Imrich Vrto

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.

Which Is the Worst-Case Nash Equilibrium?

A Nash equilibrium of a routing network represents a stable state of the network where no user finds it beneficial to unilaterally deviate from its routing strategy. In this work, we investigate the structure of such equilibria within the context of a certain game that models selfish routing for a set of n users each shipping its traffic over a network consisting of m parallel links. In particular, we are interested in identifying the worst-case Nash equilibrium – the one that maximizes social cost. Worst-case Nash equilibria were first introduced and studied in the pioneering work of Koutsoupias and Papadimitriou [9].

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).

One Sided Crossing Minimization Is NP-Hard for Sparse Graphs

The one sided crossing minimization problem consists of placing the vertices of one part of a bipartite graph on prescribed positions on a straight line and finding the positions of the vertices of the second part on a parallel line and drawing the edges as straight lines such that the number of pairwise edge crossings is minimized. This problem represents the basic building block used for drawing hierarchical graphs aesthetically or producing row-based VLSI layouts. Eades and Wormald [3] showed that the problem is NP-hard for dense graphs. Typical graphs of practical interest are usually very sparse. We prove that the problem remains NP-hard even for forests of 4-stars.

Optimal Cutwidths and Bisection Widths of 2- and 3-Dimensional Meshes

We prove exact cutwidths and bisection widths of ordinary, cylindrical and toroidal meshes. This answers an open problem in [9]. We also give upper bounds for cutwidths and bisection widths of many dimensional meshes. Furthemore, we show the exact cyclic cutwidth of 2-dimensional toroidal meshes and prove optimal upper and lower bounds for other meshes.

On VLSI layouts of the star graph and related networks

Abstract We prove that the minimal VLSI layout of the arrangement graph A ( n , k ) occupies Θ ( n !/( n - k - 1)!) 2 area. As a special case we obtain an optimal layout for the star graph S n with the area Θ ( n !) 2 . This answers an open problem posed by Akers et al. (1987). The method is also applied to the pancake graph. The results provide optimal upper and lower bounds for crossing numbers of the above graphs.

Two New Heuristics for Two-Sided Bipartite Graph Drawing

Two new heuristic strategies are studied based on heuristics for the linear arrangement problem and a stochastic hill-climbing method for the two-sided bipartite crossing number problem. These are compared to the standard heuristic for two-sided bipartite drawing based on iteration of the barycentre method. Our experiments show that they can efficiently find good solutions.

Two trees which are self-intersecting when drawn simultaneously

A current topic in graph drawing is the question how to draw two edge sets on the same vertex set, the so-called simultaneous drawing of graphs. The goal is to simultaneously find a nice drawing for both of the sets. It has been found out that only restricted classes of planar graphs can be drawn simultaneously using straight lines and without crossings within the same edge set. In this paper, we negatively answer one of the most often posted open questions namely whether any two trees with the same vertex set can be drawn simultaneously crossing-free in a straight-line way.

Towards practical deteministic write-all algorithms

The problem of performing <i>t</i> tasks on <i>n</i> asynchronous or undependable processors is a basic problem in parallel and distributed computing. We consider an abstraction of this problem called the <i>Write-All</i> problem— <i>using n processors write 1s into all locations of an array of size t</i>. The most efficient known deterministic asynchronous algorithms for this problem are due to Anderson and Woll. The first class of algorithms has <i>work</i> complexity of <i>&Ogr;</i>(<i>t</i> . <i>n</i> ^ε), for <i>n</i> ≰ <i>t</i>y and any ε > 0, and they are the best known for the full range of processors (<i>n</i> = <i>t</i>). To schedule the work of the processors, the algorithms use sets of <i>q</i> permutations on [<i>q</i>] (<i>q</i> ≰ <i>n</i>) that have certain combinatorial properties. Instantiating such an algorithm for a specific ε either requires substantial pre-processing (exponential in 1/ε²) to find the requisite permutations, or imposes a prohibitive constant (exponential in 1/ε³) hidden by the asymptotic analysis. The second class deals with the specific case of <i>t</i> = <i>n^u, u</i> ≰ 2, and these algorithms have work complexity of <i>&Ogr;</i>(<i>t</i> log <i>t</i>). They also use sets of permutations with the same combinatorial properties. However instantiating these algorithms requires exponential in <i>n</i> preprocessing to find the permutations. To alleviate this costly instantiation Kanellakis and Shvartsman proposed a simple way of computing the permutation schedules. They conjectured that their construction has the desired properties but they provided no analysis. In this paper we show, for the first time, an analysis of the properties of the set of permutations proposed by Kanellakis and Shvartsman. Our result is hybrid as it includes analytical and empirical parts. The analytical result covers a subset of the possible adversarial patterns of asynchrony. The empirical results provide strong evidence that our analysis covers the worst case scenario, and we formally state it as a conjecture. We use these results to analyze an algorithm for <i>t</i> = <i>n^u </i> (<i>u</i> ⪈ 2), tasks, that takes advantage of processor slackness and that has work <i>&Ogr;</i>(<i>t</i> log² <i>t</i>), conditioned on our conjecture. This algorithm requires only <i>&Ogr;</i>(<i>n</i> log <i>n</i>) time to instantiate it. Next we study the case for the full range of processors <i>n</i> ≰ <i>t</i>. We define a family of deterministic asynchronous <i>Write-All</i> algorithms with work <i>&Ogr;</i>(<i>t</i> . <i>n</i> ^ε) contingent upon our conjecture. We show that our method yields a faster construction of <i>&Ogr;</i>(<i>t</i> . <i>n</i> ^ε) <i>Write-All</i> algorithms than the method developed by Anderson and Woll. Finally we show that our approach yields more efficient <i>Write-all</i> algorithms as compared to the algorithms induced by the constructions of Naor and Roth for the same asymptotic work complexity.

Cutwidth of the de Bruijn graph

We prove optimal upper bound on the cutwidth of the general de Bruijn graph. Our upper bound is essentially based on a new relation between the cutwidth and the area of the VLSI layout of a graph. The relation is interesting itself as it generalizes the known relation between the area and the bisection width of graphs of bounded degrees and holds for arbitrary graphs.