Abstract
Counting and enumerating maximal and maximum independent sets are well-studied problems in graph theory. In this paper we introduce methods to count and enumerate maximal/maximum independent sets in threshold graphs and k-threshold graphs and improve former results for these problems. The results can be applied to combinatorial optimization problems, and in particular to different variations of the knapsack problem. As feasible solutions for instances of those problems correspond to independent sets in threshold graphs and k-threshold graphs, we obtain polynomial time results for special knapsack and multidimensional knapsack instances. Also, we show lower and upper bounds for the number of necessary bins in several bin packing problems.
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Notes
The class of \(\#\text{ P }\)-complete problems is a class of computationally equivalent counting problems which are at least as difficult as NP-complete problems.
A graph G with n vertices and m edges is \(\ell \)-sparse if \(m \le \ell \cdot n\). It is uniformly \(\ell \)-sparse if every subgraph of G is \(\ell \)-sparse.
For \(A=A'=\{a_1,a_2\}\), \(s_1=2\), \(s_2=7\), \(c=10\), the right condition is true, but for \(j=j'=2\) the left condition is not true.
Because of the hardness result in Yannakakis (1982), we assume that we are given k threshold graphs which cover the edge set of G.
Please note the different but equivalent definition of edges within threshold graphs in Caprara et al. (2004). By using values \(w_i\) for the vertices \(v_i\) such that \(0<w_i\le 1\), the authors in Caprara et al. (2004) define edges \(\{v_i,v_j\}\) whenever \(w_i+w_j\le 1\). For our notation see the statement (3.) of Theorem 3.
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This work was funded in part by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—388221852.
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A short version of this paper appeared in the Proceedings of the International Conference on Operations Research (OR 2017), see Gurski and Rehs (2018).
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Gurski, F., Rehs, C. Counting and enumerating independent sets with applications to combinatorial optimization problems. Math Meth Oper Res 91, 439–463 (2020). https://doi.org/10.1007/s00186-019-00696-4
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DOI: https://doi.org/10.1007/s00186-019-00696-4