Journal of Combinatorial Optimization

, Volume 36, Issue 3, pp 937–964 | Cite as

Subset sum problems with digraph constraints

  • Laurent Gourvès
  • Jérôme MonnotEmail author
  • Lydia Tlilane


We introduce and study optimization problems which are related to the well-known Subset Sum problem. In each new problem, a node-weighted digraph is given and one has to select a subset of vertices whose total weight does not exceed a given budget. Some additional constraints called digraph constraints and maximality need to be satisfied. The digraph constraint imposes that a node must belong to the solution if at least one of its predecessors is in the solution. An alternative of this constraint says that a node must belong to the solution if all its predecessors are in the solution. The maximality constraint ensures that no superset of a feasible solution is also feasible. The combination of these constraints provides four problems. We study their complexity and present some approximation results according to the type of input digraph, such as directed acyclic graphs and oriented trees.


Subset sum Maximal problems Digraph constraints Complexity Directed acyclic graphs Oriented trees PTAS 



Many thanks to the anonymous referee and the anonymous associate editor for pertinent and useful comments and suggestions.


  1. Alimonti P, Kann V (2000) Some APX-completeness results for cubic graphs. Theor Comput Sci 237(1–2):123–134MathSciNetCrossRefzbMATHGoogle Scholar
  2. Arkin EM, Bender MA, Mitchell JSB, Skiena S (2003) The lazy bureaucrat scheduling problem. Inf Comput 184(1):129–146MathSciNetCrossRefzbMATHGoogle Scholar
  3. Bang-Jensen J, Hell P (1993) Fast algorithms for finding hamiltonian paths and cycles in in-tournament digraphs. Discrete Appl Math 41(1):75–79MathSciNetCrossRefzbMATHGoogle Scholar
  4. Becker RI, Perl Y (1995) The shifting algorithm technique for the partitioning of trees. Discrete Appl Math 62(1–3):15–34MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bervoets S, Merlin V, Woeginger GJ (2015) Vote trading and subset sums. Oper Res Lett 43:99–102MathSciNetCrossRefGoogle Scholar
  6. Biggs NL, Lloyd EK, Wilson RJ (1976) Graph theory 1736–1936. Clarendon Press, OxfordzbMATHGoogle Scholar
  7. Boland N, Bley A, Fricke C, Froyland G, Sotirov R (2012) Clique-based facets for the precedence constrained knapsack problem. Math Program 133(1–2):481–511MathSciNetCrossRefzbMATHGoogle Scholar
  8. Borradaile G, Heeringa B, Wilfong GT (2012) The knapsack problem with neighbour constraints. J Discrete Algorithms 16:224–235MathSciNetCrossRefzbMATHGoogle Scholar
  9. Chlebík M, Chlebíková J (2008) Approximation hardness of dominating set problems in bounded degree graphs. Inf Comput 206(11):1264–1275MathSciNetCrossRefzbMATHGoogle Scholar
  10. Cho G, Shaw DX (1997) A depth-first dynamic programming algorithm for the tree knapsack problem. INFORMS J Comput 9(4):431–438MathSciNetCrossRefzbMATHGoogle Scholar
  11. Cieliebak M, Eidenbenz S, Pagourtzis A (2003) Composing equipotent teams. In: Fundamentals of computation theory, 14th international symposium, FCT 2003, Malmö, Sweden, August 12–15, 2003, Proceedings, pp 98–108Google Scholar
  12. Cieliebak M, Eidenbenz S, Pagourtzis A, Schlude K (2008) On the complexity of variations of equal sum subsets. Nord J Comput 14(3):151–172MathSciNetzbMATHGoogle Scholar
  13. Eggermont C, Woeginger GJ (2013) Motion planning with pulley, rope, and baskets. Theory Comput Syst 53(4):569–582MathSciNetCrossRefzbMATHGoogle Scholar
  14. Esfahbod B, Ghodsi M, Sharifi A (2003) Common-deadline lazy bureaucrat scheduling problems. In: Dehne FKHA, Sack J, Smid MHM (eds), Algorithms and data structures, 8th international workshop, WADS 2003, Ottawa, Ontario, Canada, July 30–August 1, 2003, Proceedings, vol 2748 of lecture notes in computer science, Springer, pp 59–66Google Scholar
  15. Gai L, Zhang G (2008) On lazy bureaucrat scheduling with common deadlines. J Comb Optim 15(2):191–199MathSciNetCrossRefzbMATHGoogle Scholar
  16. Garey M, Johnson D (1979) Computers and intractability, vol 174. Freeman, New YorkzbMATHGoogle Scholar
  17. Gourvès L, Monnot J, Pagourtzis A (2014) The lazy matroid problem. In: Diaz J, Lanese I, Sangiorgi D (eds), Theoretical computer science—8th IFIP TC 1/WG 2.2 international conference, TCS 2014, Rome, Italy, September 1–3, 2014. Proceedings, vol 8705 of lecture notes in computer science, Springer, pp 66–77Google Scholar
  18. Gourvès L, Monnot J, Pagourtzis AT (2013) The lazy bureaucrat problem with common arrivals and deadlines: approximation and mechanism design. In: Fundamentals of Computation Theory, Springer, pp 171–182Google Scholar
  19. Hajiaghayi MT, Jain K, Lau LC, Mandoiu II, Russell A, Vazirani VV (2006) Minimum multicolored subgraph problem in multiplex PCR primer set selection and population haplotyping. In: Alexandrov VN, van Albada GD, Sloot PMA, Dongarra J (eds) Computational science—ICCS 2006, 6th international conference, Reading, UK, May 28–31, 2006, Proceedings, Part II, vol 3992 of lecture notes in computer science, Springer, pp 758–766Google Scholar
  20. Halldórsson MM (1993) Approximating the minimum maximal independence number. Inf Process Lett 46(4):169–172MathSciNetCrossRefzbMATHGoogle Scholar
  21. Hastad J (1996) Clique is hard to approximate within \(n^{1-\varepsilon }\). In: Proceedings 37th annual symposium on foundations of computer science, 1996, IEEE, pp 627–636Google Scholar
  22. Johnson DS, Niemi KA (1983) On knapsacks, partitions, and a new dynamic programming technique for trees. Math Oper Res 8(1):1–14MathSciNetCrossRefzbMATHGoogle Scholar
  23. Kellerer H, Pferschy U, Pisinger D (2004) Knapsack problems. Springer, BerlinCrossRefzbMATHGoogle Scholar
  24. Kolliopoulos SG, Steiner G (2007) Partially ordered knapsack and applications to scheduling. Discrete Appl Math 155(8):889–897MathSciNetCrossRefzbMATHGoogle Scholar
  25. Kothari A, Suri S, Zhou Y (2005) Interval subset sum and uniform-price auction clearing. In: Wang L (ed) Computing and Combinatorics, 11th annual international conference, COCOON 2005, Kunming, China, August 16–29, 2005, Proceedings, vol 3595 of lecture notes in computer science, Springer, pp 608–620Google Scholar
  26. Manlove D (1999) On the algorithmic complexity of twelve covering and independence parameters of graphs. Discrete Appl Math 91(1–3):155–175MathSciNetCrossRefzbMATHGoogle Scholar
  27. Woeginger GJ, Yu Z (1992) On the equal-subset-sum problem. Inf Process Lett 42(6):299–302MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Laurent Gourvès
    • 1
  • Jérôme Monnot
    • 1
    Email author
  • Lydia Tlilane
    • 1
  1. 1.CNRS, LAMSADEUniversité Paris-Dauphine, PSL Research UniversityParisFrance

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