Flow Increment through Network Expansion

The network expansion problem is a very important practical optimization problem when there is a need to increment the flow through an existing network of transportation, electricity, water, gas, etc. In this problem, the flow augmentation can be achieved either by increasing the capacities on the existing arcs, or by adding new arcs to the network. Both operations are coming with an expansion cost. In this paper, the problem of finding the minimum network expansion cost so that the modified network can transport a given amount of flow from the source node to the sink node is studied. A strongly polynomial algorithm is deduced to solve the problem.

Inverse Minimum Cut Problem with Lower and Upper Bounds

The inverse minimum cut problem is one of the classical inverse optimization researches. In this paper, the inverse minimum cut with a lower and upper bounds problem is considered. The problem is to change both, the lower and upper bounds on arcs so that a given feasible cut becomes a minimum cut in the modified network and the distance between the initial vector of bounds and the modified one is minimized. A strongly polynomial algorithm to solve the problem under l1 norm is developed.

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~\cite{LSW15} runs in time \so{n^3 \cdot \eo + n^4} where \eo denotes the cost to evaluate the function on any set. For functions with range [-1,1], the best \eps-additive approximation algorithm~\cite{CLSW17} runs in time \so{n^{5/3}/\eps^{2} \cdot \eo}. In this paper we present a classical and a quantum algorithm for approximate submodular minimization. Our classical result improves on the algorithm of \cite{CLSW17} and runs in time \so{n^{3/2}/\eps^2 \cdot \eo}. Our quantum algorithm is, up to our knowledge, the first attempt to use quantum computing for submodular optimization. The algorithm runs in time \so{n^{5/4}/\eps^{5/2} \cdot \log(1/\eps) \cdot \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 \bo{\sqrt{Tn}}. Previous quantum algorithms for this problem were of complexity \bo{T\sqrt{n}}.