Walter Unger

Scheduling time-constrained communication in linear networks

We study the problem of centrally scheduling multiple messages in a linear network, when each message has both a release time and a deadline. We show that the problem of transmitting optimally many messages is NP-Hard, both when messages may be buffered in transit and when they may not be. For either case, we present efficient algorithms that produce approximately optimal schedules. In particular, our bufferless scheduling algorithm achieves throughput that is within a factor of 2 of optimal. We show that buffering can improve throughput in general by a logarithmic factor (but no more), but that in several significant special cases, such as when all messages can be released immediately, buffering can help by only a small constant factor. Finally, we show how to convert any of our centralized, offline bufferless schedules to a fully distributed online buffered schedule of equal throughput. Most of our results extend readily to ring-structured networks.

Elastic image matching is NP-complete

One fundamental problem in image recognition is to establish the resemblance of two images. This can be done by searching the best pixel to pixel mapping taking into account monotonicity and continuity constraints. We show that this problem is NP-complete by reduction from 3-SAT, thus giving evidence that the known exponential time algorithms are justified, but approximation algorithms or simplifications are necessary.

THE CUBE-CONNECTED CYCLES NETWORK IS A SUBGRAPH OF THE BUTTERFLY NETWORK

We prove the following results: (a) The Cube-Connected Cycles network of dimension n is a subgraph of the Butterfly network of dimension n. (b) The Shuffle-Exchange network of dimension n is a subgraph of the DeBruijn network of dimension n.

Embedding ladders and caterpillars into the hypercube

We present an embedding of generalized ladders as subgraphs into the hypercube. Through an embedding of caterpillars into ladders, we obtain an embedding of caterpillars into the hypercube. In this way we get almost all known results concerning the embedding of caterpillars into the hypercube. In addition we construct an embedding for some new types of caterpillars. Our results support the conjecture of Havel (1984).

Towards the notion of stability of approximation for hard optimization tasks and the traveling salesman problem

The investigation of the possibility to efficiently compute approximations of hard optimization problems is one of the central and most fruitful areas of current algorithm and complexity theory. The aim of this paper is twofold. First, we introduce the notion of stability of approximation algorithms. This notion is shown to be of practical as well as of theoretical importance, especially for the real understanding of the applicability of approximation algorithms and for the determination of the border between easy instances and hard instances of optimization problems that do not admit polynomial-time approximation. Secondly, we apply our concept to the study of the traveling salesman problem (TSP). We show how to modify the Christofides algorithm for Δ-TSP to obtain efficient approximation algorithms with constant approximation ratio for every instance of TSP that violates the triangle inequality by a multiplicative constant factor. This improves the result of Andreae and Bandelt (SIAM J. Discrete Math. 8 (1995) 1).

Optimal Embedding of Complete Binary Trees into Lines and Grid

We consider several graph embedding problems which have a lot of important applications in parallel and distributed computing and which have been unsolved so far. Our major result is that the complete binary tree can be embedded into the square grid of the same size with almost optimal dilation (up to a very small factor). To achieve this, we first state an embedding of the complete binary tree into the line with optimal dilation.

On the k-Colouring of Circle-Graphs

It is shown that the k-colouring problem for the class of circle graphs is NP-complete for k at least four. Until now this problem was still open. For circle graphs with maximum clique size k a 2k-colouring is always possible and can be found in O(n2). This provides an approximation algorithm with a factor two. Further it is proven that the k-colouring problem for circle graphs is solvable in polynomial time if the degree is bounded. The complexity of the 3-colouring problem for circle graphs remains open.

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.

An Improved Lower Bound on the Approximability of Metric TSP and Approximation Algorithms for the TSP with Sharpened Triangle Inequality

The traveling salesman problem (TSP) is one of the hardest optimization problems in NPO because it does not admit any polynomial time approximation algorithm (unless P = NP). On the other hand we have a polynomial time approximation scheme (PTAS) for the Euclidean TSP and the 3/2 -approximation algorithm of Christofides for TSP instances satisfying the triangle inequality. The main contributions of this paper are the following: (i) We essentially modify the method of Engebretsen [En99] in order to get a lower bound of 3813/3812-Ɛ on the polynomial-time approximability of the metric TSP for any Ɛ > 0. This is an improvement over the lower bound of 5381/5380 -Ɛ in [En99]. Using this approach we moreover prove a lower bound δβ on the approximability of Δβ-TSP for 1/2 < β < 1, where Δβ-TSP is a subproblem of the TSP whose input instances satisfy the β-sharpened triangle inequality cost({u, v}) ≤ β ċ (cost({u, v}) + cost({x, v})) for all vertices u, v, x. (ii) We present three different methods for the design of polynomial-time approximation algorithms for Δβ-TSP with 1/2 < β < 1, where the approximation ratio lies between 1 and 3/2, depending on β.

Optimal Algorithms for Broadcast and Gossip in the Edge-Disjoint Path Modes

The communication power of the one-way and two-way edge-disjoint path modes for broadcast and gossip is investigated. The complexity of communication algorithms is measured by the number of communication steps (rounds). The main results achieved are the following: 1. For each connected graphGnofnnodes, the complexity of broadcast inGn,Bmin(Gn), satisfies ?log2n??Bmin(Gn)??log2n?+1. The complete binary trees meet the upper bound, and all graphs containing a Hamiltonian path meet the lower bound. 2. For each connected graphGnofnnodes, the one-way (two-way) gossip complexityR(Gn) (R2(Gn)) satisfies?log2n??R2(Gn)?2·?log2n?+1,1.44...log2n?R(Gn)?2·?log2n?+2.All these lower and upper bounds are shown to be sharp up to 1. 3. All planar graphs ofnnodes and degreehhave a two-way gossip complexity of at least 1.5log2n?log2log2n?0.5log2h?8, and the two-dimensional grid ofnnodes has the gossip complexity 1.5log2n?log2log2n±O(1); i.e., two-dimensional grids are optimal gossip structures among planar graphs of bounded degree. Some upper bounds are also obtained for the one-way mode. 4. Thed-dimensional grid,d?3, ofnnodes has the two-way gossip complexity (1+1/d)·log2n?log2nlog2n±O(d).