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1. A PTAS for Bounded-Capacity Vehicle Routing in Planar Graphs

   https://doi.org/10.1007/978-3-030-24766-9_8  

   Becker, Amariah; Klein, Philip N; Schild, Aaron (July 2019, Proceedings of the 16th International Symposium on Algorithms and Data Structures, WADS 2019)

   null (Ed.)

   The Capacitated Vehicle Routing problem is to find a minimum-cost set of tours that collectively cover clients in a graph, such that each tour starts and ends at a specified depot and is subject to a capacity bound on the number of clients it can serve. In this paper, we present a polynomial-time approximation scheme (PTAS) for instances in which the input graph is planar and the capacity is bounded. Previously, only a quasipolynomial-time approximation scheme was known for these instances. To obtain this result, we show how to embed planar graphs into bounded-treewidth graphs while preserving, in expectation, the client-to-client distances up to a small additive error proportional to client distances to the depot. 

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2. Optimal Padded Decomposition For Bounded Treewidth Graphs

   https://doi.org/10.46298/theoretics.25.22  

   Filtser, Arnold; Friedrich, Tobias; Issac, Davis; Kumar, Nikhil; Le, Hung; Mallek, Nadym; Zeif, Ziena (October 2025, TheoretiCS)

   A $(β,δ,Δ)$-padded decomposition of an edge-weighted graph $G = (V,E,w)$ is a stochastic decomposition into clusters of diameter at most $$Δ$$ such that for every vertex $$v\in V$$, the probability that $$\rm{ball}_G(v,γΔ)$$ is entirely contained in the cluster containing $$v$$ is at least $$e^{-βγ}$$ for every $$γ\in [0,δ]$$. Padded decompositions have been studied for decades and have found numerous applications, including metric embedding, multicommodity flow-cut gap, multicut, and zero extension problems, to name a few. In these applications, parameter $$β$$, called the padding parameter, is the most important parameter since it decides either the distortion or the approximation ratios. For general graphs with $$n$$ vertices, $$β= Θ(\log n)$$. Klein, Plotkin, and Rao showed that $$K_r$$-minor-free graphs have padding parameter $β= O(r^3)$, which is a significant improvement over general graphs when $$r$$ is a constant. A long-standing conjecture is to construct a padded decomposition for $$K_r$$-minor-free graphs with padding parameter $$β= O(\log r)$$. Despite decades of research, the best-known result is $β= O(r)$, even for graphs with treewidth at most $$r$$. In this work, we make significant progress toward the aforementioned conjecture by showing that graphs with treewidth $$\rm{tw}$$ admit a padded decomposition with padding parameter $$O(\log \rm{tw})$$, which is tight. As corollaries, we obtain an exponential improvement in dependency on treewidth in a host of algorithmic applications: $$O(\sqrt{ \log n \cdot \log(\rm{tw})})$$ flow-cut gap, max flow-min multicut ratio of $$O(\log(\rm{tw}))$$, an $$O(\log(\rm{tw}))$$ approximation for the 0-extension problem, an $$\ell^{O(\log n)}_\infty$$ embedding with distortion $$O(\log \rm{tw})$$, and an $$O(\log \rm{tw})$$ bound for integrality gap for the uniform sparsest cut. 39 pages. This is the TheoretiCS journal version 

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3. Embedding Planar Graphs into Low-Treewidth Graphs with Applications to Efficient Approximation Schemes for Metric Problems

   https://doi.org/10.1137/1.9781611975482.66  

   Eli Fox-Epstein, Eli Klein (March 2019, Proceedings of the Thirtieth Annual {ACM-SIAM} Symposium on Discrete Algorithms)

   We show that, for any ∊ > 0, there is a deterministic embedding of edge-weighted planar graphs of diameter D into bounded-treewidth graphs. The embedding has additive error ∊D. We use this construction to obtain the first efficient bicriteria approximation schemes for weighted planar graphs addressing k-Center (equivalently d-Domination), and a metric generalization of independent set, d-independent SET. The approximation schemes employ a metric generalization of Baker's framework that is based on our embedding result. 

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4. Bandwidth vs BFS Width in Matrix Reordering, Graph Reconstruction, and Graph Drawing

   https://doi.org/10.4230/lipics.esa.2025.69  

   Eppstein, David; Goodrich, Michael T; Liu, Songyu Alfred (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Benoit, Anne; Kaplan, Haim; Wild, Sebastian; Herman, Grzegorz (Ed.)

   We provide the first approximation quality guarantees for the Cuthull-McKee heuristic for reordering symmetric matrices to have low bandwidth, and we provide an algorithm for reconstructing bounded-bandwidth graphs from distance oracles with near-linear query complexity. To prove these results we introduce a new width parameter, BFS width, and we prove polylogarithmic upper and lower bounds on the BFS width of graphs of bounded bandwidth. Unlike other width parameters, such as bandwidth, pathwidth, and treewidth, BFS width can easily be computed in polynomial time. Bounded BFS width implies bounded bandwidth, pathwidth, and treewidth, which in turn imply fixed-parameter tractable algorithms for many problems that are NP-hard for general graphs. In addition to their applications to matrix ordering, we also provide applications of BFS width to graph reconstruction, to reconstruct graphs from distance queries, and graph drawing, to construct arc diagrams of small height. 

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5. Induced subgraphs and tree decompositions XIII. Basic obstructions in H-free graphs for finite H

   https://doi.org/10.19086/aic.2024.6  

   Alecu, Bogdan; Chudnovsky, Maria; Hajebi, Sepehr; Spirkl, Sophie (November 2024, Advances in Combinatorics)

   Unlike minors, the induced subgraph obstructions to bounded treewidth come in a large variety, including, for every t ∈ N, the t-basic obstructions: the graphs Kt+1 and Kt,t, along with the subdivisions of the t-by-t wall and their line graphs. But this list is far from complete. The simplest example of a “non-basic” obstruction is due to Pohoata and Davies (independently). For every n ∈ N, they construct certain graphs of treewidth n and with no 3-basic obstruction as an induced subgraph, which we call n-arrays. Let us say a graph class G is clean if the only obstructions to bounded treewidth in G are in fact the basic ones. It follows that a full description of the induced subgraph obstructions to bounded treewidth is equivalent to a characterization of all families H of graphs for which the class of all H-free graphs is clean (a graph G is H-free if no induced subgraph of G is isomorphic to any graph in H). This remains elusive, but there is an immediate necessary condition: if H-free graphs are clean, then there are only finitely many n ∈ N such that there is an n-array which is H-free. The above necessary condition is not sufficient in general. However, the situation turns out to be different if H is finite: we prove that for every finite set H of graphs, the class of all H-free graphs is clean if and only if there is no H-free n-array except possibly for finitely many values of n. 

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