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1. 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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2. Rapid Mixing for the Hardcore Glauber Dynamics and Other Markov Chains in Bounded-Treewidth Graphs

   https://doi.org/10.4230/LIPIcs.ISAAC.2023.30  

   Eppstein, David; Frishberg, Daniel (January 2023, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Iwata, Satoru; Kakimura, Naonori (Ed.)

   We give a new rapid mixing result for a natural random walk on the independent sets of a graph G. We show that when G has bounded treewidth, this random walk - known as the Glauber dynamics for the hardcore model - mixes rapidly for all fixed values of the standard parameter λ > 0, giving a simple alternative to existing sampling algorithms for these structures. We also show rapid mixing for analogous Markov chains on dominating sets, b-edge covers, b-matchings, maximal independent sets, and maximal b-matchings. (For b-matchings, maximal independent sets, and maximal b-matchings we also require bounded degree.) Our results imply simpler alternatives to known algorithms for the sampling and approximate counting problems in these graphs. We prove our results by applying a divide-and-conquer framework we developed in a previous paper, as an alternative to the projection-restriction technique introduced by Jerrum, Son, Tetali, and Vigoda. We extend this prior framework to handle chains for which the application of that framework is not straightforward, strengthening existing results by Dyer, Goldberg, and Jerrum and by Heinrich for the Glauber dynamics on q-colorings of graphs of bounded treewidth and bounded degree. 

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3. A decidable dichotomy theorem on directed graph homomorphisms with non-negative weights

   https://doi.org/10.1109/FOCS.2010.49  

   Cai, Jin-Yi; Chen, Xi. (January 2019, Computational complexity)

   The complexity of graph homomorphism problems has been the subject of intense study for some years. In this paper, we prove a decidable complexity dichotomy theorem for the partition function of directed graph homomorphisms. Our theorem applies to all non-negative weighted forms of the problem: given any fixed matrix A with non-negative algebraic entries, the partition function ZA(G) of directed graph homomorphisms from any directed graph G is either tractable in polynomial time or #P-hard, depending on the matrix A. The proof of the dichotomy theorem is combinatorial, but involves the definition of an infinite family of graph homomorphism problems. The proof of its decidability on the other hand is algebraic and based on properties of polynomials. 

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4. A decomposition method on solving the linear arboricity conjecture

   https://doi.org/10.1002/jgt.23040  

   Chen, Guantao; Hao, Yanli; Yu, Guoning (February 2024, Journal of Graph Theory)

   Abstract Alinear forestis a disjoint union of path graphs. Thelinear arboricity of a graph, denoted by , is the least number of linear forests into which the graph can be partitioned. Clearly, for any graph of maximum degree . For the upper bound, the long‐standingLinear Arboricity Conjecture(LAC) due to Akiyama, Exoo, and Harary from 1981 asserts that . A graph is apseudoforestif each of its components contains at most one cycle. In this paper, we prove thatthe union of any two pseudoforests of maximum degree up to 3 can be decomposed into three linear forests. Combining it with a recent result of Wdowinski on the minimum number of pseudoforests into which a graph can be decomposed, we prove that the LAC holds for the following simple graph classes: ‐degenerate graphs with maximum degree , all graphs on nonnegative Euler characteristic surfaces provided the maximum degree , and graphs on negative Euler characteristic surfaces provided the maximum degree , as well as graphs with no ‐minor satisfying some conditions on maximum degrees. 

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5. Toward a Dichotomy for Approximation of H-Coloring

   https://doi.org/10.4230/LIPIcs.ICALP.2019.91  

   Rafiey, Akbar; Rafiey, Arash; Santos, Thiago. (July 2019, Leibniz international proceedings in informatics)

   Given two (di)graphs G, H and a cost function c:V(G) x V(H) -> Q_{>= 0} cup {+infty}, in the minimum cost homomorphism problem, MinHOM(H), we are interested in finding a homomorphism f:V(G)-> V(H) (a.k.a H-coloring) that minimizes sum limits_{v in V(G)}c(v,f(v)). The complexity of exact minimization of this problem is well understood [Pavol Hell and Arash Rafiey, 2012], and the class of digraphs H, for which the MinHOM(H) is polynomial time solvable is a small subset of all digraphs. In this paper, we consider the approximation of MinHOM within a constant factor. In terms of digraphs, MinHOM(H) is not approximable if H contains a digraph asteroidal triple (DAT). We take a major step toward a dichotomy classification of approximable cases. We give a dichotomy classification for approximating the MinHOM(H) when H is a graph (i.e. symmetric digraph). For digraphs, we provide constant factor approximation algorithms for two important classes of digraphs, namely bi-arc digraphs (digraphs with a conservative semi-lattice polymorphism or min-ordering), and k-arc digraphs (digraphs with an extended min-ordering). Specifically, we show that: - Dichotomy for Graphs: MinHOM(H) has a 2|V(H)|-approximation algorithm if graph H admits a conservative majority polymorphims (i.e. H is a bi-arc graph), otherwise, it is inapproximable; - MinHOM(H) has a |V(H)|^2-approximation algorithm if H is a bi-arc digraph; - MinHOM(H) has a |V(H)|^2-approximation algorithm if H is a k-arc digraph. In conclusion, we show the importance of these results and provide insights for achieving a dichotomy classification of approximable cases. Our constant factors depend on the size of H. However, the implementation of our algorithms provides a much better approximation ratio. It leaves open to investigate a classification of digraphs H, where MinHOM(H) admits a constant factor approximation algorithm that is independent of |V(H)|. 

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