Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 62: Issue 1
Graph-Like Scheduling Problems and Property B
Author:
John Machacek Department of Mathematics and Computer Science, Hampden-Sydney College, Hampden-Sydney, VA 23943, USA

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Pages:
7–20
Online Publication Date:
02 Apr 2025
Publication Date:
09 Apr 2025
Article Category:
Research Article
DOI:
https://doi.org/10.1556/012.2025.04327
Keywords:
Scheduling problems; property B; probabilistic method
Restricted access

Breuer and Klivans defined a diverse class of scheduling problems in terms of Boolean formulas with atomic clauses that are inequalities. We consider what we call graph-like scheduling problems. These are Boolean formulas that are conjunctions of disjunctions of atomic clauses (𝑥𝑖 ≠ 𝑥𝑗). These problems generalize proper coloring in graphs and hypergraphs. We focus on the existence of a solution with all 𝑥i taking the value of 0 or 1 (i.e. problems analogous to the bipartite case). When a graph-like scheduling problem has such a solution, we say it has property B just as is done for 2-colorable hypergraphs. We define the notion of a 𝜆-uniform graph-like scheduling problem for any integer partition 𝜆. Some bounds are attained for the size of the smallest 𝜆-uniform graph-like scheduling problems without B. We make use of both random and constructive methods to obtain bounds. Just as in the case of hypergraphs finding tight bounds remains an open problem.

  • [1]

    H. L. Abbott and L. Moser. On a combinatorial problem of Erdős and Hajnal. Canad. Math. Bull., 7:177–181, 1964.

  • [2]

    Sachin Aglave, V. A. Amarnath, Saswata Shannigrahi, and Shwetank Singh. Improved bounds for uniform hypergraphs without property B. Australas. J. Combin., 76(part 1):73–86, 2020.

    • Search Google Scholar
    • Export Citation
  • [3]

    Noga Alon and Joel H. Spencer. The probabilistic method. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience [John Wiley & Sons], New York, second edition, 2000. With an appendix on the life and work of Paul Erdős.

    • Search Google Scholar
    • Export Citation
  • [4]

    Jean-Christophe Aval, Nantel Bergeron, and John Machacek. New invariants for permutations, orders and graphs. Adv. in Appl. Math., 121:102080, 30, 2020.

    • Search Google Scholar
    • Export Citation
  • [5]

    Felix Bernstein. Zur theorie der trigonometrische reihen. Leipz. Ber., 60:325–328, 1908.

  • [6]

    Felix Breuer and Caroline J. Klivans. Scheduling problems. J. Combin. Theory Ser. A, 139:59–79, 2016.

  • [7]

    Danila D. Cherkashin and Jakub Kozik. A note on random greedy coloring of uniform hypergraphs. Random Structures Algorithms, 47(3):407–413, 2015.

    • Search Google Scholar
    • Export Citation
  • [8]

    Richard Conway, William L. Maxwell, and Louis W. Miller. Theory of scheduling. Addison-Wesley Pub. Co Reading, Mass, 1967.

  • [9]

    P. Erdős. On a combinatorial problem. Nordisk Mat. Tidskr., 11:5–10, 40, 1963.

  • [10]

    P. Erdős. On a combinatorial problem. II. Acta Math. Acad. Sci. Hungar., 15:445–447, 1964.

  • [11]

    P. Erdős and A. Hajnal. On a property of families of sets. Acta Math. Acad. Sci. Hungar., 12:87–123, 1961.

  • [12]

    Michael R. Garey and David S. Johnson. Computers and intractability. W. H. Freeman and Co., San Francisco, Calif., 1979. A guide to the theory of NP-completeness, A Series of Books in the Mathematical Sciences.

    • Search Google Scholar
    • Export Citation
  • [13]

    Donald W. Gillies and Jane W.-S. Liu. Scheduling tasks with and/or precedence constraints. SIAM J. Comput., 24(4):797–810, 1995.

  • [14]

    R. L. Graham, E. L. Lawler, J. K. Lenstra, and A. H. G. Rinnooy Kan. Optimization and approximation in deterministic sequencing and scheduling: a survey. In Interfaces between computer science and operations research (Proc. Sympos., Math. Centrum, Amsterdam, 1976), volume 99 of Math. Centre Tracts, pages 169–214. Math. Centrum, Amsterdam, 1978.

    • Search Google Scholar
    • Export Citation
  • [15]

    Ronald L. Graham. Combinatorial scheduling theory. In Lynn Arthur Steen, editor, Mathematics Today Twelve Informal Essays, pages 183–211. Springer New York, New York, NY, 1978.

    • Search Google Scholar
    • Export Citation
  • [16]

    Branko Grünbaum. Acyclic colorings of planar graphs. Israel J. Math., 14:390–408, 1973.

  • [17]

    Eun-Seok Kim and Marc E. Posner. Parallel machine scheduling with s-precedence constraints. IIE Transactions, 42(7):525–537, 2010.

  • [18]

    L. Lovász. Coverings and coloring of hypergraphs. In Proceedings of the Fourth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1973), pages 3–12, 1973.

    • Search Google Scholar
    • Export Citation
  • [19]

    John Machacek. Plurigraph coloring and scheduling problems. Electron. J. Combin., 24(2):Paper No. 2.29, 2017.

  • [20]

    Brendan D. McKay and Adolfo Piperno. Practical graph isomorphism, II. J. Symbolic Comput., 60:94–112, 2014.

  • [21]

    Edwin W. Miller. On a property of families of sets. C. R. Soc. Sci. Varsovie, Cl. III, 30:31–38, 1937.

  • [22]

    Patric R. J. Östergård. On the minimum size of 4-uniform hypergraphs without property 𝐵. Discrete Appl. Math., 163(part 2):199–204, 2014.

    • Search Google Scholar
    • Export Citation
  • [23]

    Jaikumar Radhakrishnan and Aravind Srinivasan. Improved bounds and algorithms for hypergraph 2-coloring. Random Structures Algorithms, 16(1):4–32, 2000.

    • Search Google Scholar
    • Export Citation
  • [24]

    A. M. Raigorodskii and D. D. Cherkashin. Extremal problems in hypergraph colouring. Uspekhi Mat. Nauk, 75(1(451)):95–154, 2020.

  • [25]

    A. M. Ra˘ıgorodski˘ı and D. A. Shabanov. The Erdős-Hajnal problem of hypergraph colorings, its generalizations, and related problems. Uspekhi Mat. Nauk, 66(5(401)):109–182, 2011.

    • Search Google Scholar
    • Export Citation
  • [26]

    Mario Sanchez. Möbius inversion as duality for Hopf monoids. Sém. Lothar. Combin., 82B:Art. 91, 12, 2020.

  • [27]

    P. D. Seymour. A note on a combinatorial problem of Erdős and Hajnal. J. London Math. Soc. (2), 8:681–682, 1974.

  • [28]

    É. Sopena. Homomorphisms and colourings of oriented graphs: an updated survey. Discrete Math., 339(7):1993–2005, 2016.

  • [29]

    Carsten Thomassen. The even cycle problem for directed graphs. J. Amer. Math. Soc., 5(2):217–229, 1992.

  • [30]

    B. Toft. On colour-critical hypergraphs. In Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vol. III, pages 1445–1457. Colloq. Math. Soc. János Bolyai, Vol. 10. 1975.

    • Search Google Scholar
    • Export Citation
  • [31]

    D. J. A. Welsh and M. B. Powell. An upper bound for the chromatic number of a graph and its application to timetabling problems. The Computer Journal, 10(1):85–86, 1967.

    • Search Google Scholar
    • Export Citation
  • [32]

    Jacob A. White. Quasisymmetric functions from combinatorial Hopf monoids and Ehrhart theory. In 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016), Discrete Math. Theor. Comput. Sci. Proc., BC, pages 1215–1226. Assoc. Discrete Math. Theor. Comput. Sci., Nancy, [2016] ©2016.

    • Search Google Scholar
    • Export Citation
  • [33]

    D. C. Wood. A technique for colouring a graph applicable to large scale timetabling problems. The Computer Journal, 12(4):317–319, 1969.

    • Search Google Scholar
    • Export Citation
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Studia Scientiarum Mathematicarum Hungarica
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0081-6906
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Studia Scientiarum Mathematicarum Hungarica
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Foundation		 1966
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ISSN 0081-6906 (Print)
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