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Towards Tight Bounds on Theta-Graphs

We present improved upper and lower bounds on the spanning ratio of $θ$-graphs with at least six cones. Given a set of points in the plane, a $θ$-graph partitions the plane around each vertex into $m$ disjoint cones, each having aperture $θ=2π/m$, and adds an edge to the `closest' vertex in each cone. We show that for any integer $k \geq 1$, $θ$-graphs with $4k+2$ cones have a spanning ratio of $1+2\sin(θ/2)$ and we provide a matching lower bound, showing that this spanning ratio tight. Next, we show that for any integer $k \geq 1$, $θ$-graphs with $4k+4$ cones have spanning ratio at most $1+2\sin(θ/2)/(\cos(θ/2)-\sin(θ/2))$. We also show that $θ$-graphs with $4k+3$ and $4k+5$ cones have spanning ratio at most $\cos(θ/4)/(\cos(θ/2)-\sin(3θ/4))$. This is a significant improvement on all families of $θ$-graphs for which exact bounds are not known. For example, the spanning ratio of the $θ$-graph with 7 cones is decreased from at most 7.5625 to at most 3.5132. These spanning proofs also imply improved upper bounds on the competitiveness of the $θ$-routing algorithm. In particular, we show that the $θ$-routing algorithm is $(1+2\sin(θ/2)/(\cos(θ/2)-\sin(θ/2)))$-competitive on $θ$-graphs with $4k+4$ cones and that this ratio is tight. Finally, we present improved lower bounds on the spanning ratio of these graphs. Using these bounds, we provide a partial order on these families of $θ$-graphs. In particular, we show that $θ$-graphs with $4k+4$ cones have spanning ratio at least $1+2\tan(θ/2)+2\tan^2(θ/2)$. This is somewhat surprising since, for equal values of $k$, the spanning ratio of $θ$-graphs with $4k+4$ cones is greater than that of $θ$-graphs with $4k+2$ cones, showing that increasing the number of cones can make the spanning ratio worse.