Abstract
We study the classical 0–1 knapsack problem with additional restrictions on pairs of items. A conflict constraint states that from a certain pair of items at most one item can be contained in a feasible solution. Reversing this condition, we obtain a forcing constraint stating that at least one of the two items must be included in the knapsack. A natural way for representing these constraints is the use of conflict (resp. forcing) graphs. By modifying a recent result of Lokstanov et al. (Proceedings of the 25th annual ACM-SIAM symposium on discrete algorithms, SODA, pp 570–581, 2014) we derive a fairly complicated FPTAS for the knapsack problem on weakly chordal conflict graphs. Next, we show that the techniques of modular decompositions and clique separators, widely used in the literature for solving the independent set problem on special graph classes, can be applied to the knapsack problem with conflict graphs. In particular, we can show that every positive approximation result for the atoms of prime graphs arising from such a decomposition carries over to the original graph. We point out a number of structural results from the literature which can be used to show the existence of an FPTAS for several graph classes characterized by the exclusion of certain induced subgraphs. Finally, a PTAS for the knapsack problem with H-minor free conflict graph is derived. This includes planar graphs and, more general, graphs of bounded genus. The PTAS is obtained by expanding a general result of Demaine et al. (Proceedings of 46th annual IEEE symposium on foundations of computer science, FOCS 2005, pp 637–646, 2005). The knapsack problem with forcing graphs can be transformed into a minimization knapsack problem with conflict graphs. It follows immediately that all our FPTAS results of the current and a previous paper carry over from conflict graphs to forcing graphs. In contrast, the forcing graph variant is already inapproximable on planar graphs.
Highlights
The classical 0–1 knapsack problem is an N P-hard discrete optimization problem known to be relatively easy to solve in practice
J Comb Optim (2017) 33:1300–1323 forcing graphs can be transformed into a minimization knapsack problem with conflict graphs
The underlying technique of this construction was developed by Fomin and Villanger (2010). It was expanded by Lokshtanov et al (2014) who recently gave a polynomial time algorithm for the weighted independent set problem on P5-free graphs. Note that this technique is crucial for K C G on weakly chordal graphs since it allows dynamic programming
Summary
The classical 0–1 knapsack problem is an N P-hard discrete optimization problem known to be relatively easy to solve in practice. 5. For the minimum weight vertex cover problem it is known that no PTAS exists, but various papers study the approximability on special graph classes. For the minimum weight vertex cover problem it is known that no PTAS exists, but various papers study the approximability on special graph classes Both independent set and vertex cover are polynomially solvable on perfect graphs. The underlying technique of this construction was developed by Fomin and Villanger (2010) It was expanded by Lokshtanov et al (2014) who recently gave a polynomial time algorithm for the weighted independent set problem on P5-free graphs. Note that this technique is crucial for K C G on weakly chordal graphs since it allows dynamic programming. This analogy breaks down for planar graphs: even finding a feasible solution for K F G on planar graphs is strongly N P-complete while there exists a PTAS for K C G
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