# Martin Tieves

6.63

· PhD MathematicsAbout

13

Research items

349

Reads

47

Citations

Research Experience

Nov 2011

- Chair of Mathematics II (for engineers)
- Aachen, Germany

Position

- Research Assistant

Description

- Research in discrete optimization and robust optimization with applications in telecommunications, e.g., network design and dimensioning, and traffic routing. I also work in interdisciplinary projects, e.g., VINO.

Education

Oct 2009 - Nov 2011

Oct 2006 - Oct 2009

Current institution

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Following

Projects

Projects (1)

Project

An equitable graph coloring is a vertex coloring, such that the size of the color classes differ by at most one. In this project, we consider a flow-based scheme for generating pruning rules for enumerative algorithms such as DSATUR. Such scheme models the extendability of a partial (equitable) coloring into an equitable coloring via network flows.
See also
https://www.math2.rwth-aachen.de/en/forschung/projekte/projekteqcolor/

Research

Research Item (13)

- Oct 2017

Flexgrid optical networking technology allows for a more flexible consumption of bandwidth. The spectrum allocation problem consists of the conflict-free assignment of consecutive spectrum space of different sizes to lightpaths. In this article, we study the computational complexity of spectrum allocation with and without demand uncertainty. First, it is shown that the problem becomes already NP-hard for cases where wavelength assignment is still polynomial time solvable. Next, five different ways to define the robust counterpart are compared. It is shown (amongst others) that on a single network edge, the two least efficient models are less computationally demanding than the other variants. A computational study using comparable integer linear programming formulations reveals that the additional slots required by these models directly depend on the restrictions of the employed technology. © 2017 Wiley Periodicals, Inc. NETWORKS, 2017

- Mar 2017

An equitable graph coloring is a proper vertex coloring of a graph G where the sizes of the color classes differ by at most one. The equitable chromatic number, denoted by \(\chi _{eq}(G),\) is the smallest number k such that G admits such equitable k-coloring. We focus on enumerative algorithms for the computation of \(\chi _{eq}(G)\) and propose a general scheme to derive pruning rules for them: We show how the extendability of a partial coloring into an equitable coloring can be modeled via network flows. Thus, we obtain pruning rules which can be checked via flow algorithms. Computational experiments show that the search tree of enumerative algorithms can be significantly reduced in size by these rules and, in most instances, such naive approach even yields a faster algorithm. Moreover, the stability, i.e., the number of solved instances within a given time limit, is greatly improved. Since the execution of flow algorithms at each node of a search tree is time consuming, we derive arithmetic pruning rules (generalized Hall-conditions) from the network model. Adding these rules to an enumerative algorithm yields an even larger runtime improvement.

- Sep 2016
- 2016 8th International Workshop on Resilient Networks Design and Modeling (RNDM)

- Jul 2016

An equitable graph coloring is a proper vertex coloring, such that the size of the color classes differ by at most one. We present a flow-based scheme for generating pruning rules for enumerative algorithms such as DSATUR. The scheme models the extendability of a partial (equitable) coloring into an equitable coloring via network flows. Computational experiments show that significant reductions of the search tree can be achieved by the derived pruning rules.

- Jun 2016

Given a graph representing a substrate (or physical) network with node and edge capacities and a set of virtual networks with node capacity demands and node-to-node traffic demands, the Virtual Network Embedding problem (VNE) calls for an embedding of (a subset of) the virtual networks onto the substrate network which maximizes the total profit while respecting the physical node and edge capacities. In this work, we investigate the computational complexity of VNE. In particular, we present a polynomial-time reduction from the maximum stable set problem which implies strong -hardness for VNE even for very special subclasses of graphs and yields a strong inapproximability result for general graphs. We also consider the special cases obtained when fixing one of the dimensions of the problem to one. We show that VNE is still strongly -hard when a single virtual network request is present or when each virtual network request consists of a single virtual node and that it is weakly -hard for the case with a single physical node.

- Mar 2016

We address the virtual network embedding problem (VNE) which, given a physical (substrate) network and a collection of virtual networks (VNs), calls for an embedding of the most profitable subset of VNs onto the physical substrate, subject to capacity constraints. In practical applications, node and link demands of the different VNs are, typically, uncertain and difficult to know a priori. To face this issue, we first model VNE as a chance-constrained Mixed-Integer Linear Program (MILP) where the uncertain demands are assumed to be random variables. We then propose a \(\varGamma \)-robust optimization approach to approximate the original chance-constrained formulation, capable of yielding solutions with a large profit that are feasible for almost all the possible realizations of the uncertain demands. To solve larger scale instances, for which the exact approach is computationally too demanding, we propose two MILP-based heuristics: a parametric one, which relies on a parameter setting chosen a priori, and an adaptive one, which does not. We conclude by reporting on extensive computational experiments where the different methods and approaches are compared.

- Jun 2015
- International Symposium on Experimental Algorithms

We propose a new cutting plane algorithm for Integer Linear Programming, which we refer to as the bound-optimal cutting plane method. The algorithm amounts to simultaneously generating k cuts which, when added to the linear programming relaxation, yield the (provably) largest bound improvement. We show that, in the general case, the corresponding cut generating problem can be cast as a Quadratically Constrained Quadratic Program. We also show that, for a large family of cuts, the latter can be reformulated as a Mixed-Integer Linear Program. We present computational experiments on the generation of bound-optimal stable set and cover inequalities for the max clique and knapsack problems. They show that, with respect to standard algorithms, the bound-optimal cutting plane method allows for a substantial reduction in the number of cuts and iterations needed to achieve either a given bound or an optimal solution.

- Mar 2015
- Design of Reliable Communication Networks (DRCN), 2015 11th International Conference on the

Given a physical substrate network and a collection of requests of virtual networks, the Virtual Network Embedding problem (VNE) calls for the embedding onto the physical substrate of a selection of virtual networks in such a way that the profit is maximized. The embedding corresponds to a virtual-to-physical mapping of nodes and links, subject to capacity constraints. Since, in practical scenarios, node and link demands are typically much smaller than the peak values specified in the virtual network requests, in this work we propose and investigate a robust optimization approach. This allows us to find solutions with a much larger profit which, at the same time, are guaranteed to be feasible with a high probability. To this end, we propose a robust Mixed-Integer Linear Programming (MILP) formulation for VNE, based on the well-known model of Γ-robustness. To solve larger scale instances, for which the exact approach is computationally too demanding, we also propose a MILP-based two-phase heuristic which relies on Γ-robustness.

- Jan 2015

Network virtualization techniques allow for the coexistence of many virtual
networks (VNs) jointly sharing the resources of an underlying substrate
network. The Virtual Network Embedding problem (VNE) arises when looking for
the most profitable set of VNs to embed onto the substrate. In this paper, we
address the offline version of the problem. We propose a Mixed-Integer Linear
Programming formulation to solve it to optimality which accounts for acceptance
and rejection of virtual network requests, allowing for both splittable and
unsplittable (single path) routing schemes. Our formulation also considers a
Rent-at-Bulk (RaB) model for the rental of substrate capacities where economies
of scale apply. To better emphasize the importance of RaB, we also compare our
method to a baseline one which only takes RaB into account a posteriori, once a
solution to VNE, oblivious to RaB, has been found. Computational experiments
show the viability of our approach, stressing the relevance of addressing RaB
directly with an exact formulation

- Jan 2015
- INFORMS Computing Society Conference

Recent advances in communication technology allow to compress data streams in communication
networks by deploying physical devices (caches) at routers, yielding a more
efficient usage of link capacities. This gives rise to the network design problem with
compression (NDPC), a generalization of the classical Network Design problem.
In this paper, we compare both problems, focusing on the computational complexity
and analyzing the differences induced by the compression aspect.
We show that the subproblem of adding compression, i.e., the compressor placement
problem (CPP), is already weakly N P-hard, even on instances where Network
Design alone is easy. We conclude with a pseudopolynomial algorithm for tree
instances and a restricted polynomial case.

- Aug 2013
- Green Computing and Communications (GreenCom), 2013 IEEE and Internet of Things (iThings/CPSCom), IEEE International Conference on and IEEE Cyber, Physical and Social Computing

Many studies have shown that energy-aware routing (EAR) can significantly reduce energy consumption of a backbone network. Redundancy Elimination (RE) techniques provide a complementary approach to reduce the amount of traffic in the network. In particular, the GreenRE model combines both techniques, offering potentially significant energy savings. We propose a concept for respecting uncertain rates of redundant traffic within the GreenRE model, closing the gap between theoretical modeling and drawn-from life data. To model redundancy rate uncertainty, the robust optimization approach of Bertsimas and Sim (2004) is adapted and the problem is formally defined as mixed integer linear program. An exemplary evaluation of this concept with real-life traffic traces and estimated fluctuations of data redundancy shows that this closer-to-reality model potentially offers significant energy savings in comparison to GreenRE and EAR.

- Jun 2013

In this paper, we enhance the MIP formulation for the Network Power Consumption problem, proposed by Giroire et al. We derive cutting planes, extending the wellknown cutset inequalities, and report on preliminary computations.

- Aug 2012

This article deals with the frequency assignment problem (FAP) in slow frequency hopping (GSM) networks, a generalization of the classical FAP. Due to symmetry in the solutions, a natural integer linear programming formulation does not yield a good solution procedure. Instead, we decompose the co-channel and adjacent channel interference minimization and develop a two-stage algorithm. The co-channel optimization problem is solved with a column generation model, whereas the second stage is solved by a cutting plane approach. Computational experiments reveal, that although no optimal solutions can be guaranteed, the approach provides promising results, both regarding practical applicability and further research potential.