A Hybrid Quantum Algorithm for Load Flow

We study a hybrid quantum algorithm for solving the AC load flow problem. The algorithm uses a quantum algorithm to compute the direction in the Newton-Raphson method. This hybrid approach offers scalability and improved convergence rates in theory

On a class of covering problems with variable capacities in wireless networks

We consider the problem of allocating clients to base stations in wireless networks. Two design decisions are the location of the base stations, and the power levels of the base stations. We model the interference, due to the increased power usage resulting in greater serving radius, as capacities that are non-increasing with respect to the covering radius. Clients have demands that are not necessarily uniform and the capacity of a facility limits the total demand that can be served by the facility. We consider three models. In the first model, the location of the base stations and the clients are fixed, and the problem is to determine the serving radius for each base station so as to serve a set of clients with maximum total profit subject to the capacity constraints of the base stations. In the second model, each client has an associated demand in addition to its profit. A fixed number of facilities have to be opened from a candidate set of locations. The goal is to serve clients so as to maximize the profit subject to the capacity constraints. In the third model, the location and the serving radius of the base stations are to be determined. There are costs associated with opening the base stations, and the goal is to open a set of base stations of minimum total cost so as to serve the entire demand subject to the capacity constraints at the base stations. We show that for the first model the problem is NP-complete even when there are only two choices for the serving radius, and the capacities are 1, 2. For the second model, we give a 1/2 approximation algorithm. For the third model, we give a column generation procedure for solving the standard linear programming model, and a randomized rounding procedure. We establish the efficacy of the column generation based rounding scheme on randomly generated instances

VERY SHORT COURSE ON LP

Given a matrix A ∈ R m×n and vectors b ∈ R m and c ∈ R n, a linear program (LP) is an optimisation problem of the form min x {c T x: Ax ≥ b}. (LP1) x is the solution vector, the value c T x is the cost associated with solution x, A is the constraint matrix, and b is called sometimes right hand side vector. An integer programming (IP) problem is a linear program in which the solution vector must be integral, i.e. all components must be integers. min x {c T x: Ax ≥ b, x ∈ Z n} (IP1) While (LP1) belongs to the complexity class P, (IP1) is an NP-hard problem. The most frequently used algorithm to solve LP problems, the simplex method, is efficient in practice but has an exponential worst case running time. A detailed discussion and a list of useful pointers in the literature regarding the complexity of LP and IP problems can be found in the book of Schrijver [72]. This section gives a brie

Information Processing Letters 79 (2001) 215–221 On computing the optimal bridge between two convex polygons

Given two convex polygons P and Q we want to find a line segment (a bridge) that connects P and Q so that the maximum distance from a point inside P across the bridge to a point inside Q is minimized. We propose a linear-time algorithm to solv

On Implementing a Two-Step Interior Point Method for Solving Linear Programs

A new two-step interior point method for solving linear programs is presented. The technique uses a convex combination of the auxiliary and central points to compute the search direction. To update the central point, we find the best value for step size such that the feasibility condition is held. Since we use the information from the previous iteration to find the search direction, the inverse of the system is evaluated only once every iteration. A detailed empirical evaluation is performed on NETLIB instances, which compares two variants of the approach to the primal-dual log barrier interior point method. Results show that the proposed method is faster. The method reduces the number of iterations and CPU time(s) by 27% and 18%, respectively, on NETLIB instances tested compared to the classical interior point algorithm