In the classical k-median problem the goal is to select a subset of at most k facilities in order to minimize the total cost of opened facilities and established connections between clients and opened facilities. We consider the capacitated version of the problem, where a single facility may only serve a limited number of clients. We construct approximation algorithms slightly violating the capacities based on rounding a fractional solution to the standard LP.It is well known that the standard LP (even in the case of uniform capacities) has unbounded integrality gap if we only allow violating capacities by a factor smaller than 2, or if we only allow violating the number of facilities by a factor smaller than 2. It is also known that violating capacities by a factor of 2 + ε is sufficient to obtain constant factor approximation of the connection cost in the case of uniform capacities. In this paper we substantially extend this result in the following two directions. On one hand, we obtain a 2+ε capacity violating algorithm to the more general kfacility location problem with uniform capacities, where opening facilities incurs a location specific opening cost. On the other hand, we show that violating capacities by a slightly bigger factor of 3 + ε is sufficient to obtain constant factor approximation of the connection cost also in the case of the non-uniform hard capacitated k-median problem.Our algorithms first use the clustering of Charikar et al. to partition the facilities into sets of total fractional opening at least 1−1/ for some fixed . Then we exploit the technique of Levi, Shmoys, and Swamy developed for the capacitated facility location problem, which is to locally group the demand from clients to * Supported by NCN 2012/07/N/ST6/03068 grant. obtain a system of single node demand instances. Next, depending on the setting, we either work with stars of facilities (for non-uniform capacities), or we use a dedicated routing tree on the demand nodes (for nonuniform opening cost), to redistribute the demand that cannot be satisfied locally within the clusters.

Abstract. We investigate the problem of drawing graphs in 2D and 3D such that their edges (or only their vertices) can be covered by few lines or planes. We insist on straight-line edges and crossing-free drawings. This problem has many connections to other challenging graph-drawing problems such as small-area or small-volume drawings, layered or track drawings, and drawing graphs with low visual complexity. While some facts about our problem are implicit in previous work, this is the first treatment of the problem in its full generality. Our contribution is as follows.-We show lower and upper bounds for the numbers of lines and planes needed for covering drawings of graphs in certain graph classes. In some cases our bounds are asymptotically tight; in some cases we are able to determine exact values. -We relate our parameters to standard combinatorial characteristics of graphs (such as the chromatic number, treewidth, maximum degree, or arboricity) and to parameters that have been studied in graph drawing (such as the track number or the number of segments appearing in a drawing). -We pay special attention to planar graphs. For example, we show that there are planar graphs that can be drawn in 3-space on a lot fewer lines than in the plane.

We study the classical NP-hard problems of finding maximum-size subsets from given sets of k terminal pairs that can be routed via edge-disjoint paths (MaxEDP) or node-disjoint paths (MaxNDP) in a given graph. The approximability of MaxEDP/MaxNDP is currently not well understood; the best known lower bound is 2 Ω( √ log n) , assuming NP DTIME (n O(log n) (or nodes) may be used by O (log n/ log log n) paths. In this paper, we strengthen the fundamental results above. We provide new bounds formulated in terms of the feedback vertex set number r of a graph, which measures its vertex deletion distance to a forest. In particular, we obtain the following results:-For MaxEDP, we give an O( √ r log(kr))-approximation algorithm. Up to a logarithmic factor, our result strengthens the best known ratio O( √ n) due to Chekuri et al., as r ≤ n. -Further, we show how to route Ω(OPT * ) pairs with congestion bounded by O(log(kr)/ log log(kr)), strengthening the bound obtained by the classic approach of Raghavan and Thompson. -For MaxNDP, we give an algorithm that gives the optimal answer in time

It is well known that any graph admits a crossing-free straight-line drawing in R 3 and that any planar graph admits the same even in R 2 . For a graph G and d ∈ {2, 3}, let ρ 1 d (G) denote the minimum number of lines in R d that together can cover all edges of a drawing of G. For d = 2, G must be planar. We investigate the complexity of computing these parameters and obtain the following hardness and algorithmic results. *

Given a set of k-colored points in the plane, we consider the problem of finding k trees such that each tree connects all points of one color class, no two trees cross, and the total edge length of the trees is minimized. For k = 1, this is the well-known Euclidean Steiner tree problem. For general k, a kρ-approximation algorithm is known, where ρ ≤ 1.21 is the Steiner ratio.We present a PTAS for k = 2, a (5/3 + ε)-approximation algorithm for k = 3, and two approximation algorithms for general k, with ratios O( √ n log k) and k + ε.