[Paper Review] Sliding k-Transmitters: Hardness and Approximation

Thérèse Biedl, Saeed Mehrabi+1 more|arXiv (Cornell University)|Jul 25, 2016

VLSI and FPGA Design Techniques3 citations

TL;DR

This paper proves that guarding an orthogonal, y-monotone polygon with the minimum number of sliding k-transmitters is NP-hard for any fixed k > 0, even in the simplest case with only horizontal transmitters. It also presents an O(1)-approximation algorithm for the problem, extending prior work on sliding cameras to the k-transmitter model.

ABSTRACT

A sliding k-transmitter in an orthogonal polygon P is a mobile guard that travels back and forth along an orthogonal line segment s inside P. It can see a point p in P if the perpendicular from p onto s intersects the boundary of P at most k times. We show that guarding an orthogonal polygon P with the minimum number of k-transmitters is NP-hard, for any fixed k>0, even if P is simple and monotone. Moreover, we give an O(1)-approximation algorithm for this problem.

Motivation & Objective

• To establish the computational complexity of the sliding k-transmitter art gallery problem in orthogonal polygons.
• To investigate whether the problem remains hard under restrictive geometric conditions, such as y-monotonicity and orthogonality.
• To develop an efficient approximation algorithm for the minimum sliding k-transmitter set problem.
• To determine whether the approximation framework for sliding cameras can be adapted to sliding k-transmitters.

Proposed method

• Reduction from minimum vertex cover in planar 2-connected graphs to construct a y-monotone orthogonal polygon with sliding k-transmitters.
• Design of vertex-gadgets using stacked boxes and vertical channels to model vertices and edges in the graph.
• Construction of edge-gadgets using thin vertical strips to represent edges, ensuring visibility constraints align with graph edges.
• Discretization of the guarding problem into a cross-hitting problem using guard-segments and pixel crosses.
• Mapping of sliding k-transmitters to maximal guard-segments along polygon edges to simplify the visibility problem.
• Adaptation of the O(1)-approximation algorithm from sliding cameras to sliding k-transmitters via the cross-hitting formulation.

Experimental results

Research questions

• RQ1 Is the sliding k-transmitter problem NP-hard even when restricted to orthogonal, y-monotone polygons?
• RQ2 Can the O(1)-approximation algorithm for sliding cameras be extended to sliding k-transmitters?
• RQ3 Does the complexity of the problem remain high when only horizontal sliding k-transmitters are allowed?
• RQ4 Are there non-trivial polygon subclasses (beyond orthogonally convex) for which the problem becomes polynomial-time solvable?

Key findings

• The problem of guarding an orthogonal, y-monotone polygon with sliding k-transmitters is NP-hard for any fixed k > 0.
• The NP-hardness holds even when only horizontal sliding k-transmitters are allowed, and the polygon is simple and monotone.
• An O(1)-approximation algorithm exists for the sliding k-transmitter problem in any orthogonal polygon, including those with holes.
• The approximation algorithm is derived by reducing the problem to a cross-hitting problem over guard-segments, analogous to the sliding camera case.
• The algorithm remains effective and simplifies when restricted to horizontal sliding k-transmitters only.
• The constant factor in the O(1)-approximation is unspecified but likely large, as it relies on ε-net constructions from prior work.