Maximum Network Flow Algorithms

The aim of the maximum network flow problem is to push as much flow as possible between two special vertices, the source and the sink satisfying the capacity constraints. For the solution of the maximum flow problem, there exists a number of algorithms. The existing algorithms can be divided into two families. First, augmenting path algorithms that satisfy the conservation constraints at intermediate vertices and the second preflow push relabel algorithms that violates the conservation constraints at the intermediate vertices resulting incoming flow more than outgoing flow.In this paper, we study different algorithms that determine the maximum flow in the static and dynamic networks.

An analytical solution to the multicommodity network flow problem with weighted random routing

We derive an analytical expression for the mean load at each node of an arbitrary undirected graph for the non-uniform multicommodity flow problem under weighted random routing. We show the mean load at each node, net of its demand and normalized by its (weighted) degree, is a constant equal to the trace of the product of two matrices: the Laplacian of the demand matrix and the generalized inverse of the graph Laplacian. For the case of uniform demand, this constant reduces to the sum of the inverses of the non-zero eigenvalues of the graph Laplacian. We note that such a closed-form expression for the network capacity for the general multicommodity network flow problem has not been reported before, and even though (weighted) random routing is not a practical procedure, it enables us to derive a (tight) upper bound for the capacity of the network under more standard routing policies. Using this new expression, we compute network capacity for a sample of demand matrices for some prototypical networks, including uniform demand (one unit between all node pairs) and broadcast demand (one unit between a source node and each other node as destination), and finally derive estimates of the mean load in some asymptotic cases.

Optimal Screening of Populations with Heterogeneous Risk Profiles Under the Availability of Multiple Tests

We study the design of large-scale group testing schemes under a heterogeneous population (i.e., subjects with potentially different risk) and with the availability of multiple tests. The objective is to classify the population as positive or negative for a given binary characteristic (e.g., the presence of an infectious disease) as efficiently and accurately as possible. Our approach examines components often neglected in the literature, such as the dependence of testing cost on the group size and the possibility of no testing, which are especially relevant within a heterogeneous setting. By developing key structural properties of the resulting optimization problem, we are able to reduce it to a network flow problem under a specific, yet not too restrictive, objective function. We then provide results that facilitate the construction of the resulting graph and finally provide a polynomial time algorithm. Our case study, on the screening of HIV in the United States, demonstrates the substantial benefits of the proposed approach over conventional screening methods. Summary of Contribution: This paper studies the problem of testing heterogeneous populations in groups in order to reduce costs and hence allow for the use of more efficient tests for high-risk groups. The resulting problem is a difficult combinatorial optimization problem that is NP-complete under a general objective. Using structural properties specific to our objective function, we show that the problem can be cast as a network flow problem and provide a polynomial time algorithm.

Knights Exchange Puzzle—Teaching the Efficiency of Modeling

Puzzles and games enhance the quality of teaching by creating an enjoyable, interactive, and playful atmosphere. The knight exchange is a famous, very old, and amusing game on the chessboard. This puzzle was used by the author to teach modeling in a mathematical programming course designed for graduate students. The aim was to teach the students the efficiency of the models. Accordingly, first, a binary programming formulation was developed. This formulation was, however, found to be inefficient, and tremendous time (i.e., more than four hours) and a large amount of processing memory were needed to solve the puzzle. The puzzle was subsequently formulated as a minimum cost network flow problem. The latter formulation outperformed the general binary formulation by solving the puzzle in less than a minute. The network formulation could also save the required processing memory. The results could help students to learn the value of modeling combinatorial optimization problems as network flows.