Finding kk Simple Shortest Paths and Cycles

The problem of finding multiple simple shortest paths in a weighted directed graph $G=(V,E)$ has many applications, and is considerably more difficult than the corresponding problem when cycles are allowed in the paths. Even for a single source-sink pair, it is known that two simple shortest paths cannot be found in time polynomially smaller than $n^3$ (where $n=|V|$) unless the All-Pairs Shortest Paths problem can be solved in a similar time bound. The latter is a well-known open problem in algorithm design. We consider the all-pairs version of the problem, and we give a new algorithm to find $k$ simple shortest paths for all pairs of vertices. For $k=2$, our algorithm runs in $O(mn + n^2 \log n)$ time (where $m=|E|$), which is almost the same bound as for the single pair case, and for $k=3$ we improve earlier bounds. Our approach is based on forming suitable path extensions to find simple shortest paths; this method is different from the `detour finding' technique used in most of the prior work on simple shortest paths, replacement paths, and distance sensitivity oracles. Enumerating simple cycles is a well-studied classical problem. We present new algorithms for generating simple cycles and simple paths in $G$ in non-decreasing order of their weights; the algorithm for generating simple paths is much faster, and uses another variant of path extensions. We also give hardness results for sparse graphs, relative to the complexity of computing a minimum weight cycle in a graph, for several variants of problems related to finding $k$ simple paths and cycles.Comment: The current version includes new results for undirected graphs. In Section 4, the notion of an (m,n) reduction is generalized to an f(m,n) reductio

Approximating the Regular Graphic TSP in near linear time

We present a randomized approximation algorithm for computing traveling salesperson tours in undirected regular graphs. Given an $n$-vertex, $k$-regular graph, the algorithm computes a tour of length at most $\left(1+\frac{7}{\ln k-O(1)}\right)n$, with high probability, in $O(nk \log k)$ time. This improves upon a recent result by Vishnoi (\cite{Vishnoi12}, FOCS 2012) for the same problem, in terms of both approximation factor, and running time. The key ingredient of our algorithm is a technique that uses edge-coloring algorithms to sample a cycle cover with $O(n/\log k)$ cycles with high probability, in near linear time. Additionally, we also give a deterministic $\frac{3}{2}+O\left(\frac{1}{\sqrt{k}}\right)$ factor approximation algorithm running in time $O(nk)$.Comment: 12 page

Even Orientations and Pfaffian graphs

We give a characterization of Pfaffian graphs in terms of even orientations, extending the characterization of near bipartite non--pfaffian graphs by Fischer and Little \cite{FL}. Our graph theoretical characterization is equivalent to the one proved by Little in \cite{L73} (cf. \cite{LR}) using linear algebra arguments

On Groupoids and Hypergraphs

We present a novel construction of finite groupoids whose Cayley graphs have large girth even w.r.t. a discounted distance measure that contracts arbitrarily long sequences of edges from the same colour class (sub-groupoid), and only counts transitions between colour classes (cosets). These groupoids are employed towards a generic construction method for finite hypergraphs that realise specified overlap patterns and avoid small cyclic configurations. The constructions are based on reduced products with groupoids generated by the elementary local extension steps, and can be made to preserve the symmetries of the given overlap pattern. In particular, we obtain highly symmetric, finite hypergraph coverings without short cycles. The groupoids and their application in reduced products are sufficiently generic to be applicable to other constructions that are specified in terms of local glueing operations and require global finite closure.Comment: Explicit completion of H in HxI (Section 2) is unstable (incompatible with restrictions), hence does not support inductive construction towards Prop. 2.17 based on Lem 2.16 as claimed. For corresponding technical result, now see arxiv:1806.08664; for discussion of main applications first announced here, now see arxiv:1709.0003

A Dichotomy Theorem for Circular Colouring Reconfiguration

The "reconfiguration problem" for circular colourings asks, given two $(p,q)$-colourings $f$ and $g$ of a graph $G$, is it possible to transform $f$ into $g$ by changing the colour of one vertex at a time such that every intermediate mapping is a $(p,q)$-colouring? We show that this problem can be solved in polynomial time for $2\leq p/q <4$ and is PSPACE-complete for $p/q\geq 4$. This generalizes a known dichotomy theorem for reconfiguring classical graph colourings.Comment: 22 pages, 5 figure