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Approximation algorithms in combinatorial scientific computing

  • Alex Pothen (a1), S. M. Ferdous (a2) and Fredrik Manne (a3)

Abstract

We survey recent work on approximation algorithms for computing degree-constrained subgraphs in graphs and their applications in combinatorial scientific computing. The problems we consider include maximization versions of cardinality matching, edge-weighted matching, vertex-weighted matching and edge-weighted $b$ -matching, and minimization versions of weighted edge cover and $b$ -edge cover. Exact algorithms for these problems are impractical for massive graphs with several millions of edges. For each problem we discuss theoretical foundations, the design of several linear or near-linear time approximation algorithms, their implementations on serial and parallel computers, and applications. Our focus is on practical algorithms that yield good performance on modern computer architectures with multiple threads and interconnected processors. We also include information about the software available for these problems.

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The work of the first two authors was supported in part by US NSF grant CCF-1637534; the US Department of Energy through grant DE-FG02-13ER26135; and the Exascale Computing Project (17-SC-20-SC), a collaborative effort of the DOE Office of Science and the NNSA.

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Achlioptas, D. and Naor, A. (2005), ‘The two possible values of the chromatic number of a random graph’, Ann. of Math. 162, 13351351.
Agrawal, A., Klein, P. N. and Ravi, R. (1993), Cutting down on fill using nested dissection: Provably good elimination orderings. In Graph Theory and Sparse Matrix Computations (George, A., Gilbert, J. R. and Liu, J. W. H., eds), Springer, pp. 3155.
Al-Herz, A. and Pothen, A. (2019), ‘A $2/3$ -approximation algorithm for vertex-weighted matching’, Discrete Appl. Math., under review. arXiv:1902.05877
Anstee, R. P. (1987), ‘A polynomial algorithm for $b$ -matchings: An alternative approach’, Inform. Process. Lett. 24, 153157.
Azad, A. and Buluç, A. (2016), Distributed memory algorithms for maximum cardinality matching on bipartite graphs. In 2016 IEEE International Parallel and Distributed Processing Symposium (IPDPS), IEEE, pp. 3242.
Azad, A., Buluç, A. and Pothen, A. (2017), ‘Computing maximum cardinality matchings in parallel on bipartite graphs via tree grafting’, IEEE Trans. Parallel Distrib. Syst. 28, 4459.
Azad, A., Buluç, A., Li, X. S., Wang, X. and Langguth, J. (2018), ‘A distributed memory approximation algorithm for maximum weight perfect bipartite matching’, SIAM J. Sci. Comput., under review. arXiv:1801.09809v1
Azad, A., Langguth, J., Fang, Y., Qi, A. and Pothen, A. (2010), Identifying rare cell populations in comparative flow cytometry. In Algorithms in Bioinformatics: International Workshop on Algorithms in Bioinformatics (WABI), Vol. 6293 of Lecture Notes in Bioinformatics, Springer, pp. 162175.
Bast, H., Mehlhorn, K., Schäfer, G. and Tamaki, H. (2006), ‘Matching algorithms are fast in sparse random graphs’, Theory Comput. Syst. 39, 314.
Bell, C. E. (1994), ‘Weighted matching with vertex weights: An application to scheduling training sessions in NASA space shuttle cockpit simulators’, Europ. J. Oper. Res. 73, 443449.
Birn, M., Osipov, V., Sanders, P., Schulz, C. and Sitchinava, N. (2013), Efficient parallel and external matching. In Euro-Par 2013 Parallel Processing, Vol. 8097 of Lecture Notes in Computer Science, Springer, pp. 659670.
Birnbaum, B. and Mathieu, C. (2008), ‘On-line bipartite matching made simple’, ACM SIGACT News 39, 8087.
Blelloch, G. E., Fineman, J. T. and Shun, J. (2012), Greedy sequential maximal independent set and matching are parallel on average. In Proceedings of the 24th Annual ACM Symposium on Parallelism in Algorithms and Architectures (SPAA ’12), ACM, pp. 308317.
Blelloch, G. E., Peng, R. and Tangwongsan, K. (2011), Linear work parallel greedy approximate set cover and variants. In Proceedings of the 23rd Annual ACM Symposium on Parallelism in Algorithms and Architectures (SPAA ’11), ACM, pp. 2332.
Boldi, P. and Vigna, S. (2004), The WebGraph framework I: Compression techniques. In Proceedings of the 13th International Conference on World Wide Web (WWW 2004), ACM, pp. 595601.
Boldi, P., Marino, A., Santini, M. and Vigna, S. (2014), BUbiNG: Massive crawling for the masses. In Proceedings of the Companion Publication of the 23rd International Conference on World Wide Web, ACM, pp. 227228.
Buluç, A. and Gilbert, J. R. (2011), ‘The Combinatorial BLAS: Design, implementation and applications’, Internat. J. High Perf. Comput. Appl. 25, 496509.
Burkard, R., Dell’Amico, M. and Martello, S. (2009), Assignment Problems, SIAM.
Cao, Y. and Sandeep, R. B. (2017), Minimum fill-in: Inapproximability and almost tight lower bounds. In Proceedings of the 28th Annual Symposium on Discrete Algorithms (SODA), SIAM, pp. 875880.
Choromanski, K. M., Jebara, T. and Tang, K. (2013), Adaptive anonymity via b-matching. In Advances in Neural Information Processing Systems (NIPS 2013) (Burges, C. J. C. et al. , eds), pp. 31923200.
Chvatal, V. (1979), ‘A greedy heuristic for the set-covering problem’, Math. Oper. Res. 4, 233235.
Cohen, J. and Castonguay, P. (2012), Efficient graph matching and coloring on GPUs. Presentation available at: http://on-demand.gputechconf.com/gtc/2012/presentations/S0332-Efficient-Graph-Matching-and-Coloring-on-GPUs.pdf
Coleman, T. F. and Pothen, A. (1987), ‘The null space problem II: Algorithms’, SIAM J. Algebraic Discrete Methods 8, 544563.
Coleman, T. F., Edenbrandt, A. and Gilbert, J. R. (1986), ‘Predicting fill for sparse orthogonal factorization’, J. Assoc. Comput. Mach. 33, 517532.
Cormen, T. H., Leiserson, C. E., Rivest, R. L. and Stein, C. (2009), Introduction to Algorithms, MIT Press.
Davis, T. and Hu, Y. (2011), ‘The University of Florida Sparse Matrix Collection’, ACM Trans. Math. Softw. 38, 1:1–1:25.
De Francisci Morales, G., Gionis, A. and Sozio, M. (2011), ‘Social content matching in MapReduce’, Proc. VLDB Endowment 4, 460469.
Derigs, U. and Metz, A. (1986), ‘On the use of optimal fractional matchings for solving the (integer) matching problem’, Computing 36, 263270.
Deveci, M., Kaya, K., Uçar, B. and Çatalyürek, Ü. (2013), GPU accelerated maximum cardinality matching algorithms for bipartite graphs. In Proceedings of 19th International Euro-Par Conference on Parallel Processing, pp. 850861.
Dezső, B., Jüttner, A. and Kovács, P. (2011), ‘LEMON: An open source C++ graph template library’, Electron. Notes Theoret. Comput. Sci. 264, 2345.
Dobrian, F., Halappanavar, M., Pothen, A. and Al-Herz, A. (2019), ‘A $2/3$ -approximation algorithm for vertex-weighted matching in bipartite graphs’, SIAM J. Sci. Comput. 41, A566A591.
Drake, D. and Hougardy, S. (2003a), Linear time local improvements for weighted matchings in graphs. In Experimental and Efficient Algorithms, (Jansen, K., Margraf, M., Mastrolilli, M. and Rolim, J., eds), Vol. 2647 of Lecture Notes in Computer Science, Springer, pp. 107119.
Drake, D. E. and Hougardy, S. (2003b), ‘A simple approximation algorithm for the weighted matching problem’, Inform. Process. Lett. 85, 211213.
Drake, D. E. and Hougardy, S. (2005), ‘A linear time approximation algorithm for weighted matchings in graphs’, ACM Trans. Algorithms 1, 107122.
Du, D., Ko, K. and Hu, X. (2012), Design and Analysis of Approximation Algorithms, Springer.
Duan, R. and Pettie, S. (2010), Approximating maximum weight matching in near-linear time. In 2010 IEEE 51st Annual Symposium on Foundations of Computer Science (FOCS ’10), IEEE, pp. 673682.
Duan, R. and Pettie, S. (2014), ‘Linear-time approximation for maximum weight matching’, J. Assoc. Comput. Mach. 61, 123.
Duff, I. S. and Koster, J. (2001), ‘On algorithms for permuting large entries to the diagonal of a sparse matrix’, SIAM J. Matrix Anal. Appl. 22, 973996.
Duff, I. S. and Uçar, B. (2012), Combinatorial problems in solving linear systems. In Combinatorial Scientific Computing (Naumann, U. and Schenk, O., eds), CRC, pp. 2168.
Duff, I. S., Kaya, K. and Uçar, B. (2011), ‘Design, implementation, and analysis of maximum transversal algorithms’, ACM Trans. Math. Softw. 38, 13:1–13:31.
Dufossé, F., Kaya, K. and Uçar, B. (2015), ‘Two approximation algorithms for bipartite matching on multicore architectures’, J. Parallel Distrib. Comput. 85, 6278.
Edmonds, J. (1965), ‘Maximum matching and a polyhedron with 0, 1-vertices’, J. Res. Nat. Bureau Standards 69B, 125130.
Fagginger Auer, B. O. and Bisseling, R. H. (2012), A GPU algorithm for greedy graph matching. In Facing the Multicore-Challenge II (Keller, R., Kramer, D. and Weiss, J., eds), Springer, pp. 108119.
Feige, U. and Kilian, J. (1998), ‘Zero knowledge and the chromatic number’, J. Comput. Systems Sci. 57, 187199.
Ferdous, S., Pothen, A. and Khan, A. (2018), New approximation algorithms for minimum weighted edge cover. In 2018 Proceedings of the Seventh SIAM Workshop on Combinatorial Scientific Computing, SIAM, pp. 97108.
Fritzson, P. (2014), Principles of Object-Oriented Modeling and Simulation with Modelica 3.3: A Cyber-Physical Approach, Wiley/IEEE.
Gabow, H. N. (1983), An efficient reduction technique for degree-constrained subgraph and bidirected network flow problems. In Proceedings of the 15th Annual ACM Symposium on the Theory of Computing (STOC ’83), ACM, pp. 448456.
Gabow, H. N. (2018), ‘Data structures for weighted matching and extensions to $b$ -matching and $f$ -factors’, ACM Trans. Algorithms 14, 39:1–39:80.
Gale, D. and Shapley, L. S. (1962), ‘College admissions and the stability of marriage’, Amer. Math. Monthly 69, 915.
Gallai, T. (1959), ‘Über extreme Punkt- und Kantenmengen’, Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae, Sectio Mathematica 2, 133138.
Gebremedhin, A. H., Manne, F. and Pothen, A. (2005), ‘What color is your Jacobian? Graph coloring for computing derivatives’, SIAM Review 47, 629705.
Gebremedhin, A. H., Tarafdar, A., Manne, F. and Pothen, A. (2007), ‘New acyclic and star coloring algorithms with application to computing Hessians’, SIAM J. Sci. Comput. 29, 10421072.
George, A. (1973), ‘Nested dissection of a finite element mesh’, SIAM J. Numer. Anal. 10, 345363.
Georgiadis, G. and Papatriantafilou, M. (2013), ‘Overlays with preferences: Distributed, adaptive approximation algorithms for matching with preference lists’, Algorithms 6, 824856.
Goemans, M. X. and Williamson, D. P. (1997), The primal–dual method for approximation algorithms and its application to network design problems. In Approximation Algorithms for NP-hard Problems (Hochbaum, D. S., ed.), PWS Publishing Co., pp. 144191.
Grötschel, M. and Holland, O. (1985), ‘Solving matching problems with linear programming’, Math. Program. 33, 243259.
Halappanavar, M., Feo, J., Villa, O., Dobrian, F. and Pothen, A. (2012), ‘Approximate weighted matching on emerging manycore and multithreaded architectures’, Internat. J. High Perf. Comput. Appl. 26, 413430.
Hall, N. G. and Hochbaum, D. S. (1986), ‘A fast approximation algorithm for the multicovering problem’, Discrete Appl. Math. 15, 3540.
Hanke, S. and Hougardy, S. (2010), New approximation algorithms for the weighted matching problem. Research report 101010, Research Institute for Discrete Mathematics, University of Bonn.
D. S. Hochbaum, ed. (1997), Approximation Algorithms for NP-hard Problems, PWS Publishing Co.
Hogg, J. and Scott, J. (2015), ‘On the use of suboptimal matchings for scaling and ordering sparse symmetric matrices’, Numer. Linear Algebra Appl. 22, 648663.
Hogg, J. and Scott, J. (2013), ‘Pivoting strategies for tough sparse indefinite systems’, ACM Trans. Math. Softw. 40, 4.
Hopcroft, J. and Karp, R. (1973), ‘An $n^{5/2}$ algorithm for maximum matchings in bipartite graphs’, SIAM J. Comput. 2, 225231.
Hougardy, S. (2009), Linear time approximation algorithms for degree constrained subgraph problems. In Research Trends in Combinatorial Optimization (Cook, W. J., Lovász, L. and Vygen, J., eds), Springer, pp. 185200.
Huang, B. C. and Jebara, T. (2011), Fast b-matching via sufficient selection belief propagation. In Proc. 14th International Conference on Artificial Intelligence and Statistics (AISTATS), pp. 361369.
Huang, D. and Pettie, S. (2017), Approximate generalized matching: $f$ -factors and $f$ -edge covers. arXiv:1706.05761
Idelberger, A. and Manne, F. (2014), New iterative algorithms for weighted matching. In Norsk Informatikkonferanse 2014. www.nik.no/publikasjoner/
Jebara, T. and Shchogolev, V. (2006), b-matching for spectral clustering. In Proceedings of the 17th European Conference on Machine Learning (ECML 2006), Vol. 4212 of Lecture Notes in Computer Science, Springer, pp. 679686.
Jebara, T., Wang, J. and Chang, S.-F. (2009), Graph construction and b-matching for semi-supervised learning. In Proceedings of the 26th Annual International Conference on Machine Learning (ICML ’09), ACM, pp. 441448.
Juedes, D. and Jones, J. (2012), ‘Coloring Jacobians revisited: A new algorithm for acyclic and star bicoloring’, Optim. Methods Softw. 27, 295309.
Kang, R. J. and McDiarmid, C. (2015), Colouring random graphs. In Topics in Chromatic Graph Theory (Wilson, R. J. and Beineke, L. W., eds), Cambridge University Press, pp. 199219.
Karp, R. M. and Sipser, M. (1981), Maximum matching in sparse random graphs. In Proceedings of the 22nd Annual Symposium on Foundations of Computer Science (SFCS 1981), pp. 364375.
Karp, R. M., Vazirani, U. and Vazirani, V. (1990), An optimal algorithm for on-line bipartite matching. In Proceedings of the 22nd Annual ACM Symposium on Theory of Computing (STOC ’90), ACM, pp. 352358.
Keiter, E. R., Thornquist, H. K., Hoekstra, R. J., Russo, T. V., Schiek, R. L. and Rankin, E. L. (2011), Parallel transistor-level circuit simulation. In Advanced Simulation and Verification of Electronic and Biological Systems (Li, P., Silveira, L. M. and Feldmann, P., eds), Springer, pp. 121.
Khan, A. and Pothen, A. (2016), A new 3/2-approximation algorithm for the b-edge cover problem. In 2016 Proceedings of the Seventh SIAM Workshop on Combinatorial Scientific Computing, SIAM, pp. 5261.
Khan, A., Choromanski, K., Pothen, A., Ferdous, S., Halappanavar, M. and Tumeo, A. (2018a), Adaptive anonymization of data using b-edge cover. In Proceedings of the International Conference for High Performance Computing, Networking, Storage, and Analysis (SC ’18), IEEE, pp. 59:1–59:11.
Khan, A. M., Gleich, D. F., Pothen, A. and Halappanavar, M. (2012), A multithreaded algorithm for network alignment via approximate matching. In Proceedings of the International Conference on High Performance Computing, Networking, Storage, and Analysis (SC ’12), IEEE, pp. 64:1–64:11.
Khan, A., Pothen, A. and Ferdous, S. M. (2018b), Parallel algorithms through approximation: b-edge cover. In 2018 IEEE International Parallel and Distributed Processing Symposium (IPDPS), IEEE, pp. 2233.
Khan, A., Pothen, A., Patwary, M. M., Halappanavar, M., Satish, N., Sundaram, N. and Dubey, P. (2016a), Designing scalable b-matching algorithms on distributed memory multiprocessors by approximation. In Proceedings of the International Conference for High Performance Computing, Networking, Storage, and Analysis (SC ’16), IEEE, pp. 773783.
Khan, A., Pothen, A., Patwary, M. M., Satish, N., Sundaram, N., Manne, F., Halappanavar, M. and Dubey, P. (2016b), ‘Efficient approximation algorithms for weighted $b$ -matching’, SIAM J. Sci. Comput. 38, S593S619.
Khuller, S., Vishkin, U. and Young, N. (1994), ‘A primal–dual parallel approximation technique applied to weighted set and vertex covers’, J. Algorithms 17, 280289.
Knoblauch, V. (2007), Marriage Matching: A conjecture of Donald Knuth. Working papers 2007-15, University of Connecticut, Department of Economics.
Kolmogorov, V. (2009), ‘BLOSSOM V: A new implementation of a minimum cost perfect matching algorithm’, Math. Prog. Comput. 1, 4367.
Kolyvakis, P., Kalousis, A., Smith, B. and Kiritsis, D. (2018), ‘Biomedical ontology alignment: An approach based on representation learning’, J. Biomed. Semantics 9, 21.
Koufogiannakis, C. and Young, N. E. (2011), ‘Distributed algorithms for covering, packing and maximum weighted matching’, Distrib. Comput. 24, 4563.
Li, X. S. and Demmel, J. W. (2003), ‘SuperLU_DIST: A scalable distributed-memory sparse direct solver for unsymmetric linear systems’, ACM Trans. Math. Softw. 29, 110140.
Lovász, L. and Plummer, M. D. (2009), Matching Theory, AMS.
Manlove, D. F. (2013), Algorithmics of Matching Under Preferences, World Scientific.
Manne, F. and Halappanavar, M. (2014), New effective multithreaded matching algorithms. In 2014 IEEE 28th International Parallel and Distributed Processing Symposium (IPDPS), IEEE, pp. 519528.
Manne, F., Naim, M., Lerring, H. and Halappanavar, M. (2016), On stable marriages and greedy matchings. In 2016 Proceedings of the Seventh SIAM Workshop on Combinatorial Scientific Computing, SIAM, pp. 92101.
Manshadi, F. M., Awerbuch, B., Gemulla, R., Khandekar, R., Mestre, J. and Sozio, M. (2013), ‘A distributed algorithm for large-scale generalized matching’, Proc. VLDB Endowment 6, 613624.
Maue, J. and Sanders, P. (2007), Engineering algorithms for approximate weighted matching. In Experimental Algorithms: 6th International Workshop on Experimental and Efficient Algorithms (WEA 2007), Vol. 4525 of Lecture Notes in Computer Science, Springer, pp. 242255.
McCormick, S. T. (1983), ‘Optimal approximation of sparse Hessians and its equivalence to a graph coloring problem’, Math. Program. 26, 153171.
McDiarmid, C. (1984), ‘Colouring random graphs’, Ann. Oper. Res. 1, 183200.
McVitie, D. G. and Wilson, L. B. (1971), ‘The stable marriage problem’, Commun. Assoc. Comput. Mach. 14, 486490.
Mehlhorn, K. and Näher, S. (1999), LEDA: A platform for combinatorial and geometric computing. www.algorithmic-solutions.com/leda/index.htm
Mehta, A. (2012), ‘Online matching and ad allocation’, Found. Trends. Theor. Comput. Sci. 8, 265368.
Mendelsohn, N. S. and Dulmage, A. L. (1958), ‘Some generalizations of the problem of distinct representatives’, Canad. J. Math. 10, 230241.
Mestre, J. (2006), Greedy in approximation algorithms. In Algorithms: 14th Annual European Symposium on Algorithms (ESA 2006), Vol. 4168 of Lecture Notes in Computer Science, Springer, pp. 528539.
Micali, S. and Vazirani, V. V. (1980), An O (√|V|⋅|E|) algorithm for finding maximum matching in general graphs. In Proceedings of the 21st Annual Symposium on Foundations of Computer Science (SFCS 1980), IEEE, pp. 1727.
Miller, D. L. and Pekny, J. F. (1995), ‘A staged primal–dual algorithm for perfect $b$ -matching with edge capacities’, ORSA J. Comput. 7, 298320.
Minoux, M. (1978), Accelerated greedy algorithms for maximizing submodular set functions. In Optimization Techniques: Proceedings of the 8th IFIP Conference on Optimization Techniques (IFIP 1977) (Stoer, J., ed.), Springer, pp. 234243.
Motwani, R. (1994), ‘Average-case analysis of algorithms for matchings and related problems’, J. Assoc. Comput. Mach. 41, 13291356.
Müller-Hannemann, M. and Schwartz, A. (2000), ‘Implementing weighted $b$ -matching algorithms: Insights from a computational study’, J. Exp. Algorithmics 5, 8.
Murphy, R. C., Wheeler, K. B., Barrett, B. W. and Ang, J. A. (2010), Introducing the Graph 500. In Proceedings of the Cray User’s Group Meeting (CUG), 2010.
Naim, M. and Manne, F. (2018), Scalable b-matching on GPUs. In Proceedings of the International Parallel and Distributed Processing Symposium Workshops (IPDPS), pp. 637646.
Naim, M., Manne, F., Halappanavar, M., Tumeo, A. and Langguth, J. (2015), Optimizing approximate weighted matching on Nvidia Kepler K40. In IEEE 22nd International Conference on High Performance Computing (HiPC 2015), pp. 105114.
Natanzon, A., Shamir, R. and Sharan, R. (2000), ‘A polynomial approximation for the minimum fill-in problem’, SIAM J. Comput. 30, 10671079.
U. Naumann and O. Schenk, eds (2012), Combinatorial Scientific Computing, CRC Press.
Norman, R. Z. and Rabin, M. O. (1959), ‘An algorithm for a minimum cover of a graph’, Proc. Amer. Math. Soc. 10, 315319.
Olschowka, M. and Neumaier, A. (1996), ‘A new pivoting strategy for Gaussian elimination’, Linear Algebra Appl. 240(suppl. C), 131151.
Padberg, M. W. and Rao, M. R. (1982), ‘Odd minimum cut-sets and $b$ -matchings’, Math. Oper. Res. 7, 6780.
Pettie, S. and Sanders, P. (2004), ‘A simpler linear time $2/3-\unicode[STIX]{x1D716}$ approximation for maximum weight matching’, Inform. Process. Lett. 91, 271276.
Pinar, A., Chow, E. and Pothen, A. (2006), ‘Combinatorial algorithms for computing column space bases that have sparse inverses’, Electron. Trans. Numer. Anal. 22, 122145.
Pothen, A. (1993), ‘Predicting the structure of sparse orthogonal factors’, Linear Algebra Appl. 194, 183203.
Pothen, A. and Fan, C.-J. (1990), ‘Computing the block triangular form of a sparse matrix’, ACM Trans. Math. Softw. 16, 303324.
Preis, R. (1999,), Linear time 1/2-approximation algorithm for maximum weighted matching in general graphs. In 1999 Proceedings of 16th Annual Symposium on Theoretical Aspects of Computer Science (STACS 99), Vol. 1563 of Lecture Notes in Computer Science, Springer, pp. 259269.
Pulleyblank, W. R. (1973), Faces of matching polyhedra. PhD thesis, Faculty of Mathematics, University of Waterloo.
Rajagopalan, S. and Vazirani, V. V. (1993), Primal–dual RNC approximation algorithms for (multi)-set (multi)-cover and covering integer programs. In Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science (SFCS ’93), IEEE, pp. 322331.
Schrijver, A. (2003), Combinatorial Optimization: Polyhedra and Efficiency, Vol. A: Paths, Flows, Matchings , Springer.
Sinkhorn, R. and Knopp, P. (1967), ‘Concerning nonnegative matrices and doubly stochastic matrices’, Pacific J. Math. 21, 343348.
Spencer, T. H. and Mayr, E. W. (1984), Node weighted matching. In Proceedings of the 11th Colloquium on Automata, Languages, and Programming (ICALP), Vol. 172 of Lecture Notes in Computer Science, Springer, pp. 454464.
Subramanya, A. and Talukdar, P. P. (2014), Graph-Based Semi-Supervised Learning, Vol. 29 of Synthesis Lectures on Artificial Intelligence and Machine Learning, Morgan & Claypool.
Tabatabaee, V., Georgiadis, L. and Tassiulas, L. (2001), ‘QoS provisioning and tracking fluid policies in input queueing switches’, IEEE/ACM Trans. Netw. 9, 605617.
Tamir, A. and Mitchell, J. S. B. (1998), ‘A maximum $b$ -matching problem arising from median location models with applications to the roommates problem’, Math. Program. 80, 171194.
Tangwongsan, K. (2011), Efficient parallel approximation algorithms. PhD thesis, Carnegie Mellon University, Pittsburgh, PA.
Vazirani, V. V. (2003), Approximation Algorithms, Springer.
Williamson, D. P. and Shmoys, D. B. (2011), The Design of Approximation Algorithms, Cambridge University Press.
Wilson, L. B. (1972), ‘An analysis of the marriage matching assignment algorithm’, BIT 12, 569575.
Yannakakis, M. (1981), ‘Computing the minimum fill-in is NP-complete’, SIAM J. Algebraic Discrete Methods 2, 7779.

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Approximation algorithms in combinatorial scientific computing

  • Alex Pothen (a1), S. M. Ferdous (a2) and Fredrik Manne (a3)

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