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

Charting the Algorithmic Complexity of Waypoint Routing

Modern computer networks support interesting new routing models in which traffic flows from a source sto a destination t can be flexibly steered through a sequence of waypoints, such as (hardware) middleboxes or (virtualized) network functions (VNFs), to create innovative network services like service chains or segment routing. While the benefits and technological challenges of providing such routing models have been articulated and studied intensively over the last years, less is known about the underlying algorithmic traffic routing problems. The goal of this paper is to provide the network community with an overview of algorithmic techniques for waypoint routing and also inform about limitations due to computational hardness. In particular, we put the waypoint routing problem into perspective with respect to classic graph theoretical problems. For example, we find that while computing a shortest path from a source s to a destination t is simple (e.g., using Dijkstra's algorithm), the problem of finding a shortest route from s to t via a single waypoint already features a deep combinatorial structure.</jats:p

Challenges for Efficient Query Evaluation on Structured Probabilistic Data

Query answering over probabilistic data is an important task but is generally intractable. However, a new approach for this problem has recently been proposed, based on structural decompositions of input databases, following, e.g., tree decompositions. This paper presents a vision for a database management system for probabilistic data built following this structural approach. We review our existing and ongoing work on this topic and highlight many theoretical and practical challenges that remain to be addressed.Comment: 9 pages, 1 figure, 23 references. Accepted for publication at SUM 201

Algebraic Methods in the Congested Clique

In this work, we use algebraic methods for studying distance computation and subgraph detection tasks in the congested clique model. Specifically, we adapt parallel matrix multiplication implementations to the congested clique, obtaining an $O(n^{1-2/\omega})$ round matrix multiplication algorithm, where $\omega < 2.3728639$ is the exponent of matrix multiplication. In conjunction with known techniques from centralised algorithmics, this gives significant improvements over previous best upper bounds in the congested clique model. The highlight results include: -- triangle and 4-cycle counting in $O(n^{0.158})$ rounds, improving upon the $O(n^{1/3})$ triangle detection algorithm of Dolev et al. [DISC 2012], -- a $(1 + o(1))$-approximation of all-pairs shortest paths in $O(n^{0.158})$ rounds, improving upon the $\tilde{O} (n^{1/2})$-round $(2 + o(1))$-approximation algorithm of Nanongkai [STOC 2014], and -- computing the girth in $O(n^{0.158})$ rounds, which is the first non-trivial solution in this model. In addition, we present a novel constant-round combinatorial algorithm for detecting 4-cycles.Comment: This is work is a merger of arxiv:1412.2109 and arxiv:1412.266

Minimizing and Computing the Inverse Geodesic Length on Trees

For any fixed measure H that maps graphs to real numbers, the MinH problem is defined as follows: given a graph G, an integer k, and a target tau, is there a set S of k vertices that can be deleted, so that H(G - S) is at most tau? In this paper, we consider the MinH problem on trees. We call H balanced on trees if, whenever G is a tree, there is an optimal choice of S such that the components of G - S have sizes bounded by a polynomial in n / k. We show that MinH on trees is Fixed-Parameter Tractable (FPT) for parameter n / k, and furthermore, can be solved in subexponential time, and polynomial space, whenever H is additive, balanced on trees, and computable in polynomial time. A particular measure of interest is the Inverse Geodesic Length (IGL), which is used to gauge the efficiency and connectedness of a graph. It is defined as the sum of inverse distances between every two vertices: IGL(G) = sum_{{u,v} subseteq V} 1/d_G(u,v). While MinIGL is W[1]-hard for parameter treewidth, and cannot be solved in 2^{o(k + n + m)} time, even on bipartite graphs with n vertices and m edges, the complexity status of the problem remains open in the case where G is a tree. We show that IGL is balanced on trees, to give a 2^O((n log n)^(5/6)) time, polynomial space algorithm. The distance distribution of G is the sequence {a_i} describing the number of vertex pairs distance i apart in G: a_i = |{{u, v}: d_G(u, v) = i}|. Given only the distance distribution, one can easily determine graph parameters such as diameter, Wiener index, and particularly, the IGL. We show that the distance distribution of a tree can be computed in O(n log^2 n) time by reduction to polynomial multiplication. We also extend the result to graphs with small treewidth by showing that the first p values of the distance distribution can be computed in 2^(O(tw(G))) n^(1 + epsilon) sqrt(p) time, and the entire distance distribution can be computed in 2^(O(tw(G))) n^{1 + epsilon} time, when the diameter of G is O(n^epsilon\u27) for every epsilon\u27 > 0