Andreas Baltz

Fast approximation of minimum multicast congestion – Implementation VERSUS Theory

The problem of minimizing the maximum edge congestion in a multicast communication network generalizes the well-known NP -hard standard routing problem. We present the presently best theoretical ap- proximation results as well as efficient implementations.

Approximation algorithms for the Euclidean bipartite TSP

We study approximation results for the Euclidean bipartite traveling salesman problem (TSP). We present the first worst-case examples, proving that the approximation guarantees of two known polynomial-time algorithms are tight. Moreover, we propose a new algorithm which displays a superior average case behavior indicated by computational experiments.

Exact and Heuristic Algorithms for the Travelling Salesman Problem with Multiple Time Windows and Hotel Selection

We introduce and study the Travelling Salesman Problem with Multiple Time Windows and Hotel Selection (TSP-MTWHS), which generalises the well-known Travelling Salesman Problem with Time Windows and the recently introduced Travelling Salesman Problem with Hotel Selection. The TSP-MTWHS consists in determining a route for a salesman (eg, an employee of a services company) who visits various customers at different locations and different time windows. The salesman may require a several-day tour during which he may need to stay in hotels. The goal is to minimise the tour costs consisting of wage, hotel costs, travelling expenses and penalty fees for possibly omitted customers. We present a mixed integer linear programming (MILP) model for this practical problem and a heuristic combining cheapest insert, 2-OPT and randomised restarting. We show on random instances and on real world instances from industry that the MILP model can be solved to optimality in reasonable time with a standard MILP solver for several small instances. We also show that the heuristic gives the same solutions for most of the small instances, and is also fast, efficient and practical for large instances.

Probabilistic analysis for a multiple depot vehicle routing problem

We give the first probabilistic analysis of the Multiple Depot Vehicle Routing Problem(MDVRP) where we are given k depots and n customers in [0,1]2. The optimization problem is to find a collection of disjoint TSP tours with minimum total length such that all customers are served and each tour contains exactly one depot(not all depots have to be used). In the random setting the depots as well as the customers are given by independently and uniformly distributed random variables in [0,1]2. We show that the asymptotic tour length is

Fast approximation of minimum multicast congestion: implementation versus theory

\alpha_{k} \sqrt{n}

The price of anarchy in selfish multicast routing

for some constant αk depending on the number of depots. If k=o(n), αk is the constant α(TSP) of the TSP problem. Beardwood, Halton, and Hammersley(1959) showed 0.62≤ α(TSP)≤ 0.93. For k=λn, λ>0, one expects that with increasing λ the MDVRP tour length decreases. We prove that this is true exhibiting lower and upper bounds on αk, which decay as fast as

Constructions of Sparse Asymmetric Connectors

(1+\lambda)^{-\frac{1}{2}}

Pobabilistic Analysis of Bipartite Traveling Salesman Problems (Extended Abstract)

. A heuristics which first clusters customers around the nearest depot and then does the TSP routing is shown to find an optimal tour almost surely.

Multicast Routing and Design of Sparse Connectors

The problem of minimizing the maximum edge congestion in a multicast communication network generalizes the well-known NP-hard standard routing problem. We present the presently best theoretical approximation results as well as efficient implementations.

On the minimum load coloring problem

We study the price of anarchy for selfish multicast routing games in directed multigraphs with latency functions on the edges, extending the known theory for the unicast situation, and exhibiting new phenomena not present in the unicast model. In the multicast model we have N commodities (or player classes), where for each i = 1,..., N, a flow from a source s i to a finite number of terminals t 1 i ,..., t ki i has to be routed such that every terminal t j i receives flow n i ∈ R≥0. One of the significant results of this paper are upper and lower bounds on the price of anarchy for edge latencies being polynomials of degree at most p with non-negative coefficients. We show an upper bound of (p+1) ν p+1 ν* in some variants of multicast routing. We also prove a lower bound of ν p , so we have upper and lower bounds that are tight up to a factor of (p+ 1)ν. Here, v and ν* are network and strategy dependent parameters reflecting the maximum/minimum consumption of the network. Both are 1 in the unicast case. Our lower bound of v p , where in the general situation we have v > 1, shows an exponential increase compared to the Roughgarden bound of O(p/lnp) for the unicast model. This exhibits the contrast to the unicast case, where we have Roughgardens (2002) result that the price of anarchy is independent of the network topology. To our knowledge this paper is the first thorough study of the price of anarchy in the multicast scenario. The approach may lead to further research extending game-theoretic network analysis to models used in applications.