Algorithms for routing in planar graphs

We present algorithms for solving routing problems for two-terminal nets in planar graphs. Our algorithms run in time O(n2) for general planar graphs and in time O(bn) for grid graphs where n is the number of vertices and b is the number of vertices on the boundary of the infinite face

Shortest disjoint paths on a grid

The well-known k-disjoint paths problem involves finding pairwise vertex-disjoint paths between k specified pairs of vertices within a given graph if they exist. In the shortest k-disjoint paths problem one looks for such paths of minimum total length. Despite nearly 50 years of active research on the k-disjoint paths problem, many open problems and complexity gaps still persist. A particularly well-defined scenario, inspired by VLSI design, focuses on infinite rectangular grids where the terminals are placed at arbitrary grid points. While the decision problem in this context remains NP-hard, no prior research has provided any positive results for the optimization version. The main result of this paper is a fixed-parameter tractable (FPT) algorithm for this scenario. It is important to stress that this is the first result achieving the FPT complexity of the shortest disjoint paths problem in any, even very restricted classes of graphs where we do not put any restriction on the placements of the terminals

The Parameterized Complexity of Coordinated Motion Planning

In Coordinated Motion Planning (CMP), we are given a rectangular-grid on which $k$ robots occupy $k$ distinct starting gridpoints and need to reach $k$ distinct destination gridpoints. In each time step, any robot may move to a neighboring gridpoint or stay in its current gridpoint, provided that it does not collide with other robots. The goal is to compute a schedule for moving the $k$ robots to their destinations which minimizes a certain objective target - prominently the number of time steps in the schedule, i.e., the makespan, or the total length traveled by the robots. We refer to the problem arising from minimizing the former objective target as CMP-M and the latter as CMP-L. Both CMP-M and CMP-L are fundamental problems that were posed as the computational geometry challenge of SoCG 2021, and CMP also embodies the famous $(n^2-1)$-puzzle as a special case. In this paper, we settle the parameterized complexity of CMP-M and CMP-L with respect to their two most fundamental parameters: the number of robots, and the objective target. We develop a new approach to establish the fixed-parameter tractability of both problems under the former parameterization that relies on novel structural insights into optimal solutions to the problem. When parameterized by the objective target, we show that CMP-L remains fixed-parameter tractable while CMP-M becomes para-NP-hard. The latter result is noteworthy, not only because it improves the previously-known boundaries of intractability for the problem, but also because the underlying reduction allows us to establish - as a simpler case - the NP-hardness of the classical Vertex Disjoint and Edge Disjoint Paths problems with constant path-lengths on grids.Comment: Short version appeared in SoCG 202

Lower and Upper-Bounds for the General Junction Routing Problem

Coordinated Science Laboratory was formerly known as Control Systems LaboratoryNational Science Foundation / ECS 84-10902Semiconductor Research Corporatio

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Disjoint paths in a rectilinear grid

SIGLEDEGerman

MAXIMUM EDGE DISJOINT PATHS AND MINIMUM UNWEIGHTED MULTICUT PROBLEMS IN GRID GRAPHS

Let G = (V, E) be an undirected graph and let L be a list of K pairs (source sk, sink tk) of terminal vertices of G (or nets). The maximum edge disjoint paths problem (MaxEDP) consists in maximizing the number of nets linked by edge disjoint paths. The related minimum multicut problem (MinUWMC) is to find a minimum set of edges whose removal separates each pair (sk, tk) in an augmented graph (i.e. where each terminal vertex is linked to the graph by a unique edge, as in [4]). Both problems are NP-hard even in planar graphs [1]. MaxEDP defined in rectilinear grids where any vertex can be a terminal is also NP-hard [3]. However, A. Frank gives in [2] necessary and sufficient conditions for the existence of K edge disjoint paths when the terminals are two-sided (i.e. they are all distinct and lie on the uppermost and lowermost lines of the grid): thus, solving MaxEDP is equivalent to removing the minimum number of nets in order to fulfill Frank’s conditions. We prove that this can be done by solving a polynomial number of linear programs having totally unimodular constraints matrices. Then, we show that, in two-sided augmented grids, MinUWMC is polynomial time solvable via linear programming, by using a duality relationship with a continuous multiflow problem. As a by-product, the gap between the optimal values of MaxEDP and MinUWMC is proved to be at most one. References [1] M.-C. Costa, L. Létocart and F. Roupin. Minimal multicut and maximal integer multiflow: A survey. To appear in EJOR (Available online). [2] A. Frank. Disjoint paths in a rectilinear grid. Combinatorica 2 (1982) pp. 361-371. [3] D. Marx. Eulerian disjoint paths problem in grid graphs is NP-complete