Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Extending the Extension: Deterministic Algorithm for Non-monotone Submodular Maximization

Niv Buchbinder, Moran Feldman

TL;DR

A new tool is introduced, designed to derandomize submodular maximization algorithms that are based on the successful “solve fractionally and then round” approach, and it allows for a deterministic implementation of both the fractionally solving step and the rounding step of the above approach.

Abstract

Maximization of submodular functions under various constraints is a fundamental problem that has been studied extensively. A powerful technique that has emerged and has been shown to be extremely effective for such problems is the following. First, a continues relaxation of the problem is obtained by relaxing the (discrete) set of feasible solutions to a convex body, and extending the discrete submodular function $f$ to a continuous function $F$ known as the multilinear extension. Then, two algorithmic steps are implemented. The first step approximately solves the relaxation by finding a fractional solution within the convex body that approximately maximizes $F$; and the second step rounds this fractional solution to a feasible integral solution. While this ``fractionally solve and then round'' approach has been a key technique for resolving many questions in the field, the main drawback of algorithms based on it is that evaluating the multilinear extension may require a number of value oracle queries to $f$ that is exponential in the size of $f$'s ground set. The only known way to tackle this issue is to approximate the value of $F$ via sampling, which makes all algorithms based on this approach inherently randomized and quite slow. In this work, we introduce a new tool, that we refer to as the extended multilinear extension, designed to derandomize submodular maximization algorithms that are based on the successful ``solve fractionally and then round'' approach. We demonstrate the effectiveness of this new tool on the fundamental problem of maximizing a submodular function subject to a matroid constraint, and show that it allows for a deterministic implementation of both the fractionally solving step and the rounding step of the above approach. As a bonus, we also get a randomized algorithm for the problem with an improved query complexity.

Extending the Extension: Deterministic Algorithm for Non-monotone Submodular Maximization

TL;DR

A new tool is introduced, designed to derandomize submodular maximization algorithms that are based on the successful “solve fractionally and then round” approach, and it allows for a deterministic implementation of both the fractionally solving step and the rounding step of the above approach.

Abstract

Maximization of submodular functions under various constraints is a fundamental problem that has been studied extensively. A powerful technique that has emerged and has been shown to be extremely effective for such problems is the following. First, a continues relaxation of the problem is obtained by relaxing the (discrete) set of feasible solutions to a convex body, and extending the discrete submodular function to a continuous function known as the multilinear extension. Then, two algorithmic steps are implemented. The first step approximately solves the relaxation by finding a fractional solution within the convex body that approximately maximizes ; and the second step rounds this fractional solution to a feasible integral solution. While this ``fractionally solve and then round'' approach has been a key technique for resolving many questions in the field, the main drawback of algorithms based on it is that evaluating the multilinear extension may require a number of value oracle queries to that is exponential in the size of 's ground set. The only known way to tackle this issue is to approximate the value of via sampling, which makes all algorithms based on this approach inherently randomized and quite slow. In this work, we introduce a new tool, that we refer to as the extended multilinear extension, designed to derandomize submodular maximization algorithms that are based on the successful ``solve fractionally and then round'' approach. We demonstrate the effectiveness of this new tool on the fundamental problem of maximizing a submodular function subject to a matroid constraint, and show that it allows for a deterministic implementation of both the fractionally solving step and the rounding step of the above approach. As a bonus, we also get a randomized algorithm for the problem with an improved query complexity.
Paper Structure (21 sections, 26 theorems, 69 equations, 6 algorithms)

This paper contains 21 sections, 26 theorems, 69 equations, 6 algorithms.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Preliminaries
  4. Vector operations:
  5. Submodular Functions:
  6. Independence Systems and Matroids:
  7. Our Problem and Dummy Elements:
  8. The Extended Multilinear Extension
  9. Properties of the Extended Multilinear Extension
  10. Deterministic Measured Continuous Greedy
  11. The Split Algorithm
  12. Deterministic Measured Continuous Greedy
  13. Randomly Decomposing the Output of the Algorithm
  14. Deterministically Rounding the Extended Multilinear Extension
  15. The subroutines Relax and Hit-Constraint
  16. ...and 6 more sections

Key Result

Theorem 1.1

There is a deterministic algorithm that given a matroid $\cM=(\cN, \cI)$, a non-negative submodular function $f\colon 2^\cN \to {\bR_{\geq 0}}$, and a parameter $\varepsilon\in(0,1)$, produces a vector $\vy\in [0,1]^{{2^\cN}}\!\!$ such that $\mathop{\mathrm{Mar}}\nolimits(\vy)\in P(\cM)$, $\mathop{

Theorems & Definitions (51)

  • Theorem 1.1: Fractional Solution
  • Theorem 1.2: Rounding
  • Corollary 1.3
  • Proposition 1.3
  • Definition 2.1
  • Lemma 2.1: Lemma 2.2 of buchbinder2014submodular
  • Corollary 2.2
  • Lemma 2.3: Proved by brualdi1969comments, and can also be found as Corollary 39.12a in schrijver2003combinatorial
  • Corollary 2.4
  • Definition 3.1: Extended Multilinear Extension
  • ...and 41 more