Katerina Potika

Resource Allocation Problems in Multifiber WDM Tree Networks

All-optical networks with multiple fibers lead to several interesting optimization problems. In this paper, we consider the problem of minimizing the total number of fibers necessary to establish a given set of requests with a bounded number w of wavelengths, and the problem of maximizing the number of accepted requests for given fibers and bounded number w of wavelengths. We study both problems in undirected tree networks T=(V,E) and present approximation algorithms with ratio 1 + 4|E|log|V|/OPT and 4 for the former and ratio 2.542 for the latter. Our results can be adapted to directed trees as well.

Line crossing minimization on metro maps

We consider the problem of drawing a set of simple paths along the edges of an embedded underlying graph G = (V,E), so that the total number of crossings among pairs of paths is minimized. This problem arises when drawing metro maps, where the embedding of G depicts the structure of the underlying network, the nodes of G correspond to train stations, an edge connecting two nodes implies that there exists a railway line which connects them, whereas the paths illustrate the lines connecting terminal stations. We call this the metro-line crossing minimization problem (MLCM). In contrast to the problem of drawing the underlying graph nicely, MLCM has received fewer attention. It was recently introduced by Benkert et. al in [4]. In this paper, as a first step towards solving MLCM in arbitrary graphs, we study path and tree networks.We examine several variations of the problem for which we develop algorithms for obtaining optimal solutions.

Routing and wavelength assignment in multifiber WDM networks with non-uniform fiber cost

Motivated by the increasing importance of multifiber WDM networks we study a routing and wavelength assignment problem in such networks. In this problem the number of wavelengths per fiber is given and the goal is to minimize the cost of fiber links that need to be reserved in order to satisfy a set of communication requests; we introduce a generalized setting where network pricing is non-uniform, that is the cost of hiring a fiber may differ from link to link. We consider two variations: undirected, which corresponds to full-duplex communication, and directed, which corresponds to one-way communication. Moreover, for rings we also study the problem in the case of pre-determined routing. We present exact or constant-ratio approximation algorithms for all the above variations in chain, ring and spider networks.

On multi-stack boundary labeling problems

The boundary labeling problem was recently introduced in [5] as a response to the problem of labeling dense point sets with large labels. In boundary labeling, we are given a rectangle R which encloses a set of n sites. Each site is associated with an axis-parallel rectangular label. The main task is to place the labels in distinct positions on the boundary of R, so that they do not overlap, and to connect each site with its corresponding label by non-intersecting polygonal lines, so called leaders. Such a label placement is referred to as legal label placement. In this paper, we study boundary labeling problems along a new line of research. We seek to obtain labelings with labels arranged on more than one stacks placed at the same side of R. We refer to problems of this type as multi-stack boundary labeling problems. We present algorithms for maximizing the uniform label size for boundary labeling with two and three stacks of labels. The key component of our algorithms is a technique that combines the merging of lists and the bounding of the search space of the solution. We also present NP-hardness results for multi-stack boundary labeling problems with labels of variable height.

Area-Feature Boundary Labeling1

Boundary labeling is a relatively new labeling method. It can be useful in automating the production of technical drawings and medical drawings, where it is common to explain certain parts of the drawing with text labels, arranged on its boundary so that other parts of the drawing are not obscured. In boundary labeling, we are given a rectangle R which encloses a set of n sites. Each site s is associated with an axis-parallel rectangular label ls. The labels must be placed in distinct positions on the boundary of R and they must be connected to their corresponding sites with polygonal lines, called leaders, so that the labels are pairwise disjoint and the leaders do not intersect each other. In this paper, we study a version of the boundary labeling problem where the sites can ‘float’ within a polygonal region. We present a polynomial time algorithm, which runs in O(n3) time and produces a labeling of minimum total leader length for labels of uniform size placed in fixed positions on the boundary of rectangle R.

Fiber Cost reduction and Wavelength minimization in multifiber WDM networks

Motivated by the increasing importance of multifiber WDM networks we study two routing and wavelength assignment problems in such networks: Fiber Cost Minimization: the number of wavelengths per fiber is given and we want to minimize the cost of fiber links that need to be reserved in order to satisfy a set of communication requests; we introduce a generalized setting where network pricing is nonuniform, that is the cost of hiring a fiber may differ from link to link. Wavelength Minimization: the number of available parallel fibers on each link is given and we want to minimize the wavelengths per fiber that are needed in order to satisfy a set of communication requests.

On a noncooperative model for wavelength assignment in multifiber optical networks

We propose and investigate Selfish Path MultiColoring games as a natural model for noncooperative wavelength assignment in multifiber optical networks. In this setting, we view the wavelength assignment process as a strategic game in which each communication request selfishly chooses a wavelength in an effort to minimize the maximum congestion that it encounters on the chosen wavelength. We measure the cost of a certain wavelength assignment as the maximum, among all physical links, number of parallel fibers employed by this assignment. We start by settling questions related to the existence and computation of and convergence to pure Nash equilibria in these games. Our main contribution is a thorough analysis of the price of anarchy of such games, that is, the worst-case ratio between the cost of a Nash equilibrium and the optimal cost. We first provide upper bounds on the price of anarchy for games defined on general network topologies. Along the way, we obtain an upper bound of 2 for games defined on star networks. We next show that our bounds are tight even in the case of tree networks of maximum degree 3, leading to nonconstant price of anarchy for such topologies. In contrast, for network topologies of maximum degree 2, the quality of the solutions obtained by selfish wavelength assignment is much more satisfactory: We prove that the price of anarchy is bounded by 4 for a large class of practically interesting games defined on ring networks.

Boundary labelling of optimal total leader length

In this paper, we consider the leader length minimization problem for boundary labelling, i.e. the problem of finding a legal leader-label placement, such that the total leader length is minimized. We present an O(n2log3n) algorithm assuming type-opoleaders (rectilinear lines with either zero or two bends) and labels of uniform size which can be attached to all four sides of rectangle R. Our algorithm supports fixed and sliding ports, i.e., the point where each leader is connected to the label (referred to as port) may be fixed or may slide along a label edge.

Selfish routing and path coloring in all-optical networks

We study routing and path coloring problems in all-optical networks as non-cooperative games. We especially focus on oblivious payment functions, that is, functions that charge a player according to her own strategy only. We first strengthen a known relation between such games and online routing and path coloring. In particular, we show that the price of anarchy of such games is lower-bounded by, and in several cases precisely equal to, the competitive ratio of appropriate modifications of the First Fit algorithm. Based on this framework we provide results for two classes of games in ring networks: in Selfish Routing and Path Coloring a player must determine both a routing and a coloring for her request, while in Selfish Path Coloring the routing is predetermined and only a coloring of requests needs to be specified. We prove specific upper and lower bounds on the price of anarchy of these games under various payment functions.

Path multicoloring with fewer colors in spiders and caterpillars

We study a recently introduced path coloring problem with applications to wavelength assignment in all-optical networks with multiple fibers. In contrast to classical path coloring, it is, in this setting, possible to assign a color more than once to paths that pass through the same edge; the number of allowed repetitions per edge is given and the goal is to minimize the number of colors used.We present algorithms and hardness results for tree topologies of special interest. Our algorithms achieve approximation ratio of 2 in spiders and 3 in caterpillars, whereas the best algorithm for trees so far, achieves an approximation ratio of 4. We also study the directed version of the problem and show that it admits a 3-approximation algorithm in caterpillars, while it can be solved exactly in spiders.