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1. On Light Spanners, Low-treewidth Embeddings and Efficient Traversing in Minor-free Graphs

   Cohen-Addad, Vincent; Filtser, Arnold; Klein, Philip N; Le, Hung (October 2020, Proceedings of the 61st Annual IEEE Symposium on Foundations of Computer Science)

   null (Ed.)

   Understanding the structure of minor-free metrics, namely shortest path metrics obtained over a weighted graph excluding a fixed minor, has been an important research direction since the fundamental work of Robertson and Seymour. A fundamental idea that helps both to understand the structural properties of these metrics and lead to strong algorithmic results is to construct a “small-complexity” graph that approximately preserves distances between pairs of points of the metric. We show the two following structural results for minor-free metrics: 1) Construction of a light subset spanner. Given a subset of vertices called terminals, and ϵ, in polynomial time we construct a sub graph that preserves all pairwise distances between terminals up to a multiplicative 1+ϵ factor, of total weight at most Oϵ(1) times the weight of the minimal Steiner tree spanning the terminals. 2) Construction of a stochastic metric embedding into low treewidth graphs with expected additive distortion ϵD. Namely, given a minor-free graph G=(V,E,w) of diameter D, and parameter ϵ, we construct a distribution D over dominating metric embeddings into treewidth-Oϵ(log n) graphs such that ∀u,v∈V, Ef∼D[dH(f(u),f(v))]≤dG(u,v)+ϵD. Our results have the following algorithmic consequences: (1) the first efficient approximation scheme for subset TSP in minor-free metrics; (2) the first approximation scheme for bounded-capacity vehicle routing in minor-free metrics; (3) the first efficient approximation scheme for bounded-capacity vehicle routing on bounded genus metrics. En route to the latter result, we design the first FPT approximation scheme for bounded-capacity vehicle routing on bounded-treewidth graphs (parameterized by the treewidth). 

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2. C-Planarity Testing of Embedded Clustered Graphs with Bounded Dual Carving-Width

   https://doi.org/10.1007/s00453-021-00839-2  

   Da Lozzo, Giordano; Eppstein, David; Goodrich, Michael T.; Gupta, Siddharth (August 2021, Algorithmica)

   Abstract For a clustered graph , i.e, a graph whose vertex set is recursively partitioned into clusters, the C-Planarity Testing problem asks whether it is possible to find a planar embedding of the graph and a representation of each cluster as a region homeomorphic to a closed disk such that (1) the subgraph induced by each cluster is drawn in the interior of the corresponding disk, (2) each edge intersects any disk at most once, and (3) the nesting between clusters is reflected by the representation, i.e., child clusters are properly contained in their parent cluster. The computational complexity of this problem, whose study has been central to the theory of graph visualization since its introduction in 1995 [Feng, Cohen, and Eades, Planarity for clustered graphs , ESA’95], has only been recently settled [Fulek and Tóth, Atomic Embeddability, Clustered Planarity, and Thickenability , to appear at SODA’20]. Before such a breakthrough, the complexity question was still unsolved even when the graph has a prescribed planar embedding, i.e, for embedded clustered graphs . We show that the C-Planarity Testing problem admits a single-exponential single-parameter FPT (resp., XP) algorithm for embedded flat (resp., non-flat) clustered graphs, when parameterized by the carving-width of the dual graph of the input. These are the first FPT and XP algorithms for this long-standing open problem with respect to a single notable graph-width parameter. Moreover, the polynomial dependency of our FPT algorithm is smaller than the one of the algorithm by Fulek and Tóth. In particular, our algorithm runs in quadratic time for flat instances of bounded treewidth and bounded face size. To further strengthen the relevance of this result, we show that an algorithm with running time O ( r ( n )) for flat instances whose underlying graph has pathwidth 1 would result in an algorithm with running time O ( r ( n )) for flat instances and with running time $$O(r(n^2) + n^2)$$ O ( r ( n 2 ) + n 2 ) for general, possibly non-flat, instances. 

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3. Packing Compact Subgraphs with Applications to Districting

   Chen, H-L; Chou, P-Y; Dharangutte, P; Gao, J; Huang, S-E; Yu, F-Y (April 2026, 7th annual Symposium on Foundations of Responsible Computing (FORC 2026))

   Packing disjoint subgraphs in a given graph is a fundamental problem with many applications. Motivated by political districting, we focus on connected subgraphs that are compact (e.g., having constant radius from a single center vertex) and that satisfy additional composition requirements, such as a minimum population/weight threshold or balanced weight types (e.g., political affiliations). We aim to maximize coverage by disjoint districts that meet these requirements. In this work, we present new results that substantially improve the previously known bounds on balanced star districts for planar and minor-free graphs (Dharangutte et al. 2025). In particular, we improve the approximation factor from to for packing balanced star districts using the exact same algorithm, but with a refined analysis. We also extend the results beyond planar graphs to minor-free graphs and an even broader family of graphs of bounded expansion. Additionally, we obtain an approximation for packing radius- districts (with a constant ) in planar and apex-minor-free graphs. In order to get a approximation on the max coverage, we show that this can be achieved if we allow a slight relaxation of the balancedness parameters (by a factor that can be made arbitrarily close to ), for bounded radius- districts on planar and apex-minor-free graphs. We show that all of these results can also be obtained if we enforce a minimum weight threshold for each district as the composition requirement, rather than balancedness. We present various results on hardness and hardness of approximation for this variant, by graph and district types. 

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4. Packing Compact Subgraphs with Applications to Districting

   https://doi.org/10.4230/lipics.forc.2026.10  

   Chen, Ho-Lin; Chou, Po-Yu; Dharangutte, Prathamesh; Gao, Jie; Huang, Shang-En; Yu, Fang-Yi (January 2026, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Lin, Huijia Rachel (Ed.)

   Abstract: Packing disjoint subgraphs in a given graph is a fundamental problem with many applications. Motivated by political districting, we focus on connected subgraphs that are compact (e.g., having constant radius from a single center vertex) and that satisfy additional composition requirements, such as a minimum population/weight threshold or balanced weight types (e.g., political affiliations). We aim to maximize coverage by disjoint districts that meet these requirements. In this work, we present new results that substantially improve the previously known bounds on balanced star districts for planar and minor-free graphs [Prathamesh Dharangutte et al., 2025]. In particular, we improve the approximation factor from O(log n) to O(1) for packing balanced star districts using the exact same algorithm, but with a refined analysis. We also extend the results beyond planar graphs to minor-free graphs and an even broader family of graphs of bounded expansion. Additionally, we obtain an O(1) approximation for packing radius-k districts (with a constant k) in planar and apex-minor-free graphs. In order to get a (1+ε) approximation on the max coverage, we show that this can be achieved if we allow a slight relaxation of the balancedness parameters (by a factor that can be made arbitrarily close to 1), for bounded radius-k districts on planar and apex-minor-free graphs. We show that all of these results can also be obtained if we enforce a minimum weight threshold for each district as the composition requirement, rather than balancedness. We present various results on hardness and hardness of approximation for this variant, by graph and district types. 

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5. Hardness and Approximation Algorithms for Balanced Districting Problems

   https://doi.org/10.4230/LIPIcs.FORC.2025.4  

   Dharangutte, Prathamesh; Gao, Jie; Huang, Shang-En; Yu, Fang-Yi (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Bun, Mark (Ed.)

   We introduce and study the problem of balanced districting, where given an undirected graph with vertices carrying two types of weights (different population, resource types, etc) the goal is to maximize the total weights covered in vertex disjoint districts such that each district is a star or (in general) a connected induced subgraph with the two weights to be balanced. This problem is strongly motivated by political redistricting, where contiguity, population balance, and compactness are essential. We provide hardness and approximation algorithms for this problem. In particular, we show NP-hardness for an approximation better than n^{1/2-δ} for any constant δ > 0 in general graphs even when the districts are star graphs, as well as NP-hardness on complete graphs, tree graphs, planar graphs and other restricted settings. On the other hand, we develop an algorithm for balanced star districting that gives an O(√n)-approximation on any graph (which is basically tight considering matching hardness of approximation results), an O(log n) approximation on planar graphs with extensions to minor-free graphs. Our algorithm uses a modified Whack-a-Mole algorithm [Bhattacharya, Kiss, and Saranurak, SODA 2023] to find a sparse solution of a fractional packing linear program (despite exponentially many variables) which requires a new design of a separation oracle specific for our balanced districting problem. To turn the fractional solution to a feasible integer solution, we adopt the randomized rounding algorithm by [Chan and Har-Peled, SoCG 2009]. To get a good approximation ratio of the rounding procedure, a crucial element in the analysis is the balanced scattering separators for planar graphs and minor-free graphs - separators that can be partitioned into a small number of k-hop independent sets for some constant k - which may find independent interest in solving other packing style problems. Further, our algorithm is versatile - the very same algorithm can be analyzed in different ways on various graph classes, which leads to class-dependent approximation ratios. We also provide a FPTAS algorithm for complete graphs and tree graphs, as well as greedy algorithms and approximation ratios when the district cardinality is bounded, the graph has bounded degree or the weights are binary. We refer the readers to the full version of the paper for complete set of results and proofs. 

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