Walking Through Waypoints

We initiate the study of a fundamental combinatorial problem: Given a capacitated graph $G=(V,E)$, find a shortest walk ("route") from a source $s\in V$ to a destination $t\in V$ that includes all vertices specified by a set $\mathscr{W}\subseteq V$: the \emph{waypoints}. This waypoint routing problem finds immediate applications in the context of modern networked distributed systems. Our main contribution is an exact polynomial-time algorithm for graphs of bounded treewidth. We also show that if the number of waypoints is logarithmically bounded, exact polynomial-time algorithms exist even for general graphs. Our two algorithms provide an almost complete characterization of what can be solved exactly in polynomial-time: we show that more general problems (e.g., on grid graphs of maximum degree 3, with slightly more waypoints) are computationally intractable

An Algorithm to Find a K5 Minor

Coordinated Science Laboratory was formerly known as Control Systems LaboratoryJoint Services Electronics Program / N00014-90-J-127

Processor Efficient Parallel Algorithms for the Two Disjoint Paths Problem, and for Finding a Kuratowski Homeomorph

We give an $NC$ algorithm for finding vertex disjoint $s_{1}, t_{1}$ and $s_{2}, t_{2}$ paths in an undirected graph $G$. An important step in solving the general problem is solving the planar case. A new structural property yields the parallelization, as well as a simpler linear time sequential algorithm for this case. We extend the algorithm to the non-planar case by giving an $NC$ algorithm for finding a Kuratowski homeomorph, and, in particular, a homeomorph of $K_{3}, 3$, in a non-planar graph. Our algorithms are processor efficient; in each case, the processor-time product of our algorithms is within a polylogarithmic factor of the best known sequential algorithm