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Register a new userNear Optimal Sized Weight Tolerant Subgraph for Single Source Shortest Path
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Added by
Diptarka Chakraborty
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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In this paper we address the problem of computing a sparse subgraph of a weighted directed graph such that the exact distances from a designated source vertex to all other vertices are preserved under bounded weight increment. Finding a small sized subgraph that preserves distances between any pair of vertices is a well studied problem. Since in the real world any network is prone to failures, it is natural to study the fault tolerant version of the above problem. Unfortunately, it turns out that there may not always exist such a sparse subgraph even under single edge failure [Demetrescu emph{et al.} 08]. However in real applications it is not always the case that a link (edge) in a network becomes completely faulty. Instead, it can happen that some links become more congested which can easily be captured by increasing weight on the corresponding edges. Thus it makes sense to try to construct a sparse distance preserving subgraph under the above weight increment model. To the best of our knowledge this problem has not been studied so far. In this paper we show that given any weighted directed graph with $n$ vertices and a source vertex, one can construct a subgraph that contains at most $e cdot (k-1)!2^kn$ many edges such that it preserves distances between the source and all other vertices as long as the total weight increment is bounded by $k$ and we are allowed to have only integer valued (can be negative) weight on each edge and also weight of an edge can only be increased by some positive integer. Next we show a lower bound of $ccdot 2^kn$, for some constant $c ge 5/4$, on the size of the subgraph. We also argue that restriction of integer valued weight and integer valued weight increment are actually essential by showing that if we remove any one of these two restrictions we may need to store $Omega(n^2)$ edges to preserve distances.
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In this paper we consider the decremental single-source shortest paths (SSSP) problem, where given a graph $G$ and a source node $s$ the goal is to maintain shortest distances between $s$ and all other nodes in $G$ under a sequence of online adversarial edge deletions. In their seminal work, Even and Shiloach [JACM 1981] presented an exact solution to the problem in unweighted graphs with only $O(mn)$ total update time over all edge deletions. Their classic algorithm was the state of the art for the decremental SSSP problem for three decades, even when approximate shortest paths are allowed. A series of results showed how to improve upon $O(mn)$ if approximation is allowed, culminating in a recent breakthrough of Henzinger, Krinninger and Nanongkai [FOCS 14], who presented a $(1+epsilon)$-approximate algorithm for undirected weighted graphs whose total update time is near linear: $O(m^{1+o(1)}log(W))$, where $W$ is the ratio of the heaviest to the lightest edge weight in the graph. In this paper they posed as a major open problem the question of derandomizing their result. Until very recently, all known improvements over the Even-Shiloach algorithm were randomized and required the assumption of a non-adaptive adversary. In STOC 2016, Bernstein and Chechik showed the first emph{deterministic} algorithm to go beyond $O(mn)$ total update time: the algorithm is also $(1+epsilon)$-approximate, and has total update time $tilde{O}(n^2)$. In SODA 2017, the same authors presented an algorithm with total update time $tilde{O}(mn^{3/4})$. However, both algorithms are restricted to undirected, unweighted graphs. We present the emph{first} deterministic algorithm for emph{weighted} undirected graphs to go beyond the $O(mn)$ bound. The total update time is $tilde{O}(n^2 log(W))$.
In two-stage robust optimization the solution to a problem is built in two stages: In the first stage a partial, not necessarily feasible, solution is exhibited. Then the adversary chooses the worst scenario from a predefined set of scenarios. In the second stage, the first-stage solution is extended to become feasible for the chosen scenario. The costs at the second stage are larger than at the first one, and the objective is to minimize the total cost paid in the two stages. We give a 2-approximation algorithm for the robust mincut problem and a ({gamma}+2)-approximation for the robust shortest path problem, where {gamma} is the approximation ratio for the Steiner tree. This improves the factors (1+sqrt2) and 2({gamma}+2) from [Golovin, Goyal and Ravi. Pay today for a rainy day: Improved approximation algorithms for demand-robust min-cut and shortest path problems. STACS 2006]. In addition, our solution for robust shortest path is simpler and more efficient than the earlier ones; this is achieved by a more direct algorithm and analysis, not using some of the standard demand-robust optimization techniques.
The minimum-weight $2$-edge-connected spanning subgraph (2-ECSS) problem is a natural generalization of the well-studied minimum-weight spanning tree (MST) problem, and it has received considerable attention in the area of network design. The latter problem asks for a minimum-weight subgraph with an edge connectivity of $1$ between each pair of vertices while the former strengthens this edge-connectivity requirement to $2$. Despite this resemblance, the 2-ECSS problem is considerably more complex than MST. While MST admits a linear-time centralized exact algorithm, 2-ECSS is NP-hard and the best known centralized approximation algorithm for it (that runs in polynomial time) gives a $2$-approximation. In this paper, we give a deterministic distributed algorithm with round complexity of $widetilde{O}(D+sqrt{n})$ that computes a $(5+epsilon)$-approximation of 2-ECSS, for any constant $epsilon>0$. Up to logarithmic factors, this complexity matches the $widetilde{Omega}(D+sqrt{n})$ lower bound that can be derived from Das Sarma et al. [STOC11], as shown by Censor-Hillel and Dory [OPODIS17]. Our result is the first distributed constant approximation for 2-ECSS in the nearly optimal time and it improves on a recent randomized algorithm of Dory [PODC18], which achieved an $O(log n)$-approximation in $widetilde{O}(D+sqrt{n})$ rounds. We also present an alternative algorithm for $O(log n)$-approximation, whose round complexity is linear in the low-congestion shortcut parameter of the network, following a framework introduced by Ghaffari and Haeupler [SODA16]. This algorithm has round complexity $widetilde{O}(D+sqrt{n})$ in worst-case networks but it provably runs much faster in many well-behaved graph families of interest. For instance, it runs in $widetilde{O}(D)$ time in planar networks and those with bounded genus, bounded path-width or bounded tree-width.
In this thesis, we present new techniques to deal with fundamental algorithmic graph problems where graphs are directed and partially dynamic, i.e. undergo either a sequence of edge insertions or deletions: - Single-Source Reachability (SSR), - Strongly-Connected Components (SCCs), and - Single-Source Shortest Paths (SSSP). These problems have recently received an extraordinary amount of attention due to their role as subproblems in various more complex and notoriously hard graph problems, especially to compute flows, bipartite matchings and cuts. Our techniques lead to the first near-optimal data structures for these problems in various different settings. Letting $n$ denote the number of vertices in the graph and by $m$ the maximum number of edges in any version of the graph, we obtain - the first randomized data structure to maintain SSR and SCCs in near-optimal total update time $tilde{O}(m)$ in a graph undergoing edge deletions. - the first randomized data structure to maintain SSSP in partially dynamic graphs in total update time $tilde{O}(n^2)$ which is near-optimal in dense graphs. - the first deterministic data structures for SSR and SCC for graphs undergoing edge deletions, and for SSSP in partially dynamic graphs that improve upon the $O(mn)$ total update time by Even and Shiloach from 1981 that is often considered to be a fundamental barrier.
The determination of time-dependent collision-free shortest paths has received a fair amount of attention. Here, we study the problem of computing a time-dependent shortest path among growing discs which has been previously studied for the instance where the departure times are fixed. We address a more general setting: For two given points $s$ and $d$, we wish to determine the function $mathcal{A}(t)$ which is the minimum arrival time at $d$ for any departure time $t$ at $s$. We present a $(1+epsilon)$-approximation algorithm for computing $mathcal{A}(t)$. As part of preprocessing, we execute $O({1 over epsilon} log({mathcal{V}_{r} over mathcal{V}_{c}}))$ shortest path computations for fixed departure times, where $mathcal{V}_{r}$ is the maximum speed of the robot and $mathcal{V}_{c}$ is the minimum growth rate of the discs. For any query departure time $t geq 0$ from $s$, we can approximate the minimum arrival time at the destination in $O(log ({1 over epsilon}) + loglog({mathcal{V}_{r} over mathcal{V}_{c}}))$ time, within a factor of $1+epsilon$ of optimal. Since we treat the shortest path computations as black-box functions, for different settings of growing discs, we can plug-in different shortest path algorithms. Thus, the exact time complexity of our algorithm is determined by the running time of the shortest path computations.
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