI-SUPSI, Lugano, Switzerland.

Mathematical Programming
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Summary

We introduce a fine-grained analysis of the correlation gap for matroid rank functions, providing improved lower bounds based on matroid rank and girth. This research enhances understanding for approximation algorithms and mechanism design.

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Area of Science:

  • Discrete Mathematics
  • Theoretical Computer Science
  • Operations Research

Background:

  • The correlation gap quantifies the ratio between two extensions of set functions to the unit cube.
  • It serves as a performance guarantee in approximation algorithms and mechanism design.
  • Existing research shows the correlation gap for monotone submodular functions is at least 1/e, with tightness in simple matroid rank functions.

Purpose of the Study:

  • To conduct a fine-grained study of the correlation gap specifically for matroid rank functions.
  • To derive improved lower bounds for the correlation gap.
  • To investigate the impact of matroid properties on the correlation gap.

Main Methods:

  • Analyzing the correlation gap of matroid rank functions.
  • Developing new lower bounds parametrized by matroid rank and girth.
  • Investigating the behavior of the correlation gap under weighted rank functions.

Main Results:

  • An improved lower bound on the correlation gap for matroid rank functions, dependent on rank and girth.
  • Demonstration that the correlation gap of a matroid's weighted rank function is minimized with uniform weights.
  • Establishing new theoretical bounds with direct algorithmic implications.

Conclusions:

  • The study provides a more nuanced understanding of the correlation gap in matroids.
  • The findings offer tighter performance guarantees for related optimization problems.
  • This research advances the fields of submodular maximization, mechanism design, and contention resolution.