Paweł Żyliński

Weakly cooperative guards in grids

We show that a minimum coverage of a grid of n segments has n–p3 weakly cooperative guards, where p3 is the size of the maximum P3-matching in the intersection graph of the grid. This makes the minimum weakly cooperative guards problem in grids NP-hard, as we prove that the maximum P3-matching problem in subcubic bipartite planar graphs is NP-hard. At last, we propose a 7/6-approximation algorithm for the minimum weakly cooperative guards problem.

Cooperative mobile guards in grids

A grid P is a connected union of vertical and horizontal segments. A mobile guard is a guard which is allowed to move along a grid segment, thus a point x is seen by a mobile guard g if either x is on the same segment as g or x is on a grid segment crossing g. A set of mobile guards is weakly cooperative if at any point on its patrol, every guard can be seen by at least one other guard. In this paper we discuss the classes of polygon-bounded grids and simple grids for which we propose a quadratic time algorithm for solving the problem of finding the minimum weakly cooperative guard set (MinWCMG). We also provide an O(nlogn) time algorithm for the MinWCMG problem in horizontally or vertically unobstructed grids. Next, we investigate complete rectangular grids with obstacles. We show that as long as both dimensions of a grid are larger than the number of obstacles k, k+2 weakly cooperative mobile guards always suffice to cover the grid. Finally, we prove that the MinWCMG problem is NP-hard even for grids in which every segment crosses at most three other segments. Consequently, the minimum k-periscope guard problem for 2D grids is NP-hard as well, and this answers the question posed by Gewali and Ntafos [L.P. Gewali, S. Ntafos, Covering grids and orthogonal polygons with periscope guards, Computational Geometry: Theory and Applications 2 (1993) 309-334].

Packing [1, Δ]-factors in graphs of small degree

Given an undirected, connected graph G with maximum degree Δ, we introduce the concept of a [1, Δ]-factor k-packing in G, defined as a set of k edge-disjoint subgraphs of G such that every vertex of G has an incident edge in at least one subgraph. The problem of deciding whether a graph admits a [1,Δ]-factor k-packing is shown to be solvable in linear time for k = 2, but NP-complete for all k≥ 3. For k = 2, the optimisation problem of minimising the total number of edges of the subgraphs of the packing is NP-hard even when restricted to subcubic planar graphs, but can in general be approximated within a factor of

Scheduling with precedence constraints: mixed graph coloring in series-parallel graphs

\frac{42\Delta -30}{35\Delta-21}

Note on covering monotone orthogonal polygons with star-shaped polygons

by reduction to the Maximum 2-Edge-Colorable Subgraph problem. Finally, we discuss implications of the obtained results for the problem of fault-tolerant guarding of a grid, which provides the main motivation for research.

Watchman routes for lines and segments

In this note we observe that the problem of mixed graph coloring can be solved in linear time for trees, which improves the quadratic algorithm of Hansen et al. [P. Hansen, J. Kuplinsky, D. de Werra, Mixed graph colorings, Math. Methods Oper. Res. 45 (1997) 145-160].

Watchman routes for lines and line segments

We consider the mixed graph coloring problem which is used for formulating scheduling problems where both incompatibility and precedence constraints can be present. We give an O(n3.376 log n) algorithm for finding an optimal schedule for a collection of jobs whose constraint relations form a mixed series-parallel graph.

An efficient algorithm for mobile guarded guards in simple grids

We consider the problem of dividing a distributed system into subsystems for parallel processing with redundancy for fault tolerance, where every subsystem has to consist of at least three units. We prove that the problem of determining the maximum number of subsystems with redundancy for fault tolerance is NP-hard even in cubic planar 2-connected system topologies. We point out that this problem is APX-hard on cubic bipartite graphs. At last, for subcubic topologies without units connected to only one other unit, we give a linear time 4/3-approximation algorithm.