Corpus ID: 17598332

On the Approximation of Submodular Functions

@article{Devanur2013OnTA,
  title={On the Approximation of Submodular Functions},
  author={Nikhil R. Devanur and S. Dughmi and Roy Schwartz and Ankit Sharma and Mohit Singh},
  journal={ArXiv},
  year={2013},
  volume={abs/1304.4948}
}
Submodular functions are a fundamental object of study in combinatorial optimization, economics, machine learning, etc. and exhibit a rich combinatorial structure. Many subclasses of submodular functions have also been well studied and these subclasses widely vary in their complexity. Our motivation is to understand the relative complexity of these classes of functions. Towards this, we consider the question of how well can one class of submodular functions be approximated by another (simpler… Expand
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References

SHOWING 1-10 OF 27 REFERENCES
Submodular Approximation: Sampling-based Algorithms and Lower Bounds
  • Zoya Svitkina, L. Fleischer
  • Mathematics, Computer Science
  • 2008 49th Annual IEEE Symposium on Foundations of Computer Science
  • 2008
TLDR
This work introduces several generalizations of classical computer science problems obtained by replacing simpler objective functions with general submodular functions, and presents an algorithm for approximately learning sub modular functions with special structure, whose guarantee is close to the lower bound. Expand
Approximating submodular functions everywhere
TLDR
The problem of approximating a non-negative, monotone, submodular function f on a ground set of size n everywhere is considered, after only poly(n) oracle queries, and it is shown that no algorithm can achieve a factor better than Ω(√n/log n), even for rank functions of a matroid. Expand
Learning submodular functions
TLDR
This paper considers PAC-style learning of submodular functions in a distributional setting and uses lossless expanders to construct a new family of matroids which can take wildly varying rank values on superpolynomially many sets; no such construction was previously known. Expand
Is Submodularity Testable?
TLDR
This work begins the study of property testing of submodularity on the boolean hypercube, and analyzes a natural tester for this problem, and proves an interesting lower bound suggesting that this tester cannot be efficient in terms of ϵ. Expand
An analysis of approximations for maximizing submodular set functions—I
TLDR
It is shown that a “greedy” heuristic always produces a solution whose value is at least 1 −[(K − 1/K]K times the optimal value, which can be achieved for eachK and has a limiting value of (e − 1)/e, where e is the base of the natural logarithm. Expand
Sketching valuation functions
TLDR
It is proved that every deterministic algorithm that accesses the function via value queries only cannot guarantee a sketching ratio better than n1−e, and it is shown that coverage functions, an interesting subclass of submodular functions, admit arbitrarily good sketches. Expand
A combinatorial strongly polynomial algorithm for minimizing submodular functions
This paper presents a combinatorial polynomial-time algorithm for minimizing submodular functions, answering an open question posed in 1981 by Grötschel, Lovász, and Schrijver. The algorithm employsExpand
From convex optimization to randomized mechanisms: toward optimal combinatorial auctions
TLDR
The authors' is the first truthful-in-expectation and polynomial-time mechanism to achieve a constant-factor approximation for an NP-hard welfare maximization problem in combinatorial auctions with heterogeneous goods and restricted valuations. Expand
Approximation Algorithms for Online Weighted Rank Function Maximization under Matroid Constraints
TLDR
The first randomized online algorithms for this problem with poly-logarithmic competitive ratio are presented, and a weighted-majority type update rule along with uncrossing properties of tight sets in the matroid polytope are extended to find an approximately optimal fractional LP solution. Expand
Testing Coverage Functions
TLDR
An algorithm is demonstrated which makes O(m|U|) queries to an oracle of a coverage function and completely reconstructs it, giving a polytime tester for succinct coverage functions for which |U| is polynomially bounded in m. Expand
...
1
2
3
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