Eight-Fifth Approximation for TSP Paths

We prove the approximation ratio 8/5 for the metric $\{s,t\}$-path-TSP problem, and more generally for shortest connected $T$-joins. The algorithm that achieves this ratio is the simple "Best of Many" version of Christofides' algorithm (1976), suggested by An, Kleinberg and Shmoys (2012), which consists in determining the best Christofides $\{s,t\}$-tour out of those constructed from a family \Fscr_{>0} of trees having a convex combination dominated by an optimal solution $x^*$ of the fractional relaxation. They give the approximation guarantee $\frac{\sqrt{5}+1}{2}$ for such an $\{s,t\}$-tour, which is the first improvement after the 5/3 guarantee of Hoogeveen's Christofides type algorithm (1991). Cheriyan, Friggstad and Gao (2012) extended this result to a 13/8-approximation of shortest connected $T$-joins, for $|T|\ge 4$. The ratio 8/5 is proved by simplifying and improving the approach of An, Kleinberg and Shmoys that consists in completing $x^*/2$ in order to dominate the cost of "parity correction" for spanning trees. We partition the edge-set of each spanning tree in \Fscr_{>0} into an $\{s,t\}$-path (or more generally, into a $T$-join) and its complement, which induces a decomposition of $x^*$. This decomposition can be refined and then efficiently used to complete $x^*/2$ without using linear programming or particular properties of $T$, but by adding to each cut deficient for $x^*/2$ an individually tailored explicitly given vector, inherent in $x^*$. A simple example shows that the Best of Many Christofides algorithm may not find a shorter $\{s,t\}$-tour than 3/2 times the incidentally common optima of the problem and of its fractional relaxation.Comment: 15 pages, corrected typos in citations, minor change

Distributed Connectivity Decomposition

We present time-efficient distributed algorithms for decomposing graphs with large edge or vertex connectivity into multiple spanning or dominating trees, respectively. As their primary applications, these decompositions allow us to achieve information flow with size close to the connectivity by parallelizing it along the trees. More specifically, our distributed decomposition algorithms are as follows: (I) A decomposition of each undirected graph with vertex-connectivity $k$ into (fractionally) vertex-disjoint weighted dominating trees with total weight $\Omega(\frac{k}{\log n})$, in $\widetilde{O}(D+\sqrt{n})$ rounds. (II) A decomposition of each undirected graph with edge-connectivity $\lambda$ into (fractionally) edge-disjoint weighted spanning trees with total weight $\lceil\frac{\lambda-1}{2}\rceil(1-\varepsilon)$, in $\widetilde{O}(D+\sqrt{n\lambda})$ rounds. We also show round complexity lower bounds of $\tilde{\Omega}(D+\sqrt{\frac{n}{k}})$ and $\tilde{\Omega}(D+\sqrt{\frac{n}{\lambda}})$ for the above two decompositions, using techniques of [Das Sarma et al., STOC'11]. Moreover, our vertex-connectivity decomposition extends to centralized algorithms and improves the time complexity of [Censor-Hillel et al., SODA'14] from $O(n^3)$ to near-optimal $\tilde{O}(m)$. As corollaries, we also get distributed oblivious routing broadcast with $O(1)$-competitive edge-congestion and $O(\log n)$-competitive vertex-congestion. Furthermore, the vertex connectivity decomposition leads to near-time-optimal $O(\log n)$-approximation of vertex connectivity: centralized $\widetilde{O}(m)$ and distributed $\tilde{O}(D+\sqrt{n})$. The former moves toward the 1974 conjecture of Aho, Hopcroft, and Ullman postulating an $O(m)$ centralized exact algorithm while the latter is the first distributed vertex connectivity approximation

Streaming Complexity of Spanning Tree Computation

The semi-streaming model is a variant of the streaming model frequently used for the computation of graph problems. It allows the edges of an n-node input graph to be read sequentially in p passes using Õ(n) space. If the list of edges includes deletions, then the model is called the turnstile model; otherwise it is called the insertion-only model. In both models, some graph problems, such as spanning trees, k-connectivity, densest subgraph, degeneracy, cut-sparsifier, and (Δ+1)-coloring, can be exactly solved or (1+ε)-approximated in a single pass; while other graph problems, such as triangle detection and unweighted all-pairs shortest paths, are known to require Ω̃(n) passes to compute. For many fundamental graph problems, the tractability in these models is open. In this paper, we study the tractability of computing some standard spanning trees, including BFS, DFS, and maximum-leaf spanning trees. Our results, in both the insertion-only and the turnstile models, are as follows. Maximum-Leaf Spanning Trees: This problem is known to be APX-complete with inapproximability constant ρ ∈ [245/244, 2). By constructing an ε-MLST sparsifier, we show that for every constant ε > 0, MLST can be approximated in a single pass to within a factor of 1+ε w.h.p. (albeit in super-polynomial time for ε ≤ ρ-1 assuming P ≠ NP) and can be approximated in polynomial time in a single pass to within a factor of ρ_n+ε w.h.p., where ρ_n is the supremum constant that MLST cannot be approximated to within using polynomial time and Õ(n) space. In the insertion-only model, these algorithms can be deterministic. BFS Trees: It is known that BFS trees require ω(1) passes to compute, but the naïve approach needs O(n) passes. We devise a new randomized algorithm that reduces the pass complexity to O(√n), and it offers a smooth tradeoff between pass complexity and space usage. This gives a polynomial separation between single-source and all-pairs shortest paths for unweighted graphs. DFS Trees: It is unknown whether DFS trees require more than one pass. The current best algorithm by Khan and Mehta [STACS 2019] takes Õ(h) passes, where h is the height of computed DFS trees. Note that h can be as large as Ω(m/n) for n-node m-edge graphs. Our contribution is twofold. First, we provide a simple alternative proof of this result, via a new connection to sparse certificates for k-node-connectivity. Second, we present a randomized algorithm that reduces the pass complexity to O(√n), and it also offers a smooth tradeoff between pass complexity and space usage.ISSN:1868-896

Near-Optimal Distributed Approximation of Minimum-Weight Connected Dominating Set

This paper presents a near-optimal distributed approximation algorithm for the minimum-weight connected dominating set (MCDS) problem. The presented algorithm finds an $O(\log n)$ approximation in $\tilde{O}(D+\sqrt{n})$ rounds, where $D$ is the network diameter and $n$ is the number of nodes. MCDS is a classical NP-hard problem and the achieved approximation factor $O(\log n)$ is known to be optimal up to a constant factor, unless P=NP. Furthermore, the $\tilde{O}(D+\sqrt{n})$ round complexity is known to be optimal modulo logarithmic factors (for any approximation), following [Das Sarma et al.---STOC'11].Comment: An extended abstract version of this result appears in the proceedings of 41st International Colloquium on Automata, Languages, and Programming (ICALP 2014