In this paper, we show new strongly polynomial work-depth tradeoffs for computing single-source shortest paths (SSSP) in non-negatively weighted directed graphs in parallel. Most importantly, we prove that directed SSSP can be solved within \(\tilde O(m+n^{2-\epsilon})\) work and \(\tilde O(n^{1-\epsilon})\) depth for some positive \(\epsilon \lt 0\). In particular, for dense graphs with non-negative real weights, we provide the first nearly work-efficient strongly polynomial algorithm with sublinear depth.

For a fixed integer t ⩾ 1, a (t-)long claw, denoted S_{t,t,t}, is the unique tree with three leaves, each at distance exactly t from the vertex of degree three. Majewski et al. [ICALP 2022, ACM ToCT 2024] proved an analog of the Gyárfás' path argument for S_{t,t,t}-free graphs: given an n-vertex S_{t,t,t}-free graph, one can delete neighborhoods of 𝒪(log n) vertices so that the remainder admits an extended strip decomposition (an appropriate generalization of partition into connected components) into particles of multiplicatively smaller size. In this work, we refine the argument of Majewski et al. to its arguably final form: we show that a constant number of neighborhoods suffice. The statement of Majewski et al. is one of the two pillars of a recent quasi-polynomial time algorithm for Maximum Weight Independent Set in S_{t,t,t}-free graphs [Gartland et al., STOC 2024]; our work immediately improves the quasi-polynomial function in the running time bound. Furthermore, our result significantly simplifies known polynomial-time algorithms for Maximum Weight Independent Set in S_{t,t,t}-free graphs with an additional sparsity assumption such as bounded degree or excluding a fixed biclique as a subgraph.

We present a deterministic fully-dynamic data structure for maintaining information about the cut-vertices in a graph; i.e. the vertices whose removal would disconnect the graph. Our data structure supports insertion and deletion of edges, as well as queries to whether a pair of connected vertices are either biconnected, or can be separated by a cutvertex, and in the latter case we support access to separating cutvertices. All update operations are supported in amortized O(log2 n log2 log n) time, and queries take worst-case O(log n log2 log n) time. Note that these time bounds match the current best for deterministic dynamic connectivity up to log log n factors. The previous best algorithm for biconnectivity had an update time of O(logλ n log log n) by Thorup [STOC'00], based on the O(logλ μ n) data structure by Holm, de Lichtenberg, and Thorup [STOC'98]. We obtain our improved running time by a series of reductions from the original problem into well-defined data structure problems. While we do indeed apply the well-known techniques for improving running time of two-edge connectivity [STOC'00, SODA'18], surprisingly, these techniques alone do not lead to an update time of Õ(log³ n), let alone the Õ(log2 n) we give as a final result. Our contributions include a formally defined transient expose operation, which can be thought of as a cheaper read-only expose operation on a top tree. For each vertex in the graph, we maintain a data structure over its neighbors, and in this data structure we apply biasing (twice) to save an Õ(log n) factor (twice, so two Õ(log n) factors). One of these biasing techniques is a new, simple biased disjoint sets data structure, which may be of independent interest. Moreover, in this neighborhood data structure, we facilitate that the vertex can select two VIP neighbors that get special treatment, corresponding to its potentially two neighbors on an exposed path, improving an otherwise log n-time operation down to constant time. It is this combination of VIP neighbors with the transient expose operation that saves an Õ(log n)-factor from another bottleneck. Combining these technical contributions with the well-known techniques for two-edge connectivity [STOC'00, SODA'18], we obtain the desired update times of O(log2 n log2 log n). The near-linear query time follows directly from the usage of transient expose.

Exact computation of shortest paths in weighted graphs has been traditionally studied in one of two settings. First, one can assume that the edge weights are real numbers and all the performed operations on reals (typically comparisons and additions) take constant time. Classical Dijkstra's and Bellman-Ford algorithms have been described in this setting.

The classic technique of Baker [J. ACM ‘94] is the most fundamental approach for designing approximation schemes on planar, or more generally topologically-constrained graphs, and it has been applied in a myriad of different variants and settings throughout the last 30 years. In this work we propose a dynamic variant of Baker's technique, where instead of finding an approximate solution in a given static graph, the task is to design a data structure for maintaining an approximate solution in a fully dynamic graph, that is, a graph that is changing over time by edge deletions and edge insertions. Specifically, we address the two most basic problems — Maximum Weight Independent Set and Minimum Weight Dominating Set — and we prove the following: for a fully dynamic n-vertex planar graph G, one can

We present a data structure that for a dynamic graph G that is updated by edge insertions and deletions, maintains a tree decomposition of G of width at most $6 k+5$ under the promise that the treewidth of G never grows above k. The amortized update time is $\mathcal{O}_{k}\left(2^{\sqrt{\log n} \log \log n}\right)$, where n is the vertex count of G and the $\mathcal{O}_{k}(\cdot)$ notation hides factors depending on k. In addition, we also obtain the dynamic variant of Courcelle’s Theorem: for any fixed property $\varphi$ expressible in the CMSO2logic, the data structure can maintain whether G satisfies $\varphi$ within the same time complexity bounds. To a large extent, this answers a question posed by Bodlaender [WG 1993].

We describe a polynomial-time algorithm which, given a graphGwith treewidtht, approximates the pathwidth ofGto within a ratio of \(O(t\sqrt {\log t})\) . This is the first algorithm to achieve anf(t)-approximation for some functionf. Our approach builds on the following key insight: every graph with large pathwidth has large treewidth or contains a subdivision of a large complete binary tree. Specifically, we show that every graph with pathwidth at leastth+2 has treewidth at leasttor contains a subdivision of a complete binary tree of heighth+1. The boundth+2 is best possible up to a multiplicative constant. This result was motivated by, and implies (withc=2), the following conjecture of Kawarabayashi and Rossman (SODA’18): there exists a universal constantcsuch that every graph with pathwidth Ω(kc) has treewidth at leastkor contains a subdivision of a complete binary tree of heightk. Our main technical algorithm takes a graphGand some (not necessarily optimal) tree decomposition ofGof widtht′ in the input, and it computes in polynomial time an integerh, a certificate thatGhas pathwidth at leasth, and a path decomposition ofGof width at most (t′+1)h+1. The certificate is closely related to (and implies) the existence of a subdivision of a complete binary tree of heighth. The approximation algorithm for pathwidth is then obtained by combining this algorithm with the approximation algorithm of Feige, Hajiaghayi, and Lee (STOC’05) for treewidth.

The treedepth of a graph G is the least possible depth of an elimination forest of G: a rooted forest on the same vertex set where every pair of vertices adjacent in G is bound by the ancestor/descendant relation. We propose an algorithm that given a graph G and an integer d, either finds an elimination forest of G of depth at most d or concludes that no such forest exists; thus the algorithm decides whether the treedepth of G is at most d. The running time is 2^𝒪(d²)⋅n^𝒪(1) and the space usage is polynomial in n. Further, by allowing randomization, the time and space complexities can be improved to 2^𝒪(d²)⋅n and d^𝒪(1)⋅n, respectively. This improves upon the algorithm of Reidl et al. [ICALP 2014], which also has time complexity 2^𝒪(d²)⋅n, but uses exponential space.

We study the Many-Visits Traveling Salesman Problem, where given a number k(v) for each of n cities and pairwise (possibly asymmetric) integer distances, one has to find an optimal tour that visits each city v exactly k(v) times. The currently fastest algorithm is due to Berger, Kozma, Mnich and Vincze [SODA 2019, TALG 2020] and runs in time and space O⁎(5n). They also show a polynomial-space algorithm running in time O(16n+o(n)). In this work, we show three main results: A randomized polynomial-space algorithm running in time O⁎(2nD), where D is the maximum distance between two cities. By using standard methods, this results in a (1+ϵ)-approximation running in time O⁎(2nϵ−1). A tight analysis of Berger et al.'s exponential-space algorithm, resulting in an O⁎(4n) running time bound. A new polynomial-space algorithm, running in time O(7.88n).

In this work, we present the first linear time deterministic algorithm computing the 4-edge-connected components of an undirected graph. First, we show an algorithm listing all 3-edge-cuts in a given 3-edge-connected graph, and then we use the output of this algorithm in order to determine the 4-edge-connected components of the graph.

We present a data structure that in a dynamic graph of treedepth at most d, which is modified over time by edge insertions and deletions, maintains an optimum-height elimination forest. The data structure achieves worst-case update time , which matches the best known parameter dependency in the running time of a static fpt algorithm for computing the treedepth of a graph. This improves a result of Dvořák et al. [ESA 2014], who for the same problem achieved update time f(d) for some non-elementary (i.e. tower-exponential) function f. As a by-product, we improve known upper bounds on the sizes of minimal obstructions for having treedepth d from doubly-exponential in d to dO(d). As applications, we design new fully dynamic parameterized data structures for detecting long paths and cycles in general graphs. More precisely, for a fixed parameter k and a dynamic graph G, modified over time by edge insertions and deletions, our data structures maintain answers to the following queries: Does G contain a simple path on k vertices? Does G contain a simple cycle on at least k vertices? In the first case, the data structure achieves amortized update time . In the second case, the amortized update time is . In both cases we assume access to a dictionary on the edges of G.

We describe a polynomial-time algorithm which, given a graph G with treewidth t, approximates the pathwidth of G to within a ratio of . This is the first algorithm to achieve an f(t)-approximation for some function f. Our approach builds on the following key insight: every graph with large pathwidth has large treewidth or contains a subdivision of a large complete binary tree. Specifically, we show that every graph with pathwidth at least th + 2 has treewidth at least t or contains a subdivision of a complete binary tree of height h + 1. The bound th + 2 is best possible up to a multiplicative constant. This result was motivated by, and implies (with c = 2), the following conjecture of Kawarabayashi and Rossman (SODA'18): there exists a universal constant c such that every graph with pathwidth Ω(kc) has treewidth at least k or contains a subdivision of a complete binary tree of height k. Our main technical algorithm takes a graph G and some (not necessarily optimal) tree decomposition of G of width t′ in the input, and it computes in polynomial time an integer h, a certificate that G has pathwidth at least h, and a path decomposition of G of width at most (t′ + 1)h + 1. The certificate is closely related to (and implies) the existence of a subdivision of a complete binary tree of height h. The approximation algorithm for pathwidth is then obtained by combining this algorithm with the approximation algorithm of Feige, Hajiaghayi, and Lee (STOC'05) for treewidth.

Treedepth, a more restrictive graph width parameter than treewidth and pathwidth, plays a major role in the theory of sparse graph classes. We show that there exists a constant $C$ such that for all positive integers $a,b$ and a graph $G$, if the treedepth of $G$ is at least $Cab$, then the treewidth of $G$ is at least $a$ or $G$ contains a subcubic (i.e., of maximum degree at most 3) tree of treedepth at least $b$ as a subgraph. As a direct corollary, we obtain that every graph of treedepth $\Omega(k^3)$ either is of treewidth at least $k$, contains a subdivision of full binary tree of depth $k$, or contains a path of length $2^k$. This improves the bound of $\Omega(k^5 \log^2 k)$ of Kawarabayashi and Rossman [Proceedings of the 2018 Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 234--246]. We also show an application of our techniques for approximation algorithms of treedepth: given a graph $G$ of treedepth $k$ and treewidth $t$, one can in polynomial time compute a treedepth decomposition of $G$ of width $\mathcal{O}(kt \log^{3/2} t)$. This improves upon a bound of $\mathcal{O}(kt^2 \log t)$ stemming from a tradeoff between known results. The main technical ingredient in our result is a proof that every tree of treedepth $d$ contains a subcubic subtree of treedepth at least $d \cdot \log_3 ((1+\sqrt{5})/2)$.

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