IA Scholar Query: Finding Light Spanners in Bounded Pathwidth Graphs
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Internet Archive Scholar query results feedeninfo@archive.orgThu, 15 Sep 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Parameterized algorithmics for time-evolving structures: temporalizing and multistaging
https://scholar.archive.org/work/uyaoernzvvb7rmwysi2x6b5yzi
The thesis studies temporal graph problems and multistage problems. Since these problems typically are computationally hard, the focus is on developing fast exact (FPT-)algorithms. Temporal graph problems. A temporal graph is a graph whose edge set changes over time. Here, an edge at a specific time step is called time-edge. One of our main contributions is the introduction of a set of parameters tailored for temporal graph problems. We focus mainly on four problems on temporal graphs. Minimizing Reachability by Delaying. Given a temporal graph, a set of source vertices, and three integers k, r, and δ, the problem Minimizing Temporal Reachability by Delaying asks whether we can delay at most k time-edges by δ time steps (i.e., moving the edges δ time steps into the future) such that the sources can reach at most r vertices via temporal paths (i.e., paths using edges appearing in non-decreasing time-order). Our main contribution here is an algorithm running in O(r!k|G|) time, where |G| is the size of the temporal graph. This stands in contrast to the W[1]-hardness when parameterized by r for the problem of deleting instead of delaying time-edges. Restless Temporal Paths. A restless temporal path is a temporal path that can stay only a bounded amount of time at one vertex. Our main contribution here is a randomized algorithm to find a length-at-most-k restless temporal path from vertex s to vertex z in 4^ℓ |G|^O(1) time, where ℓ is the difference between k and the length of the shortest temporal path from s to z. Moreover, we show that finding these restless temporal paths is fixed-parameter tractable when parameterized by the timed feedback vertex number (that is, a temporal version of the classical feedback vertex number introduced in this thesis). This stands in contrast to the W[1]-hardness when parameterized by the feedback vertex number of the underlying graph. Temporal Separation. A temporal separator is a vertex set that intersects the vertices of all temporal paths between two distinguished vertices. We co [...]Philipp Zschoche, Technische Universität Berlin, Rolf Niedermeierwork_uyaoernzvvb7rmwysi2x6b5yziThu, 15 Sep 2022 00:00:00 GMTLow Treewidth Embeddings of Planar and Minor-Free Metrics
https://scholar.archive.org/work/rplsrxdjojh7rknmvn2nr3mqsy
Cohen-Addad, Filtser, Klein and Le [FOCS'20] constructed a stochastic embedding of minor-free graphs of diameter D into graphs of treewidth O_ϵ(log n) with expected additive distortion +ϵ D. Cohen-Addad et al. then used the embedding to design the first quasi-polynomial time approximation scheme (QPTAS) for the capacitated vehicle routing problem. Filtser and Le [STOC'21] used the embedding (in a different way) to design a QPTAS for the metric Baker's problems in minor-free graphs. In this work, we devise a new embedding technique to improve the treewidth bound of Cohen-Addad et al. exponentially to O_ϵ(loglog n)^2. As a corollary, we obtain the first efficient PTAS for the capacitated vehicle routing problem in minor-free graphs. We also significantly improve the running time of the QPTAS for the metric Baker's problems in minor-free graphs from n^O_ϵ(log(n)) to n^O_ϵ(loglog(n))^3. Applying our embedding technique to planar graphs, we obtain a deterministic embedding of planar graphs of diameter D into graphs of treewidth O((loglog n)^2)/ϵ) and additive distortion +ϵ D that can be constructed in nearly linear time. Important corollaries of our result include a bicriteria PTAS for metric Baker's problems and a PTAS for the vehicle routing problem with bounded capacity in planar graphs, both run in almost-linear time. The running time of our algorithms is significantly better than previous algorithms that require quadratic time. A key idea in our embedding is the construction of an (exact) emulator for tree metrics with treewidth O(loglog n) and hop-diameter O(loglog n). This result may be of independent interest.Arnold Filtser, Hung Lework_rplsrxdjojh7rknmvn2nr3mqsyTue, 29 Mar 2022 00:00:00 GMT