Farhad Shahrokhi

The maximum concurrent flow problem

The maximum concurrent flow problem (MCFP) is a multicommodity flow problem in which every pair of entities can send and receive flow concurrently. The ratio of the flow supplied between a pair of entities to the predefined demand for that pair is called throughput and must be the same for all pairs of entities for a concurrent flow. The MCFP objective is to maximize the throughput, subject to the capacity constraints. We develop a fully polynomial-time approximation scheme for the MCFP for the case of arbitrary demands and uniform capacity. Computational results are presented. It is shown that the problem of associating costs (distances) to the edges so as to maximize the minimum-cost of routing the concurrent flow is the dual of the MCFP. A path-cut type duality theorem to expose the combinatorial structure of the MCFP is also derived. Our duality theorems are proved constructively for arbitrary demands and uniform capacity using the algorithm. Applications include packet-switched networks [1, 4, 8], and cluster analysis [16].

Sparsest cuts and bottlenecks in graphs

Abstract The problem of determining a sparsest cut in a graph is characterized and its computation shown to be NP-hard. A class of sparsest cuts, termed bottlenecks, is characterized by a dual relation to a particular polynomial time computable multicommodity flow problem. Efficient computational techniques for determining bottlenecks in a broad class of instances are presented.

On Bipartite Drawings and the Linear Arrangement Problem

The bipartite crossing number problem is studied and a connection between this problem and the linear arrangement problem is established. A lower bound and an upper bound for the optimal number of crossings are derived, where the main terms are the optimal arrangement values. Two polynomial time approximation algorithms for the bipartite crossing number are obtained. The performance guarantees are O(log n) and O(log2 n) times the optimal, respectively, for a large class of bipartite graphs on n vertices. No polynomial time approximation algorithm which could generate a provably good solution had been known. For a tree, a formula is derived that expresses the optimal number of crossings in terms of the optimal value of the linear arrangement and the degrees, resulting in an O(n1.6) time algorithm for computing the bipartite crossing number. The problem of computing a maximum weight biplanar subgraph of an acyclic graph is also studied and a linear time algorithm for solving it is derived. No polynomial time algorithm for this problem was known, and the unweighted version of the problem had been known to be NP-hard, even for planar bipartite graphs of degree at most 3.

The book crossing number of a graph

Let G be a graph on n vertices and m edges. The book crossing number of G is defined as the minimum number of edge crossings when the vertices of G are placed on the spine of a k-page book and edges are drawn on pages, such that each edge is contained by one page. Our main results are two polynomial time algorithms to generate near optimal drawing of G on books. The first algorithm give an O(log2n) times optimal solution, on small number of pages, under some restrictions. This algorithm also gives rise to the first polynomial time algorithm for approximating the rectilinear crossing number so that the coordinates of vertices in the plane are small integers, thus resolving a recent open question concerning the rectilinear crossing number. Moreover, using this algorithm we improve the best known upper bounds on the rectilinear crossing number. The second algorithm generates a drawing of G with O(m2/k2) crossings on k pages. This is within a constant multiplicative factor from our general lower bound of Ω(m3/n2k2), provided that m = Ψ(n2).

On Crossing Sets, Disjoint Sets, and Pagenumber

Let G=(V,E) be a t-partite graph with n vertices and m edges, where the partite sets are given. We present an O(n2m1.5) time algorithm to construct drawings of G in the plane so that the size of the largest set of pairwise crossing edges and, at the same time, the size of the largest set of disjoint (pairwise noncrossing) edges are Ot·m. As an application we embed G in a book of Ot·m pages, in O(n2m1.5) time, resolving an open question for the pagenumber problem. A similar result is obtained for the dual of the pagenumber or the queuenumber. Our algorithms are obtained by derandomizing a probabilistic proof.

Applications of the crossing number

We show that any graph of <italic>n</italic> vertices that can be drawn in the plane with no <italic>k</italic>+1 pairwise crossing edges has at most <italic>c<subscrpt>k</subscrpt>n</italic>log<supscrpt>2<italic>k</italic>−2</supscrpt><italic>n</italic> edges. This gives a partial answer to a dual version of a well-known problem of Avital-Hanani, Erdo&huml;s, Kupitz, Perles, and others. We also construct two point sets {<italic>p</italic><subscrpt>1</subscrpt>,…,<italic>p<subscrpt>n</subscrpt></italic>}, {<italic>q</italic><subscrpt>1</subscrpt>,…,<italic>q<subscrpt>n</subscrpt></italic>} in the plane such that any piecewise linear one-to-one mapping <italic>f</italic>:R<supscrpt>2</supscrpt>→R<supscrpt>2</supscrpt> with <italic>f(p<subscrpt>i</subscrpt>)=q<subscrpt>i</subscrpt></italic> (1≤<italic>i</italic>≤<italic>n</italic>) is composed of at least &OHgr;(<italic>n</italic><supscrpt>2</supscrpt>) linear pieces. It follows from a recent result of Souvaine and Wenger that this bound is asymptotically tight. Both proofs are based on a relation between the crossing number and the bisection width of a graph.

Intersection of Curves and Crossing Number of C m × C n on Surfaces

Abstract. Let

Crossing numbers of graphs, lower bound techniques and algorithms: A survey

({\cal K}_1, {\cal K}_2)

On Canonical Concurrent Flows, Crossing Number and Graph Expansion

be two families of closed curves on a surface

Drawings of graphs on surfaces with few crossings

{\cal S}