Manuel Cáceres

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.

Manuel Cáceres, Sándor Kisfaludi-Bak, Saeed Odak

The shifting technique of Hochbaum and Maass [J.ACM'85] produces PTASes with the fastest known running times $n^{O(1/\varepsilon^{d-1})}$ for several $d$-dimensional geometric problems. However, it is only known, due to Marx [FOCS'07], that these algorithms are indeed optimal for dimension $d=2$. We show that these running times are optimal under Gap-ETH for every constant dimension. More precisely, we develop a framework that enables us to prove the conditional optimality of the shifting algorithms for several problems on unit ball graphs, such as maximum independent set, maximum induced forest, and others, as well as for the problem of piercing unit balls. Our framework is built using the cube wiring theorem of De Berg et al. [SICOMP'20] and the reduction steps of Marx and Sidiropoulos [SoCG'14] to create a convenient maximization version of geometric CSP that can be used as a basis for reductions.