research-article

An Exact Method for the Minimum Feedback Arc Set Problem

Authors:

Hermann Schichl,

Arnold Neumaier,

Tobias AchterbergAuthors Info & Claims

Journal of Experimental Algorithmics (JEA), Volume 26

Article No.: 1.4, Pages 1 - 28

https://doi.org/10.1145/3446429

Published: 23 April 2021 Publication History

Abstract

A feedback arc set of a directed graph G is a subset of its arcs containing at least one arc of every cycle in G. Finding a feedback arc set of minimum cardinality is an NP-hard problem called the minimum feedback arc set problem. Numerically, the minimum set cover formulation of the minimum feedback arc set problem is appropriate as long as all simple cycles in G can be enumerated. Unfortunately, even those sparse graphs that are important for practical applications often have Ω (2ⁿ) simple cycles. Here we address precisely such situations: An exact method is proposed for sparse graphs that enumerates simple cycles in a lazy fashion and iteratively extends an incomplete cycle matrix. In all cases encountered so far, only a tractable number of cycles has to be enumerated until a minimum feedback arc set is found. The practical limits of the new method are evaluated on a test set containing computationally challenging sparse graphs, relevant for industrial applications. The 4,468 test graphs are of varying size and density and suitable for testing the scalability of exact algorithms over a wide range.

References

[1]

Tobias Achterberg. 2007. Constraint Integer Programming. Ph.D. Dissertation. Technische Universität Berlin.

[2]

Tobias Achterberg. 2009. SCIP: Solving constraint integer programs. Mathematical Programming Computation 1, 1 (July 2009), 1–41.

[3]

N. Alon. 2006. Ranking tournaments. SIAM Journal on Discrete Mathematics 20, 1 (2006), 137–142.

Digital Library

[4]

Sanjeev Arora and Boaz Barak. 2009. Computational Complexity: A Modern Approach. Cambridge University Press; New York, NY.

Digital Library

[5]

S. Arora, A. Frieze, and H. Kaplan. 1996. A new rounding procedure for the assignment problem with applications to dense graph arrangement problems. In Proceedings of the 1996 37th Annual Symposium on Foundations of Computer Science. 21–30.

[6]

Aspen Technology, Inc. 2009. EO and SM variables and synchronization. In Aspen Simulation Workbook, Version Number: V7.1. Burlington, MA, 110.

[7]

P. Austrin, R. Manokaran, and C. Wenner. 2013. On the NP-hardness of approximating ordering constraint satisfaction problems. In Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques. Lecture Notes in Computer Science, Vol. 8096. Springer, 26–41.

[8]

A. Baharev. 2019. Exact and Heuristic Methods for Tearing. Retrieved February 10, 2021 from https://sdopt-tearing.readthedocs.io.

[9]

R. W. Barkley and R. L. Motard. 1972. Decomposition of nets. Chemical Engineering Journal 3 (1972), 265–275.

[10]

Oliver Bastert and Christian Matuszewski. 2001. Layered drawings of digraphs. In Drawing Graphs—Methods and Models, Michael Kaufman and Dorothea Wagner (Eds.). Springer, Berlin, Germany, 87–120.

[11]

Bonnie Berger and Peter W. Shor. 1990. Approximation algorithms for the maximum acyclic subgraph problem. In Proceedings of the 1st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’90). 236–243.

[12]

Livio Bertacco, Lorenzo Brunetta, and Matteo Fischetti. 2008. The linear ordering problem with cumulative costs. European Journal of Operational Research 189, 3 (2008), 1345–1357.

[13]

Lorenz T. Biegler, Ignacio E. Grossmann, and Arthur W. Westerberg. 1997. Systematic Methods of Chemical Process Design. Prentice Hall PTR, Upper Saddle River, NJ.

[14]

N. L. Book and W. F. Ramirez. 1984. Structural analysis and solution of systems of algebraic design equations. AIChE Journal 30, 4 (1984), 609–622.

[15]

F. J. Brandenburg and K. Hanauer. 2011. Sorting Heuristics for the Feedback Arc Set Problem. Technical Report MIP-1104. Department of Informatics and Mathematics, University of Passau, Germany.

[16]

Saskia Bublitz, Erik Esche, Gregor Tolksdorf, Volker Mehrmann, and Jens-Uwe Repke. 2017. Analysis and decomposition for improved convergence of nonlinear process models in chemical engineering. Chemie Ingenieur Technik 89, 11 (2017), 1503–1514.

[17]

Pierre Charbit, Stéphan Thomassé, and Anders Yeo. 2007. The minimum feedback arc set problem is NP-hard for tournaments. Combinatorics, Probability and Computing 16, 1 (2007), 1–4.

Digital Library

[18]

M. Charikar, K. Makarychev, and Y. Makarychev. 2007. On the advantage over random for maximum acyclic subgraph. In Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS’07). IEEE, Los Alamitos, CA, 625–633.

[19]

Jianer Chen, Yang Liu, Songjian Lu, Barry O’Sullivan, and Igor Razgon. 2008. A fixed-parameter algorithm for the directed feedback vertex set problem. Journal of the ACM 55, 5 (2008), Article 21, 19 pages.

Digital Library

[20]

James H. Christensen. 1970. The structuring of process optimization. AIChE Journal 16, 2 (1970), 177–184.

[21]

J. H. Christensen and D. F. Rudd. 1969. Structuring design computations. AIChE Journal 15 (1969), 94–100.

[22]

V. Chvatal. 1979. A greedy heuristic for the set-covering problem. Mathematics of Operations Research 4, 3 (1979), 233–235.

Digital Library

[23]

Thomas H. Cormen, Clifford Stein, Ronald L. Rivest, and Charles E. Leiserson. 2009. Introduction to Algorithms (3rd ed.). MIT Press, Cambridge, MA.

Digital Library

[24]

David Coudert, Afonso Ferreira, and Xavier Muñoz. 1999. OTIS-based multi-hop multi-OPS lightwave networks. In Parallel and Distributed Processing: 11th IPPS/SPDP’99 Workshops Held in Conjunction with the 13th International Parallel Processing Symposium and 10th Symposium on Parallel and Distributed Processing San Juan, Puerto Rico, USA, April 12–16, 1999 Proceedings. Springer, Berlin, Germany, 897–910.

[25]

G. Dantzig, R. Fulkerson, and S. Johnson. 1954. Solution of a large-scale traveling-salesman problem. Journal of the Operations Research Society of America 2, 4 (1954), 393–410.

[26]

Dassault Systèmes AB. 2016. Advanced modelica support. In Dymola—Dynamic Modeling Laboratory: User Manual. Vol. 2. Dassault Systèmes AB, 399--445.

[27]

Camil Demestrescu and Irene Finocchi. 2001. Breaking cycles for minimizing crossings. ACM Journal of Experimental Algorithmics 6 (Dec. 2001), Article 2.

[28]

Camil Demetrescu and Irene Finocchi. 2003. Combinatorial algorithms for feedback problems in directed graphs. Information Processing Letters 86, 3 (2003), 129–136.

Digital Library

[29]

Narsingh Deo. 1974. Graph Theory with Applications to Engineering and Computer Science. Prentice Hall, Englewood Cliffs, NJ.

Digital Library

[30]

A. C. Dimian, C. S. Bildea, and A. A. Kiss. 2014. Steady-state flowsheeting. In Integrated Design and Simulation of Chemical Processes (2nd ed.). Computer-Aided Chemical Engineering, Vol. 35. Elsevier Amsterdam, Netherlands, 73--125.

[31]

R. G. Downey and M. R. Fellows. 2013. In Fundamentals of Parameterized Complexity, D. Gries and F. B. Schneider (Eds.). Springer-Verlag, London, UK.

[32]

D. Z. Du and F. K. Hwang. 1988. Generalized de Bruijn digraphs. Networks 18, 1 (1988), 27–38.

Digital Library

[33]

P. Eades and X. Lin. 1995. A new heuristic for the feedback arc set problem. Australasian Journal of Combinatorics 35, 4 (1995), 15–25.

[34]

Peter Eades, Xuemin Lin, and W. F. Smyth. 1993. A fast and effective heuristic for the feedback arc set problem. Information Processing Letters 47, 6 (1993), 319–323.

Digital Library

[35]

P. Erdős and A. Rényi. 1959. On random graphs I. Publicationes Mathematicae 6 (1959), 290–297.

[36]

G. Even, J. (Seffi) Naor, B. Schieber, and M. Sudan. 1998. Approximating minimum feedback sets and multicuts in directed graphs. Algorithmica 20, 2 (1998), 151–174.

[37]

F. V. Fomin and D. Kratsch. 2010. Measure and conquer. In Exact Exponential Algorithms. Springer, Berlin, Germany, 101–124.

[38]

L. R. Ford and D. R. Fulkerson. 1958. A suggested computation for maximal multi-commodity network flows. Management Science 5, 1 (1958), 97–101.

Digital Library

[39]

R. S. Garfinkel and G. L. Nemhauser. 1972. Integer Programming. Wiley, New York, NY.

[40]

P. L. Genna and R. L. Motard. 1975. Optimal decomposition of process networks. AIChE Journal 21, 4 (1975), 656–663.

[41]

E. N. Gilbert. 1959. Random graphs. Annals of Mathematical Statistics 30 (1959), 1141–1144.

[42]

Fred Glover, T. Klastorin, and D. Kongman. 1974. Optimal weighted ancestry relationships. Management Science 20, 8 (1974), 1190–1193.

Digital Library

[43]

Martin Grötschel, Michael Jünger, and Gerhard Reinelt. 1984. A cutting plane algorithm for the linear ordering problem. Operations Research 32, 6 (1984), 1195–1220.

Digital Library

[44]

M. Grötschel, M. Jünger, and G. Reinelt. 1985. Acyclic Subdigraphs and Linear Orderings: Polytopes, Facets, and a Cutting Plane Algorithm. Springer, Dordrecht, Netherlands, 217–264.

[45]

Martin Grötschel, Michael Jünger, and Gerhard Reinelt. 1985. On the acyclic subgraph polytope. Mathematical Programming 33, 1 (Sept. 1985), 28–42.

Digital Library

[46]

G. Guardabassi. 1971. A note on minimal essential sets. IEEE Transactions on Circuit Theory 18, 5 (1971), 557–560.

[47]

T. Gundersen. 1982. Decomposition of Large Scale Chemical Engineering Systems. Ph.D. Dissertation. Department of Chemical Engineering, University of Trondheim, Norway.

[48]

T. Gundersen and T. Hertzberg. 1983. Partitioning and tearing of networks applied to process flowsheeting. Modeling, Identification and Control 4, 3 (1983), 139–165.

[49]

Prem K. Gupta, Arthur W. Westerberg, John E. Hendry, and Richard R. Hughes. 1974. Assigning output variables to equations using linear programming. AIChE Journal 20, 2 (1974), 397–399.

[50]

Gurobi. 2014. Gurobi Optimizer Version 6.0. Houston, Texas: Gurobi Optimization, Inc., May 2015. Software Program. Retrieved February 10, 2021 from http://www.gurobi.com.

[51]

V. Guruswami, J. Håstad, R. Manokaran, P. Raghavendra, and M. Charikar. 2011. Beating the random ordering is hard: Every ordering CSP is approximation resistant. SIAM Journal on Computing 40, 3 (2011), 878–914.

Digital Library

[52]

V. Guruswami, R. Manokaran, and P. Raghavendra. 2008. Beating the random ordering is hard: Inapproximability of maximum acyclic subgraph. In Proceedings of the 2008 IEEE 49th Annual Symposium on Foundations of Computer Science (FOCS’08). 573–582.

[53]

Aric A. Hagberg, Daniel A. Schult, and Pieter J. Swart. 2008. Exploring network structure, dynamics, and function using NetworkX. In Proceedings of the 7th Python in Science Conference (SciPy’08). 11–15.

[54]

Michael Hecht. 2018. Exact localisations of feedback sets. Theory of Computing Systems 62, 5 (July 2018), 1048–1084.

Digital Library

[55]

R. Hernandez and R. W. H. Sargent. 1979. A new algorithm for process flowsheeting. Computers & Chemical Engineering 3, 1–4 (1979), 363–371.

[56]

V. Hlaváček. 1977. Analysis of a complex plant-steady state and transient behavior. Computers & Chemical Engineering 1, 1 (1977), 75–100.

[57]

M. Imase and M. Itoh. 1981. Design to minimize diameter on building-block network. IEEE Transactions on Computers C-30, 6 (1981), 439–442.

Digital Library

[58]

M. Imase and M. Itoh. 1983. A design for directed graphs with minimum diameter. IEEE Transactions on Computers C-32, 8 (1983), 782–784.

Digital Library

[59]

Makoto Imase, Terunao Soneoka, and Keiji Okada. 1986. Fault-tolerant processor interconnection networks. Systems and Computers in Japan 17, 8 (1986), 21–30.

[60]

Iaroslav Ispolatov and Sergei Maslov. 2008. Detection of the dominant direction of information flow and feedback links in densely interconnected regulatory networks. BMC Bioinformatics 9 (2008), 424.

[61]

Y. V. S. Jain and J. M. Eakman. 1971. Identification of process flow networks. In Proceedings of the AIChE 68th National Meeting.

[62]

Donald B. Johnson. 1975. Finding all the elementary circuits of a directed graph. SIAM Journal on Computing 4, 1 (1975), 77–84.

[63]

David S. Johnson. 1973. Approximation algorithms for combinatorial problems. In Proceedings of the 5th Annual ACM Symposium on Theory of Computing (STOC’73). ACM, New York, NY, 38–49.

Digital Library

[64]

R. Kaas. 1981. A branch and bound algorithm for the acyclic subgraph problem. European Journal of Operational Research 8, 4 (1981), 355–362.

[65]

M. Frans Kaashoek and David R. Karger. 2003. Koorde: A simple degree-optimal distributed hash table. In Peer-to-Peer Systems II: Second International Workshop, IPTPS 2003, Berkeley, CA, USA, February 21-22, 2003. Revised Papers. Springer, Berlin, Germany, 98–107.

[66]

Viggo Kann. 1992. On the Approximability of NP-Complete Optimization Problems. Ph.D. Dissertation. Royal Institute of Technology Stockholm.

[67]

Richard M. Karp. 1972. Reducibility among combinatorial problems. In Complexity of Computer Computations, R. E. Miller, J. W. Thatcher, and J. D. Bohlinger (Eds.). Springer, 85–103.

[68]

W. H. Kautz. 1968. Bounds on directed (d, k) graphs. In Theory of Cellular Logic Networks and Machines. Technical Report AFCRL-68-0668 Final Report. Air Force Cambridge Research Laboratories, Cambridge, MA, 20–28.

[69]

John G. Kemeny. 1959. Mathematics without numbers. Daedalus 88, 4 (1959), 577–591. http://www.jstor.org/stable/20026529

[70]

Claire Kenyon-Mathieu and Warren Schudy. 2007. How to rank with few errors. In Proceedings of the 39th Annual ACM Symposium on Theory of Computing (STOC’07). ACM, New York, NY, 95–103.

Digital Library

[71]

Subhash Khot. 2002. On the power of unique 2-prover 1-round games. In Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC’02). ACM, New York, NY, 767–775.

[72]

W. Lee, J. H. Christensen, and D. F. Rudd. 1966. Design variable selection to simplify process calculations. AIChE Journal 12, 6 (1966), 1104–1115.

[73]

W. Lee and D. F. Rudd. 1966. On the ordering of recycle calculations. AIChE Journal 12, 6 (1966), 1184–1190.

[74]

H. W. Lenstra. 1973. The Acyclic Subgraph Problem. Technical Report BW 26/73. Faculteit der Wiskunde en Natuurwetenschappen, Mathematisch Centrum, Amsterdam. http://hdl.handle.net/1887/2116

[75]

W. K. Lewis and G. L. Matheson. 1932. Studies in distillation. Industrial & Engineering Chemistry 24 (1932), 494–498.

[76]

Dongsheng Li, Xicheng Lu, and Jinshu Su. 2004. Graph-theoretic analysis of Kautz topology and DHT schemes. In Network and Parallel Computing: IFIP International Conference, NPC 2004, Wuhan, China, October 18-20, 2004. Proceedings. Springer, Berlin, Germany, 308–315.

[77]

L. Lovász. 1975. On the ratio of optimal integral and fractional covers. Discrete Mathematics 13, 4 (1975), 383–390.

Digital Library

[78]

C. L. Lucchesi. 1976. A Minimax Equality for Directed Graphs. Ph.D. Dissertation. University of Waterloo, Ontario.

[79]

C. L. Lucchesi and D. H. Younger. 1978. A minimax theorem for directed graphs. Journal of the London Mathematical Society 2, 17 (1978), 369–374.

[80]

Richard S. H. Mah. 1990. Computation sequence in process flowsheet calculations. In Chemical Process Structures and Information Flows, Richard S. H. Mah (Ed.). Butterworth-Heinemann, 125–183.

[81]

Rafael Martí and Gerhard Reinelt. 2011. The Linear Ordering Problem: Exact and Heuristic Methods in Combinatorial Optimization. Applied Mathematical Sciences, Vol. 175. Springer-Verlag, Berlin, Germany.

[82]

Ronald E. Miller and Peter D. Blair. 2009. Input-Output Analysis: Foundations and Extensions (2nd ed.). Cambridge University Press.

[83]

J. E. Mitchell and B. Borehers. 2000. Solving linear ordering problems with a combined interior point/simplex cutting plane algorithm. In High Performance Optimization, H. Frenk, K. Roos, T. Terlaky, and S. Zhang (Eds.). Applied Optimization, Vol. 33. Springer-Science+Business Media, B.V., Dordrecht, Netherlands, 349–366.

[84]

Modelica. 2018. Modelica and the Modelica Association. Retrieved October 14, 2018 from https://www.modelica.org/.

[85]

Modelon AB. 2018. JModelica.org User Guide, Version 2.2. Retrieved October 14, 2018 from https://jmodelica.org/downloads/UsersGuide.pdf.

[86]

J. M. Montagna and O. A. Iribarren. 1988. Optimal computation sequence in the simulation of chemical plants. Computers & Chemical Engineering 12, 1 (1988), 71–79.

[87]

Rodolphe L. Motard and Arthur W. Westerberg. 1981. Exclusive tear sets for flowsheets. AIChE Journal 27 (1981), 725–732.

[88]

OpenModelica. 2018. OpenModelica User’s Guide. Retrieved October 14, 2018 from https://openmodelica.org/doc/OpenModelicaUsersGuide/latest/omchelptext.html.

[89]

T. Orenstein, Z. Kohavi, and I. Pomeranz. 1995. An optimal algorithm for cycle breaking in directed graphs. Journal of Electronic Testing 7, 1–2 (1995), 71–81.

Digital Library

[90]

Christos H. Papadimitriou and Mihalis Yannakakis. 1991. Optimization, approximation, and complexity classes. Journal of Computer and System Sciences 43, 3 (1991), 425–440.

[91]

S. Park and S. B. Akers. 1992. An efficient method for finding a minimal feedback arc set in directed graphs. In Proceedings of the 1992 IEEE International Symposium on Circuits and Systems (ISCAS’92), Vol. 4. 1863–1866.

[92]

T. K. Pho and L. Lapidus. 1973. Topics in computer-aided design: Part I. An optimum tearing algorithm for recycle systems. AIChE Journal 19, 6 (1973), 1170–1181.

[93]

Vijaya Ramachandran. 1988. Finding a minimum feedback arc set in reducible flow graphs. Journal of Algorithms 9, 3 (1988), 299–313.

Digital Library

[94]

Venkatesh Raman and Saket Saurabh. 2007. Improved fixed parameter tractable algorithms for two “edge” problems: MAXCUT and MAXDAG. Informations Processing Letters 104, 2 (2007), 65–72.

Digital Library

[95]

Venkatesh Raman, Saket Saurabh, and Somnath Sikdar. 2007. Efficient exact algorithms through enumerating maximal independent sets and other techniques. Theory of Computing Systems 41, 3 (Oct. 2007), 563–587.

Digital Library

[96]

J. R. Roach, B. K. O’Neill, and D. A. Hocking. 1997. A new synthetic method for stream tearing in process systems analysis. Chemical Engineering Communications 161, 1 (1997), 1–14.

[97]

D. K. Pradhan S. M. Reddy, and J. G. Kuhl. 1980. Directed Graphs with Minimal Diameter and Maximal Connectivity. Technical Report. School of Engineering, Oakland University, RochesterMI.

[98]

R. Saket and M. Sviridenko. 2012. New and improved bounds for the minimum set cover problem. In Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques. Lecture Notes in Computer Science, Vol. 7408. Springer, 288–300.

[99]

G. Sander. 1999. Graph layout for applications in compiler construction. Theoretical Computer Science 217, 2 (1999), 175–214.

Digital Library

[100]

R. W. H. Sargent. 1978. The decomposition of systems of procedures and algebraic equations. In Numerical Analysis. Lecture Notes in Mathematics, Vol. 630. Springer, 158–178.

[101]

R. W. H. Sargent and A. W. Westerberg. 1964. Speed-up in chemical engineering design. Transactions of the Institution of Chemical Engineers 42 (1964), 190–197.

[102]

Benno Schwikowski and Ewald Speckenmeyer. 2002. On enumerating all minimal solutions of feedback problems. Discrete Applied Mathematics 117, 1–3 (2002), 253–265.

[103]

P. D. Seymour. 1995. Packing directed circuits fractionally. Combinatorica 15, 2 (1995), 281–288.

[104]

Patrick Slater. 1961. Inconsistencies in a schedule of paired comparisons. Biometrika 48, 3–4 (1961), 303–312. http://www.jstor.org/stable/2332752

[105]

Ljiljana Spadavecchia. 2006. A Network-Based Asynchronous Architecture for Cryptographic Devices. Ph.D. Dissertation. College of Science and Engineering, School of Informatics, University of Edinburgh. http://hdl.handle.net/1842/860

[106]

Mark A. Stadtherr. 1979. Maintaining sparsity in process design calculations. AIChE Journal 25, 4 (1979), 609–615.

[107]

M. A. Stadtherr, W. A. Gifford, and L. E. Scriven. 1974. Efficient solution of sparse sets of design equations. Chemical Engineering Science 29, 4 (1974), 1025–1034.

[108]

M. A. Stadtherr and E. S. Wood. 1984. Sparse matrix methods for equation-based chemical process flowsheeting–I: Reordering phase. Computers & Chemical Engineering 8, 1 (1984), 9–18.

[109]

K. Sugiyama, S. Tagawa, and M. Toda. 1981. Methods for visual understanding of hierarchical system structures. IEEE Transactions on Systems, Man, and Cybernetics 11, 2 (Feb. 1981), 109–125.

[110]

R. E. Tarjan. 1972. Depth first search and linear graph algorithms. SIAM Journal on Computing 1 (1972), 146–160.

[111]

E. W. Thiele and R. L. Geddes. 1933. Computation of distillation apparatus for hydrocarbon mixtures. Industrial & Engineering Chemistry 25 (1933), 289–295.

[112]

R. S. Upadhye and E. A. Grens. 1972. An efficient algorithm for optimum decomposition of recycle systems. AIChE Journal 18 (1972), 533–539.

[113]

Leslie G. Valiant. 1979. The complexity of enumeration and reliability problems. SIAM Journal on Computing 8, 3 (1979), 410–421.

Digital Library

[114]

G. V. Varma, K. H. Lau, and D. L. Ulrichson. 1993. A new tearing algorithm for process flowsheeting. Computers & Chemical Engineering 17, 4 (1993), 355–360.

[115]

A. W. Westerberg and F. C. Edie. 1971. Computer-aided design, part 2: An approach to convergence and tearing in the solution of sparse equation sets. Chemical Engineering Journal 2, 1 (1971), 17–25.

[116]

Daniel R. Zerbino and Ewan Birney. 2008. Velvet: Algorithms for de novo short read assembly using de Bruijn graphs. Genome Research 18 (2008), 821–829.

[117]

L. Zhou, Z. Han, and K. Yu. 1988. A new strategy of net decomposition in process simulation. Computers & Chemical Engineering 12, 6 (1988), 581–588.

Cited By

Xiong ZZhou YXiao MKhoussainov B(2024)Finding small feedback arc sets on large graphsComputers & Operations Research10.1016/j.cor.2024.106724169(106724)Online publication date: Sep-2024
https://doi.org/10.1016/j.cor.2024.106724
Biallach HBouhtou MKumbria KNace DTomaszewski A(2024)Virtual network function reconfiguration in 5G networks: An optimization perspectiveNetworks10.1002/net.2221283:4(673-691)Online publication date: 8-Feb-2024
https://doi.org/10.1002/net.22212
Kudelić R(2023)A Short Sketch of Solid Algorithms for Feedback Arc SetProceedings of Eighth International Congress on Information and Communication Technology10.1007/978-981-99-3243-6_5(51-59)Online publication date: 25-Jul-2023
https://doi.org/10.1007/978-981-99-3243-6_5
Show More Cited By

Index Terms

An Exact Method for the Minimum Feedback Arc Set Problem
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
      1. Combinatorial optimization
    2. Graph theory
      1. Graph algorithms
      2. Paths and connectivity problems
  2. Mathematical analysis
    1. Mathematical optimization
      1. Discrete optimization
2. Theory of computation
  1. Design and analysis of algorithms
    1. Graph algorithms analysis
    2. Mathematical optimization
      1. Mixed discrete-continuous optimization
        Integer programming

Recommendations

Minimum feedback vertex set and acyclic coloring

In the feedback vertex set problem, the aim is to minimize, in a connected graph G = (V, E), the cardinality of the set V(G) ⊆ V, whose removal induces an acyclic subgraph. In this paper, we show an interesting relationship between the minimum feedback ...
On the Minimum Feedback Vertex Set Problem: Exact and Enumeration Algorithms
Parameterized and Exact Algorithms

We present a time $\mathcal {O}(1.7548^{n})$algorithm finding a minimum feedback vertex set in an undirected graph on n vertices. We also prove that a graph on n vertices can contain at most 1.8638ⁿ minimal feedback vertex sets and that there exist ...
On the Minimum Feedback Vertex Set Problem: Exact and Enumeration Algorithms

We present a time $\mathcal {O}(1.7548^{n})$ algorithm finding a minimum feedback vertex set in an undirected graph on n vertices. We also prove that a graph on n vertices can contain at most 1.8638n minimal feedback vertex sets and that there exist ...

Comments

Information & Contributors

Information

Published In

cover image ACM Journal of Experimental Algorithmics

ACM Journal of Experimental Algorithmics Volume 26, Issue

December 2021

479 pages

ISSN:1084-6654

EISSN:1084-6654

DOI:10.1145/3446425

Issue’s Table of Contents

Copyright © 2021 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 23 April 2021

Accepted: 01 December 2020

Revised: 01 October 2020

Received: 01 March 2019

Published in JEA Volume 26

Author Tags

Qualifiers

Research-article
Research
Refereed

Funding Sources

Austrian Research Promotion Agency (FFG)
Austrian Science Fund

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

8
Total Citations
View Citations
513
Total Downloads

Downloads (Last 12 months)161
Downloads (Last 6 weeks)21

Reflects downloads up to 28 Jul 2024

Other Metrics

View Author Metrics

Citations

Cited By

Xiong ZZhou YXiao MKhoussainov B(2024)Finding small feedback arc sets on large graphsComputers & Operations Research10.1016/j.cor.2024.106724169(106724)Online publication date: Sep-2024
https://doi.org/10.1016/j.cor.2024.106724
Biallach HBouhtou MKumbria KNace DTomaszewski A(2024)Virtual network function reconfiguration in 5G networks: An optimization perspectiveNetworks10.1002/net.2221283:4(673-691)Online publication date: 8-Feb-2024
https://doi.org/10.1002/net.22212
Kudelić R(2023)A Short Sketch of Solid Algorithms for Feedback Arc SetProceedings of Eighth International Congress on Information and Communication Technology10.1007/978-981-99-3243-6_5(51-59)Online publication date: 25-Jul-2023
https://doi.org/10.1007/978-981-99-3243-6_5
Grötschel MJünger MReinelt G(2022)Comments on “An Exact Method for the Minimum Feedback Arc Set Problem”ACM Journal of Experimental Algorithmics10.1145/354500127(1-4)Online publication date: 27-Jul-2022
https://dl.acm.org/doi/10.1145/3545001
Biallach HBouhtou MNace DMimouna M(2022)An Efficient Heuristic for the Virtual Network Function Reconfiguration Problem2022 12th International Workshop on Resilient Networks Design and Modeling (RNDM)10.1109/RNDM55901.2022.9927661(1-7)Online publication date: 19-Sep-2022
https://doi.org/10.1109/RNDM55901.2022.9927661
Kudelić RKudelić R(2022)Having the Right ToolFeedback Arc Set10.1007/978-3-031-10515-9_4(123-126)Online publication date: 10-Jul-2022
https://doi.org/10.1007/978-3-031-10515-9_4
Kudelić RKudelić R(2022)Papers and AlgorithmsFeedback Arc Set10.1007/978-3-031-10515-9_3(19-120)Online publication date: 10-Jul-2022
https://doi.org/10.1007/978-3-031-10515-9_3
Kudelić RKudelić R(2022)Feedback Arc SetFeedback Arc Set10.1007/978-3-031-10515-9_1(3-14)Online publication date: 10-Jul-2022
https://doi.org/10.1007/978-3-031-10515-9_1

View Options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Article

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

HTML Format

View this article in HTML Format.

Media

Figures

Other

Tables

View Issue’s Table of Contents