The knapsack problem with neighbour constraints

We study a constrained version of the knapsack problem in which dependencies between items are given by the adjacencies of a graph. In the 1-neighbour knapsack problem, an item can be selected only if at least one of its neighbours is also selected. In the all-neighbours knapsack problem, an item can be selected only if all its neighbours are also selected. We give approximation algorithms and hardness results when the vertices have both uniform and arbitrary weight and profit functions, and when the dependency graph is directed and undirected.

[1]  Vijay V. Vazirani,et al.  Approximation Algorithms , 2001, Springer Berlin Heidelberg.

[2]  Samir Khuller,et al.  The Budgeted Maximum Coverage Problem , 1999, Inf. Process. Lett..

[3]  Vahab S. Mirrokni,et al.  Maximizing Nonmonotone Submodular Functions under Matroid or Knapsack Constraints , 2009, SIAM J. Discret. Math..

[4]  A. Russell,et al.  The Minimum k-Colored Subgraph Problem in Haplotyping and DNA Primer Selection , 2004 .

[5]  Paolo Toth,et al.  Knapsack Problems: Algorithms and Computer Implementations , 1990 .

[6]  Hadas Shachnai,et al.  Maximizing submodular set functions subject to multiple linear constraints , 2009, SODA.

[7]  Stavros G. Kolliopoulos,et al.  Partially-Ordered Knapsack and Applications to Scheduling , 2002, ESA.

[8]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[9]  Maxim Sviridenko,et al.  A note on maximizing a submodular set function subject to a knapsack constraint , 2004, Oper. Res. Lett..

[10]  Richard M. Karp,et al.  Reducibility Among Combinatorial Problems , 1972, 50 Years of Integer Programming.

[11]  Andreas S. Schulz,et al.  Revisiting the Greedy Approach to Submodular Set Function Maximization , 2007 .

[12]  O. Goldschmidt,et al.  Note: On the set-union knapsack problem , 1994 .

[13]  Oscar H. Ibarra,et al.  Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems , 1975, JACM.

[14]  Natashia Boland,et al.  Clique-based facets for the precedence constrained knapsack problem , 2012, Math. Program..

[15]  Hans Kellerer,et al.  Knapsack problems , 2004 .

[16]  D. S. Johnson,et al.  On Knapsacks, Partitions, and a New Dynamic Programming Technique for Trees , 1983, Math. Oper. Res..

[17]  Robert Krauthgamer,et al.  Polylogarithmic inapproximability , 2003, STOC '03.