Submodular Approximation: Sampling-based Algorithms and Lower Bounds

We introduce several generalizations of classical computer science problems obtained by replacing simpler objective functions with general submodular functions. The new problems include submodular load balancing, which generalizes load balancing or minimum-makespan scheduling, submodular sparsest cu...

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Bibliographic Details
Published in:SIAM journal on computing Vol. 40; no. 6; pp. 1715 - 1737
Main Authors: Svitkina, Zoya, Fleischer, Lisa
Format: Journal Article
Language:English
Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2011
Subjects:
Algorithmics. Computability. Computer arithmetics
Algorithms
Applied sciences
Approximation
Computer science
Computer science; control theory; systems
Computer systems performance. Reliability
Exact sciences and technology
Mathematical analysis
Mathematics
Miscellaneous
Optimization
Sciences and techniques of general use
Software
Special functions
Theoretical computing
Monotonic function
Polynomial
Lower bound
information-theoretic lower bounds
Approximation
Query
Scheduling
Makespan
68W25
90C27
approximation algorithms
Approximation algorithm
Special function
N order
submodular functions
Computer science
Upper bound
Graph cut
Function minimization
Objective function
Sampling
Load balancing
Oracle
Information theory
ISSN:0097-5397, 1095-7111
Online Access:Get full text
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Summary:We introduce several generalizations of classical computer science problems obtained by replacing simpler objective functions with general submodular functions. The new problems include submodular load balancing, which generalizes load balancing or minimum-makespan scheduling, submodular sparsest cut and submodular balanced cut, which generalize their respective graph cut problems, as well as submodular function minimization with a cardinality lower bound. We establish upper and lower bounds for the approximability of these problems with a polynomial number of queries to a function-value oracle. The approximation guarantees that most of our algorithms achieve are of the order of $\sqrt{{n}/{\ln n}}$. We show that this is the inherent difficulty of the problems by proving matching lower bounds. We also give an improved lower bound for the problem of approximating a monotone submodular function everywhere. In addition, we present an algorithm for approximating submodular functions with a special structure, whose guarantee is close to the lower bound. Although quite restrictive, the class of functions with this structure includes the ones that are used for lower bounds both by us and in previous work. [PUBLICATION ABSTRACT]
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ISSN:0097-5397
1095-7111
DOI:10.1137/100783352