Streaming Algorithms for Connectivity Augmentation

We study the $k$-connectivity augmentation problem ($k$-CAP) in the single-pass streaming model. Given a $(k-1)$-edge connected graph $G=(V,E)$ that is stored in memory, and a stream of weighted edges $L$ with weights in $\{0,1,\dots,W\}$, the goal is to choose a minimum weight subset $L'\subseteq L$ such that $G'=(V,E\cup L')$ is $k$-edge connected. We give a $(2+\epsilon)$-approximation algorithm for this problem which requires to store $O(\epsilon^{-1} n\log n)$ words. Moreover, we show our result is tight: Any algorithm with better than $2$-approximation for the problem requires $\Omega(n^2)$ bits of space even when $k=2$. This establishes a gap between the optimal approximation factor one can obtain in the streaming vs the offline setting for $k$-CAP. We further consider a natural generalization to the fully streaming model where both $E$ and $L$ arrive in the stream in an arbitrary order. We show that this problem has a space lower bound that matches the best possible size of a spanner of the same approximation ratio. Following this, we give improved results for spanners on weighted graphs: We show a streaming algorithm that finds a $(2t-1+\epsilon)$-approximate weighted spanner of size at most $O(\epsilon^{-1} n^{1+1/t}\log n)$ for integer $t$, whereas the best prior streaming algorithm for spanner on weighted graphs had size depending on $\log W$. Using our spanner result, we provide an optimal $O(t)$-approximation for $k$-CAP in the fully streaming model with $O(nk + n^{1+1/t})$ words of space. Finally we apply our results to network design problems such as Steiner tree augmentation problem (STAP), $k$-edge connected spanning subgraph ($k$-ECSS), and the general Survivable Network Design problem (SNDP). In particular, we show a single-pass $O(t\log k)$-approximation for SNDP using $O(kn^{1+1/t})$ words of space, where $k$ is the maximum connectivity requirement

Approximating the Smallest Spanning Subgraph for 2-Edge-Connectivity in Directed Graphs

Let $G$ be a strongly connected directed graph. We consider the following three problems, where we wish to compute the smallest strongly connected spanning subgraph of $G$ that maintains respectively: the $2$-edge-connected blocks of $G$ (\textsf{2EC-B}); the $2$-edge-connected components of $G$ (\textsf{2EC-C}); both the $2$-edge-connected blocks and the $2$-edge-connected components of $G$ (\textsf{2EC-B-C}). All three problems are NP-hard, and thus we are interested in efficient approximation algorithms. For \textsf{2EC-C} we can obtain a $3/2$-approximation by combining previously known results. For \textsf{2EC-B} and \textsf{2EC-B-C}, we present new $4$-approximation algorithms that run in linear time. We also propose various heuristics to improve the size of the computed subgraphs in practice, and conduct a thorough experimental study to assess their merits in practical scenarios

Approximability of Connected Factors

Finding a d-regular spanning subgraph (or d-factor) of a graph is easy by Tutte's reduction to the matching problem. By the same reduction, it is easy to find a minimal or maximal d-factor of a graph. However, if we require that the d-factor is connected, these problems become NP-hard - finding a minimal connected 2-factor is just the traveling salesman problem (TSP). Given a complete graph with edge weights that satisfy the triangle inequality, we consider the problem of finding a minimal connected $d$-factor. We give a 3-approximation for all $d$ and improve this to an (r+1)-approximation for even d, where r is the approximation ratio of the TSP. This yields a 2.5-approximation for even d. The same algorithm yields an (r+1)-approximation for the directed version of the problem, where r is the approximation ratio of the asymmetric TSP. We also show that none of these minimization problems can be approximated better than the corresponding TSP. Finally, for the decision problem of deciding whether a given graph contains a connected d-factor, we extend known hardness results.Comment: To appear in the proceedings of WAOA 201

Shorter tours and longer detours: Uniform covers and a bit beyond

Motivated by the well known four-thirds conjecture for the traveling salesman problem (TSP), we study the problem of {\em uniform covers}. A graph $G=(V,E)$ has an $\alpha$-uniform cover for TSP (2EC, respectively) if the everywhere $\alpha$ vector (i.e. $\{\alpha\}^{E}$) dominates a convex combination of incidence vectors of tours (2-edge-connected spanning multigraphs, respectively). The polyhedral analysis of Christofides' algorithm directly implies that a 3-edge-connected, cubic graph has a 1-uniform cover for TSP. Seb\H{o} asked if such graphs have $(1-\epsilon)$-uniform covers for TSP for some $\epsilon > 0$. Indeed, the four-thirds conjecture implies that such graphs have 8/9-uniform covers. We show that these graphs have 18/19-uniform covers for TSP. We also study uniform covers for 2EC and show that the everywhere 15/17 vector can be efficiently written as a convex combination of 2-edge-connected spanning multigraphs. For a weighted, 3-edge-connected, cubic graph, our results show that if the everywhere 2/3 vector is an optimal solution for the subtour linear programming relaxation, then a tour with weight at most 27/19 times that of an optimal tour can be found efficiently. Node-weighted, 3-edge-connected, cubic graphs fall into this category. In this special case, we can apply our tools to obtain an even better approximation guarantee. To extend our approach to input graphs that are 2-edge-connected, we present a procedure to decompose an optimal solution for the subtour relaxation for TSP into spanning, connected multigraphs that cover each 2-edge cut an even number of times. Using this decomposition, we obtain a 17/12-approximation algorithm for minimum weight 2-edge-connected spanning subgraphs on subcubic, node-weighted graphs

Spanning trees short or small

We study the problem of finding small trees. Classical network design problems are considered with the additional constraint that only a specified number $k$ of nodes are required to be connected in the solution. A prototypical example is the $k$MST problem in which we require a tree of minimum weight spanning at least $k$ nodes in an edge-weighted graph. We show that the $k$MST problem is NP-hard even for points in the Euclidean plane. We provide approximation algorithms with performance ratio $2\sqrt{k}$ for the general edge-weighted case and $O(k^{1/4})$ for the case of points in the plane. Polynomial-time exact solutions are also presented for the class of decomposable graphs which includes trees, series-parallel graphs, and bounded bandwidth graphs, and for points on the boundary of a convex region in the Euclidean plane. We also investigate the problem of finding short trees, and more generally, that of finding networks with minimum diameter. A simple technique is used to provide a polynomial-time solution for finding $k$-trees of minimum diameter. We identify easy and hard problems arising in finding short networks using a framework due to T. C. Hu.Comment: 27 page