Topologically Trivial Closed Walks in Directed Surface Graphs

Let $G$ be a directed graph with $n$ vertices and $m$ edges, embedded on a surface $S$, possibly with boundary, with first Betti number $\beta$. We consider the complexity of finding closed directed walks in $G$ that are either contractible (trivial in homotopy) or bounding (trivial in integer homology) in $S$. Specifically, we describe algorithms to determine whether $G$ contains a simple contractible cycle in $O(n+m)$ time, or a contractible closed walk in $O(n+m)$ time, or a bounding closed walk in $O(\beta (n+m))$ time. Our algorithms rely on subtle relationships between strong connectivity in $G$ and in the dual graph $G^*$; our contractible-closed-walk algorithm also relies on a seminal topological result of Hass and Scott. We also prove that detecting simple bounding cycles is NP-hard. We also describe three polynomial-time algorithms to compute shortest contractible closed walks, depending on whether the fundamental group of the surface is free, abelian, or hyperbolic. A key step in our algorithm for hyperbolic surfaces is the construction of a context-free grammar with $O(g^2L^2)$ non-terminals that generates all contractible closed walks of length at most L, and only contractible closed walks, in a system of quads of genus $g\ge2$. Finally, we show that computing shortest simple contractible cycles, shortest simple bounding cycles, and shortest bounding closed walks are all NP-hard.Comment: 30 pages, 18 figures; fixed several minor bugs and added one figure. An extended abstraction of this paper will appear at SOCG 201

Computing the Girth of a Planar Graph in Linear Time

The girth of a graph is the minimum weight of all simple cycles of the graph. We study the problem of determining the girth of an n-node unweighted undirected planar graph. The first non-trivial algorithm for the problem, given by Djidjev, runs in O(n^{5/4} log n) time. Chalermsook, Fakcharoenphol, and Nanongkai reduced the running time to O(n log^2 n). Weimann and Yuster further reduced the running time to O(n log n). In this paper, we solve the problem in O(n) time.Comment: 20 pages, 7 figures, accepted to SIAM Journal on Computin

Single Source - All Sinks Max Flows in Planar Digraphs

Let G = (V,E) be a planar n-vertex digraph. Consider the problem of computing max st-flow values in G from a fixed source s to all sinks t in V\{s}. We show how to solve this problem in near-linear O(n log^3 n) time. Previously, no better solution was known than running a single-source single-sink max flow algorithm n-1 times, giving a total time bound of O(n^2 log n) with the algorithm of Borradaile and Klein. An important implication is that all-pairs max st-flow values in G can be computed in near-quadratic time. This is close to optimal as the output size is Theta(n^2). We give a quadratic lower bound on the number of distinct max flow values and an Omega(n^3) lower bound for the total size of all min cut-sets. This distinguishes the problem from the undirected case where the number of distinct max flow values is O(n). Previous to our result, no algorithm which could solve the all-pairs max flow values problem faster than the time of Theta(n^2) max-flow computations for every planar digraph was known. This result is accompanied with a data structure that reports min cut-sets. For fixed s and all t, after O(n^{3/2} log^{3/2} n) preprocessing time, it can report the set of arcs C crossing a min st-cut in time roughly proportional to the size of C.Comment: 25 pages, 4 figures; extended abstract appeared in FOCS 201

Finding shortest non-trivial cycles in directed graphs on surfaces

Let D be a weighted directed graph cellularly embedded in a surface of genus g, orientable or not, possibly with boundary. We describe algorithms to compute a shortest non-contractible and a shortest surface non-separating cycle in D. This generalizes previous results that only dealt with undirected graphs. Our first algorithm computes such cycles in O(n 2 log n) time, where n is the total number of vertices and edges of D, thus matching the complexity of the best known algorithm in the undirected case. It revisits and extends Thomassen’s 3-path condition; the technique applies to other families of cycles as well. We also give an algorithm with subquadratic complexity in the complexity of the input graph, if g is fixed. Specifically, we can solve the problem in O ( √ g n 3/2 log n) time, using a divide-and-conquer technique that simplifies the graph while preserving the topological properties of its cycles. A variant runs in O(ng log g + nlog 2 n) for graphs of bounded treewidth

Finding shortest non-trivial cycles in directed graphs on surfaces

Let $D$ be a weighted directed graph cellularly embedded in a surface of genus $g$, orientable or not, possibly with boundary.  We describe algorithms to compute shortest non-contractible and shortest surface non-separating cycles in $D$, generalizing previous results that dealt with undirected graphs.Our first algorithm computes such cycles in $O(n^2\log n)$ time, where $n$ is the total number of vertices and edges of $D$, thus matching the complexity of the best general algorithm in the undirected case.  It revisits and extends Thomassen's 3-path condition; the technique applies to other families of cycles as well.We also provide more efficient algorithms in special cases, such as graphs with small genus or bounded treewidth, using a divide-and-conquer technique that simplifies the graph while preserving the topological properties of its cycles.  Finally, we give an efficient output-sensitive algorithm, whose running time depends on the length of the shortest non-contractible or non-separating cycle