Biological invasions from ballast water are a severe environmental threat and exceedingly costly to society. We identify global hot spots of invasion based on worldwide patterns of ship traffic. We then estimate the rate of port-to-port invasion using gravity models for spatial interactions, and we identify bottlenecks to the regional exchange of species using the Ford-Fulkerson algorithm for n...

Column generation refers to linear programming (LP) algorithms designed to solve problems in which there are a huge number of variables compared to the number of constraints and the simplex algorithm step of determining whether the current basic solution is optimal or finding a variable to enter the basis is done by solving an optimization problem rather than by enumeration. To the best of my k...

A finite metric (or more properly semimetric) on n points is a non-negative vector d = (dij) 1 ≤ i < j ≤ n that satisfies the triangle inequality: dij ≤ dik + d jk . The L (or Manhattan )distance | |x − y| |1 between two vectors x = (xi) and y = (yi) in R is given by | |x − y| |1 = 1≤i≤m Σ |xi − yi |. A metric d is L − embeddable if there exist vectors z1, z2, . . . , zn in R for some m, such t...

The optimal path cover problem is to find a minimum number of vertex disjoint paths which together cover all the vertices of the graph. Finding an optimal path cover for an arbitrary graph is known to be NP-complete [3]. However, polynornial-time algorithms exist for trees [4], cacti [4] and for interval graphs [9]. The solution presented in [2] For circular-arc graphs is known to be wrong. In ...

A digraph whose degree sequence has a unique vertex labeled realization is called threshold. In this paper we present several characterizations of threshold digraphs and their degree sequences, and show these characterizations to be equivalent. One of the characterizations is new, and allows for a shorter proof of the equivalence of the two known characterizations as well as proving the final c...

In the last lecture, we analyzed the problem of finding the maximum flow in a graph, and how it can be efficiently solved using the Ford-Fulkerson algorithm. We also came across the Min Cut-Max Flow Theorem which relates the size of the maximum flow to the size of the minimal cut in the graph. In this lecture, we will be seeing how various different problems can be solved by reducing the proble...

We review two papers that are of historical interest for combinatorial optimization: an article of A.N. Tolstŏı from 1930, in which the transportation problem is studied, and a negative cycle criterion is developed and applied to solve a (for that time) large-scale (10×68) transportation problem to optimality; and an, until recently secret, RAND report of T. E. Harris and F. S. Ross from 1955, ...