Spanning trees of 3-uniform hypergraphs

Masbaum and Vaintrob's "Pfaffian matrix tree theorem" implies that counting spanning trees of a 3-uniform hypergraph (abbreviated to 3-graph) can be done in polynomial time for a class of "3-Pfaffian" 3-graphs, comparable to and related to the class of Pfaffian graphs. We prove a complexity result for recognizing a 3-Pfaffian 3-graph and describe two large classes of 3-Pfaffian 3-graphs -- one of these is given by a forbidden subgraph characterization analogous to Little's for bipartite Pfaffian graphs, and the other consists of a class of partial Steiner triple systems for which the property of being 3-Pfaffian can be reduced to the property of an associated graph being Pfaffian. We exhibit an infinite set of partial Steiner triple systems that are not 3-Pfaffian, none of which can be reduced to any other by deletion or contraction of triples. We also find some necessary or sufficient conditions for the existence of a spanning tree of a 3-graph (much more succinct than can be obtained by the currently fastest polynomial-time algorithm of Gabow and Stallmann for finding a spanning tree) and a superexponential lower bound on the number of spanning trees of a Steiner triple system.Comment: 34 pages, 9 figure

Fixed-Parameter Algorithms for Rectilinear Steiner tree and Rectilinear Traveling Salesman Problem in the plane

Given a set $P$ of $n$ points with their pairwise distances, the traveling salesman problem (TSP) asks for a shortest tour that visits each point exactly once. A TSP instance is rectilinear when the points lie in the plane and the distance considered between two points is the $l_1$ distance. In this paper, a fixed-parameter algorithm for the Rectilinear TSP is presented and relies on techniques for solving TSP on bounded-treewidth graphs. It proves that the problem can be solved in $O\left(nh7^h\right)$ where $h \leq n$ denotes the number of horizontal lines containing the points of $P$. The same technique can be directly applied to the problem of finding a shortest rectilinear Steiner tree that interconnects the points of $P$ providing a $O\left(nh5^h\right)$ time complexity. Both bounds improve over the best time bounds known for these problems.Comment: 24 pages, 13 figures, 6 table

Connectivity Oracles for Graphs Subject to Vertex Failures

We introduce new data structures for answering connectivity queries in graphs subject to batched vertex failures. A deterministic structure processes a batch of $d\leq d_{\star}$ failed vertices in $\tilde{O}(d^3)$ time and thereafter answers connectivity queries in $O(d)$ time. It occupies space $O(d_{\star} m\log n)$. We develop a randomized Monte Carlo version of our data structure with update time $\tilde{O}(d^2)$, query time $O(d)$, and space $\tilde{O}(m)$ for any failure bound $d\le n$. This is the first connectivity oracle for general graphs that can efficiently deal with an unbounded number of vertex failures. We also develop a more efficient Monte Carlo edge-failure connectivity oracle. Using space $O(n\log^2 n)$, $d$ edge failures are processed in $O(d\log d\log\log n)$ time and thereafter, connectivity queries are answered in $O(\log\log n)$ time, which are correct w.h.p. Our data structures are based on a new decomposition theorem for an undirected graph $G=(V,E)$, which is of independent interest. It states that for any terminal set $U\subseteq V$ we can remove a set $B$ of $|U|/(s-2)$ vertices such that the remaining graph contains a Steiner forest for $U-B$ with maximum degree $s$

Join-Reachability Problems in Directed Graphs

For a given collection G of directed graphs we define the join-reachability graph of G, denoted by J(G), as the directed graph that, for any pair of vertices a and b, contains a path from a to b if and only if such a path exists in all graphs of G. Our goal is to compute an efficient representation of J(G). In particular, we consider two versions of this problem. In the explicit version we wish to construct the smallest join-reachability graph for G. In the implicit version we wish to build an efficient data structure (in terms of space and query time) such that we can report fast the set of vertices that reach a query vertex in all graphs of G. This problem is related to the well-studied reachability problem and is motivated by emerging applications of graph-structured databases and graph algorithms. We consider the construction of join-reachability structures for two graphs and develop techniques that can be applied to both the explicit and the implicit problem. First we present optimal and near-optimal structures for paths and trees. Then, based on these results, we provide efficient structures for planar graphs and general directed graphs

Speeding-up Dynamic Programming with Representative Sets - An Experimental Evaluation of Algorithms for Steiner Tree on Tree Decompositions

Dynamic programming on tree decompositions is a frequently used approach to solve otherwise intractable problems on instances of small treewidth. In recent work by Bodlaender et al., it was shown that for many connectivity problems, there exist algorithms that use time, linear in the number of vertices, and single exponential in the width of the tree decomposition that is used. The central idea is that it suffices to compute representative sets, and these can be computed efficiently with help of Gaussian elimination. In this paper, we give an experimental evaluation of this technique for the Steiner Tree problem. A comparison of the classic dynamic programming algorithm and the improved dynamic programming algorithm that employs the table reduction shows that the new approach gives significant improvements on the running time of the algorithm and the size of the tables computed by the dynamic programming algorithm, and thus that the rank based approach from Bodlaender et al. does not only give significant theoretical improvements but also is a viable approach in a practical setting, and showcases the potential of exploiting the idea of representative sets for speeding up dynamic programming algorithms

Secluded Connectivity Problems

Consider a setting where possibly sensitive information sent over a path in a network is visible to every {neighbor} of the path, i.e., every neighbor of some node on the path, thus including the nodes on the path itself. The exposure of a path $P$ can be measured as the number of nodes adjacent to it, denoted by $N[P]$. A path is said to be secluded if its exposure is small. A similar measure can be applied to other connected subgraphs, such as Steiner trees connecting a given set of terminals. Such subgraphs may be relevant due to considerations of privacy, security or revenue maximization. This paper considers problems related to minimum exposure connectivity structures such as paths and Steiner trees. It is shown that on unweighted undirected $n$-node graphs, the problem of finding the minimum exposure path connecting a given pair of vertices is strongly inapproximable, i.e., hard to approximate within a factor of $O(2^{\log^{1-\epsilon}n})$ for any $\epsilon>0$ (under an appropriate complexity assumption), but is approximable with ratio $\sqrt{\Delta}+3$, where $\Delta$ is the maximum degree in the graph. One of our main results concerns the class of bounded-degree graphs, which is shown to exhibit the following interesting dichotomy. On the one hand, the minimum exposure path problem is NP-hard on node-weighted or directed bounded-degree graphs (even when the maximum degree is 4). On the other hand, we present a polynomial algorithm (based on a nontrivial dynamic program) for the problem on unweighted undirected bounded-degree graphs. Likewise, the problem is shown to be polynomial also for the class of (weighted or unweighted) bounded-treewidth graphs

Spectral rigidity of automorphic orbits in free groups

It is well-known that a point $T\in cv_N$ in the (unprojectivized) Culler-Vogtmann Outer space $cv_N$ is uniquely determined by its \emph{translation length function} $||.||_T:F_N\to\mathbb R$. A subset $S$ of a free group $F_N$ is called \emph{spectrally rigid} if, whenever $T,T'\in cv_N$ are such that $||g||_T=||g||_{T'}$ for every $g\in S$ then $T=T'$ in $cv_N$. By contrast to the similar questions for the Teichm\"uller space, it is known that for $N\ge 2$ there does not exist a finite spectrally rigid subset of $F_N$. In this paper we prove that for $N\ge 3$ if $H\le Aut(F_N)$ is a subgroup that projects to an infinite normal subgroup in $Out(F_N)$ then the $H$-orbit of an arbitrary nontrivial element $g\in F_N$ is spectrally rigid. We also establish a similar statement for $F_2=F(a,b)$, provided that $g\in F_2$ is not conjugate to a power of $[a,b]$. We also include an appended corrigendum which gives a corrected proof of Lemma 5.1 about the existence of a fully irreducible element in an infinite normal subgroup of of $Out(F_N)$. Our original proof of Lemma 5.1 relied on a subgroup classification result of Handel-Mosher, originally stated by Handel-Mosher for arbitrary subgroups $H\le Out(F_N)$. After our paper was published, it turned out that the proof of the Handel-Mosher subgroup classification theorem needs the assumption that $H$ be finitely generated. The corrigendum provides an alternative proof of Lemma~5.1 which uses the corrected, finitely generated, version of the Handel-Mosher theorem and relies on the 0-acylindricity of the action of $Out(F_N)$ on the free factor complex (due to Bestvina-Mann-Reynolds). A proof of 0-acylindricity is included in the corrigendum.Comment: Included a corrigendum which gives a corrected proof of Lemma 5.1 about the existence of a fully irreducible element in an infinite normal subgroup of of Out(F_N). Note that, because of the arXiv rules, the corrigendum and the original article are amalgamated into a single pdf file, with the corrigendum appearing first, followed by the main body of the original articl