DROPS

Document

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.

Cite as

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)

Copy BibTex To Clipboard

@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}
}

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.

Cite as

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)

Copy BibTex To Clipboard

@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.

Cite as

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)

Copy BibTex To Clipboard

@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}
}

Search Results

Documents authored by Mumey, Brendan

Practical Minimum Path Cover

Abstract

Cite as

Width Helps and Hinders Splitting Flows

Abstract

Cite as

Flow Decomposition with Subpath Constraints

Abstract

Cite as

Thanks for your feedback!

Could not send message