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Register a new userPolynomial Time $k$-Shortest Multi-Criteria Prioritized and All-Criteria-Disjoint Paths
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Added by
Manish Kumar
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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The shortest secure path (routing) problem in communication networks has to deal with multiple attack layers e.g., man-in-the-middle, eavesdropping, packet injection, packet insertion, etc. Consider different probabilities for each such attack over an edge, probabilities that can differ across edges. Furthermore, usage of a single shortest path (for routing) implies possible traffic bottleneck, which should be avoided if possible, which we term pathneck security avoidance. Finding all Pareto-optimal solutions for the multi-criteria single-source single-destination shortest secure path problem with non-negative edge lengths might yield a solution with an exponential number of paths. In the first part of this paper, we study specific settings of the multi-criteria shortest secure path problem, which are based on prioritized multi-criteria and on $k$-shortest secure paths. In the second part, we show a polynomial-time algorithm that, given an undirected graph $G$ and a pair of vertices $(s,t)$, finds prioritized multi-criteria $2$-disjoint (vertex/edge) shortest secure paths between $s$ and $t$. In the third part of the paper, we introduce the $k$-disjoint all-criteria-shortest secure paths problem, which is solved in time $O(min(k|E|, |E|^{3/2}))$.
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We study the decremental All-Pairs Shortest Paths (APSP) problem in undirected edge-weighted graphs. The input to the problem is an $n$-vertex $m$-edge graph $G$ with non-negative edge lengths, that undergoes a sequence of edge deletions. The goal is to support approximate shortest-path queries: given a pair $x,y$ of vertices of $G$, return a path $P$ connecting $x$ to $y$, whose length is within factor $alpha$ of the length of the shortest $x$-$y$ path, in time $tilde O(|E(P)|)$, where $alpha$ is the approximation factor of the algorithm. APSP is one of the most basic and extensively studied dynamic graph problems. A long line of work culminated in the algorithm of [Chechik, FOCS 2018] with near optimal guarantees for the oblivious-adversary setting. Unfortunately, adaptive-adversary setting is still poorly understood. For unweighted graphs, the algorithm of [Henzinger, Krinninger and Nanongkai, FOCS 13, SICOMP 16] achieves a $(1+epsilon)$-approximation with total update time $tilde O(mn/epsilon)$; the best current total update time of $n^{2.5+O(epsilon)}$ is achieved by the deterministic algorithm of [Chuzhoy, Saranurak, SODA21], with $2^{O(1/epsilon)}$-multiplicative and $2^{O(log^{3/4}n/epsilon)}$-additive approximation. To the best of our knowledge, for arbitrary non-negative edge weights, the fastest current adaptive-update algorithm has total update time $O(n^{3}log L/epsilon)$, achieving a $(1+epsilon)$-approximation. Here, L is the ratio of longest to shortest edge lengths. Our main result is a deterministic algorithm for decremental APSP in undirected edge-weighted graphs, that, for any $Omega(1/loglog m)leq epsilon< 1$, achieves approximation factor $(log m)^{2^{O(1/epsilon)}}$, with total update time $Oleft (m^{1+O(epsilon)}cdot (log m)^{O(1/epsilon^2)}cdot log Lright )$.
Given a directed weighted graph $G=(V,E)$ undergoing vertex insertions emph{and} deletions, the All-Pairs Shortest Paths (APSP) problem asks to maintain a data structure that processes updates efficiently and returns after each update the distance matrix to the current version of $G$. In two breakthrough results, Italiano and Demetrescu [STOC 03] presented an algorithm that requires $tilde{O}(n^2)$ emph{amortized} update time, and Thorup showed in [STOC 05] that emph{worst-case} update time $tilde{O}(n^{2+3/4})$ can be achieved. In this article, we make substantial progress on the problem. We present the following new results: (1) We present the first deterministic data structure that breaks the $tilde{O}(n^{2+3/4})$ worst-case update time bound by Thorup which has been standing for almost 15 years. We improve the worst-case update time to $tilde{O}(n^{2+5/7}) = tilde{O}(n^{2.71..})$ and to $tilde{O}(n^{2+3/5}) = tilde{O}(n^{2.6})$ for unweighted graphs. (2) We present a simple deterministic algorithm with $tilde{O}(n^{2+3/4})$ worst-case update time ($tilde{O}(n^{2+2/3})$ for unweighted graphs), and a simple Las-Vegas algorithm with worst-case update time $tilde{O}(n^{2+2/3})$ ($tilde{O}(n^{2 + 1/2})$ for unweighted graphs) that works against a non-oblivious adversary. Both data structures require space $tilde{O}(n^2)$. These are the first exact dynamic algorithms with truly-subcubic update time emph{and} space usage. This makes significant progress on an open question posed in multiple articles [COCOON01, STOC 03, ICALP04, Encyclopedia of Algorithms 08] and is critical to algorithms in practice [TALG 06] where large space usage is prohibitive. Moreover, they match the worst-case update time of the best previous algorithms and the second algorithm improves upon a Monte-Carlo algorithm in a weaker adversary model with the same running time [SODA 17].
A localized method to distribute paths on random graphs is devised, aimed at finding the shortest paths between given source/destination pairs while avoiding path overlaps at nodes. We propose a method based on message-passing techniques to process global information and distribute paths optimally. Statistical properties such as scaling with system size and number of paths, average path-length and the transition to the frustrated regime are analysed. The performance of the suggested algorithm is evaluated through a comparison against a greedy algorithm.
In the Disjoint Paths problem, the input is an undirected graph $G$ on $n$ vertices and a set of $k$ vertex pairs, ${s_i,t_i}_{i=1}^k$, and the task is to find $k$ pairwise vertex-disjoint paths connecting $s_i$ to $t_i$. The problem was shown to have an $f(k)n^3$ algorithm by Robertson and Seymour. In modern terminology, this means that Disjoint Paths is fixed parameter tractable (FPT), parameterized by the number of vertex pairs. This algorithm is the cornerstone of the entire graph minor theory, and a vital ingredient in the $g(k)n^3$ algorithm for Minor Testing (given two undirected graphs, $G$ and $H$ on $n$ and $k$ vertices, respectively, the objective is to check whether $G$ contains $H$ as a minor). All we know about $f$ and $g$ is that these are computable functions. Thus, a challenging open problem in graph algorithms is to devise an algorithm for Disjoint Paths where $f$ is single exponential. That is, $f$ is of the form $2^{{sf poly}(k)}$. The algorithm of Robertson and Seymour relies on topology and essentially reduces the problem to surface-embedded graphs. Thus, the first major obstacle that has to be overcome in order to get an algorithm with a single exponential running time for Disjoint Paths and {sf Minor Testing} on general graphs is to solve Disjoint Paths in single exponential time on surface-embedded graphs and in particular on planar graphs. Even when the inputs to Disjoint Paths are restricted to planar graphs, a case called the Planar Disjoint Paths problem, the best known algorithm has running time $2^{2^{O(k)}}n^2$. In this paper, we make the first step towards our quest for designing a single exponential time algorithm for Disjoint Paths by giving a $2^{O(k^2)}n^{O(1)}$-time algorithm for Planar Disjoint Paths.
Consider the following distance query for an $n$-node graph $G$ undergoing edge insertions and deletions: given two sets of nodes $I$ and $J$, return the distances between every pair of nodes in $Itimes J$. This query is rather general and captures sever
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