Pawel Zylinski

Offline variants of the "lion and man" problem

Consider the following survival problem:Given a set of k trajectories (paths) with maximum unit speed in a boundedregion over a (long) time interval [0,T], find another trajectory (if itexists) subject to the same maximum unit speed limit, that avoids (that is, stays at a safe distance of)each of the other trajectories over the entire time interval. We call this variant the continuous model of the survival problem. The discrete model of this problem is: Given the trajectories (paths) of k point robots in a graph over a (long)time interval 0,1,2,...,T, find a trajectory (path) for anotherrobot, that avoids each of the other k at any time instance in thegiven time interval. We introduce the notions of survival number of a region,and that of a graph, respectively, as the maximum number oftrajectories which can be avoided in the region (resp. graph). We give the first estimates on the survival number of the n x n grid Gn, and also devise an efficient algorithm for the corresponding safepath planning problem in arbitrary graphs. We then show that our estimates on the survival number of Gn%on the number of paths that can be avoided in Gn can be extended for the survival number of a bounded (square) region.In the final part of our paper, we consider other related offlinequestions, such as the maximum number of men problem and the spy problem.

Linear-time 3-approximation algorithm for the r-star covering problem

The complexity status of the minimum r-star cover problem for orthogonal polygons had been open for many years, until 2004 when Ch. Worman and J. M. Keil proved it to be polynomially tractable (Polygon decomposition and the orthogonal art gallery problem, IJCGA 17(2) (2007), 105-138). However, since their algorithm has O(n17)-time complexity, where O(·) hides a polylogarithmic factor, and thus it is not practical, in this paper we present a linear-time 3-approximation algorithm. Our approach is based upon the novel partition of an orthogonal polygon into so-called o-star-shaped orthogonal polygons.

An approximation algorithm for maximum P 3 -packing in subcubic graphs

We give a linear time 4/3-approximation algorithm for the problem of finding the maximum number of vertex-disjoint paths of order 3 in subcubic graphs without pendant vertices, which improves previously known results [K. Kawarabayashi, H. Matsuda, Y. Oda, K. Ota, Path factors in cubic graphs, Journal of Graph Theory 39 (2002) 188-193; A. Kelmans, D. Mubayi, How many disjoint 2-edge paths must a cubic graph have?, Journal of Graph Theory 45 (2004) 57-79].

Approximation Algorithms for Buy-at-Bulk Geometric Network Design

The buy-at-bulk network design problem has been extensively studied in the general graph model. In this paper we consider the geometric version of the problem, where all points in a Euclidean space are candidates for network nodes. We present the first general approach for geometric versions of basic variants of the buy-at-bulk network design problem. It enables us to obtain quasi-polynomial-time approximation schemes for basic variants of the buy-at-bulk geometric network design problem with polynomial total demand. Then, for instances with few sinks and low capacity links, we design very fast polynomial-time low-constant approximations algorithms.

Clearing Connections by Few Agents

We study the problem of clearing connections by agents placed at some vertices in a directed graph. The agents can move only along directed paths. The objective is to minimize the number of agents guaranteeing that any pair of vertices can be connected by a underlying undirected path that can be cleared by the agents. We provide several results on the hardness, approximability and parameterized complexity of the problem. In particular, we show it to be: NP-hard, 2-approximable in polynomial-time, and solvable exactly in O(αn 322 α ) time, where α is the number of agents in the solution. In addition, we give a simple linear-time algorithm optimally solving the problem in digraphs whose underlying graphs are trees. Finally, we discuss a related problem, where the task is to clear with a minimum number of agents a subgraph of the underlying graph containing its spanning tree. We show that this problem also admits a 2-approximation in polynomial time.

Forming a connected network in a grid by asynchronous and oblivious robots

Consider an orthogonal grid of streets and avenues in a Manhattan-like city populated by stationary sensor modules at some crossings and mobile robots that can serve as relays of information that the modules exchange. Both module-module and module-robot communication is limited to a straight line of sight along a row or a column of the grid. We present a number of distributed algorithms for the robots to establish a connected network of a given set S of modules by moving to suitable locations in the grid and serving as relays. It is shown that the number of robots required to connect the modules depends not only on the number c of connected components in the visibility graph of S, but also on the degree of symmetry in S. In most cases, our algorithms use the worst case optimal number of robots for a given c.