Multiple knapsack-constrained monotone DR-submodular maximization on distributive lattice Continuous Greedy Algorithm on Median Complex

We consider a problem of maximizing a monotone DR-submodular function under multiple order-consistent knapsack constraints on a distributive lattice. Because a distributive lattice is used to represent a dependency constraint, the problem can represent a dependency constrained version of a submodula...

Full description

Saved in:
Bibliographic Details
Published in:Mathematical programming Vol. 194; no. 1-2; pp. 85 - 119
Main Authors: Maehara, Takanori, Nakashima, So, Yamaguchi, Yutaro
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.07.2022
Subjects:
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Full Length Paper
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Theoretical
90C27 (Mathematical programming; Combinatorial Optimization)
Distributive lattices
Continuous greedy
Median complex
Submodular maximization
68W25 (Algorithms in computer science; Approximation Algorithm)
06D99 (Distributive lattices; None of the above, but in this section)
ISSN:0025-5610, 1436-4646
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We consider a problem of maximizing a monotone DR-submodular function under multiple order-consistent knapsack constraints on a distributive lattice. Because a distributive lattice is used to represent a dependency constraint, the problem can represent a dependency constrained version of a submodular maximization problem on a set. We propose a ( 1 - 1 / e )-approximation algorithm for this problem. To achieve this result, we generalize the continuous greedy algorithm to distributive lattices: We choose a median complex as a continuous relaxation of the distributive lattice and define the multilinear extension on it. We show that the median complex admits special curves, named uniform linear motions . The multilinear extension of a DR-submodular function is concave along a positive uniform linear motion, which is a key property used in the continuous greedy algorithm.
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-021-01620-7