Alexandru I. Tomescu

Manuel Cáceres, Andreas Grigorjew, Wanchote Po Jiamjitrak et al.

Given an acyclic digraph $G = (V,E)$ and a positive integer $k$, the problem of Maximum Coverage $k$-Antichains (resp. Chains) denoted as MA-$k$ (resp. MC-$k$) asks to find $k$ sets of pairwise unreachable vertices, known as antichains (resp. $k$ subsequences of paths, known as chains), maximizing the number $α_k$ (resp. $β_k$) of vertices covered by these antichains (resp. chains). While MC-$k$ was solved in almost optimal $|E|^{1+o(1)}$ time~[Kogan and Parter, ICALP'22], the fastest algorithms for MA-$k$ are a $(k|E|)^{1+o(1)}$-time solution and a $|E|^{1+o(1)}$-time $1/2$ approximation~[Kogan and Parter, ESA'24]. We obtain the following for MA-$k$: - An algorithm running in $|E|^{1+o(1)}$ time, and an algorithm running in parameterized near-linear $\tilde{O}(α_k |E|)$ time. Our algorithms are simple solutions exploiting a paths-based proof of the Greene-Kleitman theorems leveraged by the greedy algorithm for set cover as well as recent advances in fast algorithms for flows and shortest paths. - An approximation algorithm running in parameterized linear time $O(α_1^2|V| + (α_1+k)|E|)$ with approximation ratio of $(1-1/e) > 0.63 > 1/2$, beating the state-of-the-art $1/2$ approximation. Our solution uses greedy for antichains and a simple strategy to amortize the cost of computing consecutive maximum antichains. We complement these results with two examples (one for chains and one for antichains) showing that, for every $k \ge 2$, greedy misses the tight $1/e$ portion of the optimal coverage for chains, and a $1/4$ portion for antichains. We also show that greedy is a $Ω(\log{|V|})$ factor away from minimality when required to cover all vertices: previously unknown for sets of chains or antichains.

Francisco Sena, Aleksandr Politov, Corentin Moumard et al.

A fundamental algorithmic problem in computational biology is to find all subgraphs of a given type (superbubbles, snarls, and ultrabubbles) in a directed or bidirected input graph. These correspond to regions of genetic variation and are useful in analyzing collections of genomes. We present the first linear-time algorithms for identifying all snarls and all ultrabubbles, resolving problems open since 2018. The algorithm for snarls relies on a new linear-size representation of all snarls with respect to the number of vertices in the graph. We employ the well-known SPQR-tree decomposition, which encodes all 2-separators of a biconnected graph. After several dynamic-programming-style traversals of this tree, we maintain key properties (such as acyclicity) that allow us to decide whether a given 2-separator defines a subgraph to be reported. A crucial ingredient for linear-time complexity is that acyclicity of linearly many subgraphs can be tested simultaneously via the problem of computing all arcs in a directed graph whose removal renders it acyclic (so-called feedback arcs). As such, we prove a fundamental result for bidirected graphs, that may be of independent interest: all feedback arcs can be computed in linear time for tipless bidirected graphs, while in general this is at least as hard as matrix multiplication, assuming the k-Clique Conjecture. Our results form a unified framework that also yields a completely different linear-time algorithm for finding all superbubbles. Although some of the results are technically involved, the underlying ideas are conceptually simple, and may extend to other bubble-like subgraphs. More broadly, our work contributes to the theoretical foundations of computational biology and advances a growing line of research that uses SPQR-tree decompositions as a general tool for designing efficient algorithms, beyond their traditional role in graph drawing.