
Minimizing approximately submodular functions
The problem of minimizing a submodular function is well studied; several...
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Nearoptimal Approximate Discrete and Continuous Submodular Function Minimization
In this paper we provide improved running times and oracle complexities ...
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Convex Analysis and Optimization with Submodular Functions: a Tutorial
Setfunctions appear in many areas of computer science and applied mathe...
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Improved (Provable) Algorithms for the Shortest Vector Problem via Bounded Distance Decoding
The most important computational problem on lattices is the Shortest Vec...
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Efficient Algorithms for Searching the Minimum Information Partition in Integrated Information Theory
The ability to integrate information in the brain is considered to be an...
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Joint MBestDiverse Labelings as a Parametric Submodular Minimization
We consider the problem of jointly inferring the Mbest diverse labeling...
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An Approximation Algorithm for Riskaverse Submodular Optimization
We study the problem of incorporating risk while making combinatorial de...
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Quantum and Classical Algorithms for Approximate Submodular Function Minimization
Submodular functions are set functions mapping every subset of some ground set of size n into the real numbers and satisfying the diminishing returns property. Submodular minimization is an important field in discrete optimization theory due to its relevance for various branches of mathematics, computer science and economics. The currently fastest strongly polynomial algorithm for exact minimization [LSW15] runs in time O(n^3 ·EO + n^4) where EO denotes the cost to evaluate the function on any set. For functions with range [1,1], the best ϵadditive approximation algorithm [CLSW17] runs in time O(n^5/3/ϵ^2·EO). In this paper we present a classical and a quantum algorithm for approximate submodular minimization. Our classical result improves on the algorithm of [CLSW17] and runs in time O(n^3/2/ϵ^2 ·EO). Our quantum algorithm is, up to our knowledge, the first attempt to use quantum computing for submodular optimization. The algorithm runs in time O(n^5/4/ϵ^5/2·(1/ϵ) ·EO). The main ingredient of the quantum result is a new method for sampling with high probability T independent elements from any discrete probability distribution of support size n in time O(√(Tn)). Previous quantum algorithms for this problem were of complexity O(T√(n)).
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