Diameter of Cayley graphs of permutation groups generated by transposition trees

Let $\Gamma$ be a Cayley graph of the permutation group generated by a transposition tree $T$ on $n$ vertices. In an oft-cited paper \cite{Akers:Krishnamurthy:1989} (see also \cite{Hahn:Sabidussi:1997}), it is shown that the diameter of the Cayley graph $\Gamma$ is bounded as \diam(\Gamma) \le \max_{\pi \in S_n}{c(\pi)-n+\sum_{i=1}^n \dist_T(i,\pi(i))}, where the maximization is over all permutations $\pi$, $c(\pi)$ denotes the number of cycles in $\pi$, and \dist_T is the distance function in $T$. In this work, we first assess the performance (the sharpness and strictness) of this upper bound. We show that the upper bound is sharp for all trees of maximum diameter and also for all trees of minimum diameter, and we exhibit some families of trees for which the bound is strict. We then show that for every $n$, there exists a tree on $n$ vertices, such that the difference between the upper bound and the true diameter value is at least $n-4$. Observe that evaluating this upper bound requires on the order of $n!$ (times a polynomial) computations. We provide an algorithm that obtains an estimate of the diameter, but which requires only on the order of (polynomial in) $n$ computations; furthermore, the value obtained by our algorithm is less than or equal to the previously known diameter upper bound. This result is possible because our algorithm works directly with the transposition tree on $n$ vertices and does not require examining any of the permutations (only the proof requires examining the permutations). For all families of trees examined so far, the value $\beta$ computed by our algorithm happens to also be an upper bound on the diameter, i.e. \diam(\Gamma) \le \beta \le \max_{\pi \in S_n}{c(\pi)-n+\sum_{i=1}^n \dist_T(i,\pi(i))}.Comment: This is an extension of arXiv:1106.535

Hardness of Token Swapping on Trees

Given a graph where every vertex has exactly one labeled token, how can we most quickly execute a given permutation on the tokens? In (sequential) token swapping, the goal is to use the shortest possible sequence of swaps, each of which exchanges the tokens at the two endpoints of an edge of the graph. In parallel token swapping, the goal is to use the fewest rounds, each of which consists of one or more swaps on the edges of a matching. We prove that both of these problems remain NP-hard when the graph is restricted to be a tree. These token swapping problems have been studied by disparate groups of researchers in discrete mathematics, theoretical computer science, robot motion planning, game theory, and engineering. Previous work establishes NP-completeness on general graphs (for both problems), constant-factor approximation algorithms, and some poly-time exact algorithms for simple graph classes such as cliques, stars, paths, and cycles. Sequential and parallel token swapping on trees were first studied over thirty years ago (as "sorting with a transposition tree") and over twenty-five years ago (as "routing permutations via matchings"), yet their complexities were previously unknown. We also show limitations on approximation of sequential token swapping on trees: we identify a broad class of algorithms that encompass all three known polynomial-time algorithms that achieve the best known approximation factor (which is 2) and show that no such algorithm can achieve an approximation factor less than 2

Groups and Geometries

[no abstract available

IST Austria Thesis

This thesis considers two examples of reconfiguration problems: flipping edges in edge-labelled triangulations of planar point sets and swapping labelled tokens placed on vertices of a graph. In both cases the studied structures – all the triangulations of a given point set or all token placements on a given graph – can be thought of as vertices of the so-called reconfiguration graph, in which two vertices are adjacent if the corresponding structures differ by a single elementary operation – by a flip of a diagonal in a triangulation or by a swap of tokens on adjacent vertices, respectively. We study the reconfiguration of one instance of a structure into another via (shortest) paths in the reconfiguration graph. For triangulations of point sets in which each edge has a unique label and a flip transfers the label from the removed edge to the new edge, we prove a polynomial-time testable condition, called the Orbit Theorem, that characterizes when two triangulations of the same point set lie in the same connected component of the reconfiguration graph. The condition was first conjectured by Bose, Lubiw, Pathak and Verdonschot. We additionally provide a polynomial time algorithm that computes a reconfiguring flip sequence, if it exists. Our proof of the Orbit Theorem uses topological properties of a certain high-dimensional cell complex that has the usual reconfiguration graph as its 1-skeleton. In the context of token swapping on a tree graph, we make partial progress on the problem of finding shortest reconfiguration sequences. We disprove the so-called Happy Leaf Conjecture and demonstrate the importance of swapping tokens that are already placed at the correct vertices. We also prove that a generalization of the problem to weighted coloured token swapping is NP-hard on trees but solvable in polynomial time on paths and stars

Embedding Schemes for Interconnection Networks.

Graph embeddings play an important role in interconnection network and VLSI design. Designing efficient embedding strategies for simulating one network by another and determining the number of layers required to build a VLSI chip are just two of the many areas in which graph embeddings are used. In the area of network simulation we develop efficient, small dilation embeddings of a butterfly network into a different size and/or type of butterfly network. The genus of a graph gives an indication of how many layers are required to build a circuit. We have determined the exact genus for the permutation network called the star graph, and have given a lower bound for the genus of the permutation network called the pancake graph. The star graph has been proposed as an alternative to the binary hypercube and, therefore, we compare the genus of the star graph with that of the binary hypercube. Another type of embedding that is helpful in determining the number of layers is a book embedding. We develop upper and lower bounds on the pagenumber of a book embedding of the k-ary hypercube along with an upper bound on the cumulative pagewidth