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DOI: 10.4230/LIPIcs.SEA.2024.3

Practical Minimum Path Cover

Authors: Manuel Cáceres, Brendan Mumey, Santeri Toivonen, and Alexandru I. Tomescu

Published in: LIPIcs, Volume 301, 22nd International Symposium on Experimental Algorithms (SEA 2024)

Abstract

Computing a minimum path cover (MPC) of a directed acyclic graph (DAG) is a fundamental problem with a myriad of applications, including reachability. Although it is known how to solve the problem by a simple reduction to minimum flow, recent theoretical advances exploit this idea to obtain algorithms parameterized by the number of paths of an MPC, known as the width. These results obtain fast [Mäkinen et al., TALG 2019] and even linear time [Cáceres et al., SODA 2022] algorithms in the small-width regime. In this paper, we present the first publicly available high-performance implementation of state-of-the-art MPC algorithms, including the parameterized approaches. Our experiments on random DAGs show that parameterized algorithms are orders-of-magnitude faster on dense graphs. Additionally, we present new fast pre-processing heuristics based on transitive edge sparsification. We show that our heuristics improve MPC-solvers by orders of magnitude.

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Manuel Cáceres, Brendan Mumey, Santeri Toivonen, and Alexandru I. Tomescu. Practical Minimum Path Cover. In 22nd International Symposium on Experimental Algorithms (SEA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 301, pp. 3:1-3:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{caceres_et_al:LIPIcs.SEA.2024.3,
  author =	{C\'{a}ceres, Manuel and Mumey, Brendan and Toivonen, Santeri and Tomescu, Alexandru I.},
  title =	{{Practical Minimum Path Cover}},
  booktitle =	{22nd International Symposium on Experimental Algorithms (SEA 2024)},
  pages =	{3:1--3:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-325-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{301},
  editor =	{Liberti, Leo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2024.3},
  URN =		{urn:nbn:de:0030-drops-203687},
  doi =		{10.4230/LIPIcs.SEA.2024.3},
  annote =	{Keywords: minimum path cover, directed acyclic graph, maximum flow, parameterized algorithms, edge sparsification, algorithm engineering}
}

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Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2023.31

Minimum Chain Cover in Almost Linear Time

Authors: Manuel Cáceres

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract

A minimum chain cover (MCC) of a k-width directed acyclic graph (DAG) G = (V, E) is a set of k chains (paths in the transitive closure) of G such that every vertex appears in at least one chain in the cover. The state-of-the-art solutions for MCC run in time Õ(k(|V|+|E|)) [Mäkinen et at., TALG], O(T_{MF}(|E|) + k|V|), O(k²|V| + |E|) [Cáceres et al., SODA 2022], Õ(|V|^{3/2} + |E|) [Kogan and Parter, ICALP 2022] and Õ(T_{MCF}(|E|) + √k|V|) [Kogan and Parter, SODA 2023], where T_{MF}(|E|) and T_{MCF}(|E|) are the running times for solving maximum flow (MF) and minimum-cost flow (MCF), respectively. In this work we present an algorithm running in time O(T_{MF}(|E|) + (|V|+|E|)log k). By considering the recent result for solving MF [Chen et al., FOCS 2022] our algorithm is the first running in almost linear time. Moreover, our techniques are deterministic and derive a deterministic near-linear time algorithm for MCC if the same is provided for MF. At the core of our solution we use a modified version of the mergeable dictionaries [Farach and Thorup, Algorithmica], [Iacono and Özkan, ICALP 2010] data structure boosted with the SIZE-SPLIT operation and answering queries in amortized logarithmic time, which can be of independent interest.

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Manuel Cáceres. Minimum Chain Cover in Almost Linear Time. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 31:1-31:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{caceres:LIPIcs.ICALP.2023.31,
  author =	{C\'{a}ceres, Manuel},
  title =	{{Minimum Chain Cover in Almost Linear Time}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{31:1--31:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.31},
  URN =		{urn:nbn:de:0030-drops-180834},
  doi =		{10.4230/LIPIcs.ICALP.2023.31},
  annote =	{Keywords: Minimum chain cover, directed acyclic graph, minimum flow, flow decomposition, mergeable dictionaries, amortized running time}
}

Document

DOI: 10.4230/LIPIcs.ESA.2022.31

Width Helps and Hinders Splitting Flows

Authors: Manuel Cáceres, Massimo Cairo, Andreas Grigorjew, Shahbaz Khan, Brendan Mumey, Romeo Rizzi, Alexandru I. Tomescu, and Lucia Williams

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract

Minimum flow decomposition (MFD) is the NP-hard problem of finding a smallest decomposition of a network flow X on directed graph G into weighted source-to-sink paths whose superposition equals X. We focus on a common formulation of the problem where the path weights must be non-negative integers and also on a new variant where these weights can be negative. We show that, for acyclic graphs, considering the width of the graph (the minimum number of s-t paths needed to cover all of its edges) yields advances in our understanding of its approximability. For the non-negative version, we show that a popular heuristic is a O(log |X|)-approximation (|X| being the total flow of X) on graphs satisfying two properties related to the width (satisfied by e.g., series-parallel graphs), and strengthen its worst-case approximation ratio from Ω(√m) to Ω(m / log m) for sparse graphs, where m is the number of edges in the graph. For the negative version, we give a (⌈log ║X║⌉+1)-approximation (║X║ being the maximum absolute value of X on any edge) using a power-of-two approach, combined with parity fixing arguments and a decomposition of unitary flows (║X║ ≤ 1) into at most width paths. We also disprove a conjecture about the linear independence of minimum (non-negative) flow decompositions posed by Kloster et al. [ALENEX 2018], but show that its useful implication (polynomial-time assignments of weights to a given set of paths to decompose a flow) holds for the negative version.

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Manuel Cáceres, Massimo Cairo, Andreas Grigorjew, Shahbaz Khan, Brendan Mumey, Romeo Rizzi, Alexandru I. Tomescu, and Lucia Williams. Width Helps and Hinders Splitting Flows. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 31:1-31:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{caceres_et_al:LIPIcs.ESA.2022.31,
  author =	{C\'{a}ceres, Manuel and Cairo, Massimo and Grigorjew, Andreas and Khan, Shahbaz and Mumey, Brendan and Rizzi, Romeo and Tomescu, Alexandru I. and Williams, Lucia},
  title =	{{Width Helps and Hinders Splitting Flows}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{31:1--31:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.31},
  URN =		{urn:nbn:de:0030-drops-169695},
  doi =		{10.4230/LIPIcs.ESA.2022.31},
  annote =	{Keywords: Flow decomposition, approximation algorithms, graph width}
}

Document

DOI: 10.4230/LIPIcs.WABI.2021.16

Flow Decomposition with Subpath Constraints

Authors: Lucia Williams, Alexandru I. Tomescu, and Brendan Mumey

Published in: LIPIcs, Volume 201, 21st International Workshop on Algorithms in Bioinformatics (WABI 2021)

Abstract

Flow network decomposition is a natural model for problems where we are given a flow network arising from superimposing a set of weighted paths and would like to recover the underlying data, i.e., decompose the flow into the original paths and their weights. Thus, variations on flow decomposition are often used as subroutines in multiassembly problems such as RNA transcript assembly. In practice, we frequently have access to information beyond flow values in the form of subpaths, and many tools incorporate these heuristically. But despite acknowledging their utility in practice, previous work has not formally addressed the effect of subpath constraints on the accuracy of flow network decomposition approaches. We formalize the flow decomposition with subpath constraints problem, give the first algorithms for it, and study its usefulness for recovering ground truth decompositions. For finding a minimum decomposition, we propose both a heuristic and an FPT algorithm. Experiments on RNA transcript datasets show that for instances with larger solution path sets, the addition of subpath constraints finds 13% more ground truth solutions when minimal decompositions are found exactly, and 30% more ground truth solutions when minimal decompositions are found heuristically.

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Lucia Williams, Alexandru I. Tomescu, and Brendan Mumey. Flow Decomposition with Subpath Constraints. In 21st International Workshop on Algorithms in Bioinformatics (WABI 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 201, pp. 16:1-16:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{williams_et_al:LIPIcs.WABI.2021.16,
  author =	{Williams, Lucia and Tomescu, Alexandru I. and Mumey, Brendan},
  title =	{{Flow Decomposition with Subpath Constraints}},
  booktitle =	{21st International Workshop on Algorithms in Bioinformatics (WABI 2021)},
  pages =	{16:1--16:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-200-6},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{201},
  editor =	{Carbone, Alessandra and El-Kebir, Mohammed},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2021.16},
  URN =		{urn:nbn:de:0030-drops-143695},
  doi =		{10.4230/LIPIcs.WABI.2021.16},
  annote =	{Keywords: Flow decomposition, subpath constraints, RNA-Seq}
}

4 Search Results for "Mumey, Brendan"

Practical Minimum Path Cover

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Minimum Chain Cover in Almost Linear Time

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Width Helps and Hinders Splitting Flows

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Flow Decomposition with Subpath Constraints

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