The input for the hotlink assignment problem consists of a node weighted directed acyclic graph with a designated root node r. The goal is to minimize the weighted shortest path length rooted at r by adding a restricted number of outgoing arcs (hotlinks) to each node.

Approximation algorithms for NP-hard optimization problems have been widely studied for over three decades. Most of these measure the quality of the solution produced by taking the ratio of the cost of the solution produced by the algorithm to the cost of an optimal solution. In certain cases, this ratio may not be very meaningful-- for example, if the ratio of the worst solution to the best solution is at most some constant ff, then an approximation algorithm with factor ff may in fact yield the worst solution! To overcome this hurdle (among others), several authors have independently suggested the use of a different measure which we call z-approximation. An algorithm is an ff z-approximation if it runs in polynomial time, and produces a solution whose distance from the optimal one is at most ff times the distance between the optimal solution and the worst possible solution. The results known so far about z-approximations are either of the inapproximability type or rather straightforward observations. We design polynomial time algorithms for several fundamental discrete optimization problems, in particular we obtain a z-approximation factor of 1 2 for the directed traveling salesman problem (TSP) (with no triangle inequality assumption). For the undirected TSP this improves to

We introduce a new problem that was motivated by a (more complicated) problem arising in a robotized assembly environment. The bin coloring problem is to pack unit size colored items into bins, such that the maximum number of different colors per bin is minimized. Each bin has size B 2 N. The packing process is subject to the constraint that at any moment in time at most q 2 N bins are partially filled. Moreover, bins may only be closed if they are filled completely. An online algorithm must pack each item must be packed without knowledge of any future items. We investigate the existence of competitive online algorithms for the bin coloring problem. We prove an upper bound of 3q 1 and a lower bound of 2q for the competitive ratio of a natural greedy-type algorithm, and show that surprisingly a trivial algorithm which uses only one open bin has a strictly better competitive ratio of 2q 1. Moreover, we show that any deterministic algorithm has a competitive ratio q) and that randomization does not improve this lower bound even when the adversary is oblivious.

We present a mu lti-scale layou algorithm for the aesthetic drawing ofuV#860L32 graphs with straight-line edges. The algorithm is extremely fast, and is capable of drawing graphs that aresu--SVV tially larger than those we haveencou tered in prior work. For example, the paper contains a drawing of a graph with over 15,000 vertices. Also we achieve "optimal" drawings of 1000 vertex graphs in ab ou 1 second. The proposed algorithm embodies a new mu lti-scale scheme for drawing graphs, which was motivated by the earlier muS8820L4# algorithm of Hadany and Harel [HH99]. In principle, it cou# significantly improve the speed of essentially any force-directed method (regardless of that method's ability of drawing weighted graphs or the continu0 y of its cost-fuL436280 1 Introducti7 AgraphG(V,E) is an abstract structure that is used to model a relation over a set V ofen tities. Graph drawin is acon ven tion# tool for the visualization of relationq in]4#]Rq]#] an its usefulnRq depenV on its readability, that is, the capability of con veyin the meanfi# of the diagram quickly # A shorter version appeared in Proc. GraphD awing 2000, LNCS 1984, pp. 183--196, Springer Verlag, 2000. clearly. In recen t years, man y algorithms for drawin graphs automatically were proposed (the state of the art is surveyedcomprehen]8 ely in 99, KW01]). Wecon#E trateon the problem of drawin an unF8ERqfi4 graph with straight-lin edges. In this case the problem reduces to that of positionRq the vertices bydeterminfiF a mappin L : V R . A populargenrR# approach to this problem is the force-directed technRVEF which in troduces a heuristic costfunR]F9 (an energy) of themappin L, which (hopefully) achieves its min4 umwhen the layout isn ice. Varian ts of this approach di#er in the defin9fiFR of t...