The Exact Weighted Independent Set Problem in Perfect Graphs and Related Classes

The Knapsack Problem with Conflict Pair Constraints on Bipartite Graphs and Extensions

“…In KPCC, if we restrict the budget constraint to be satisfied as an equality, we obtain an instance of the exact knapsack problem with conflict pair constraints (KPCC-exact). The feasibility version of this problem was investigated by Milanic and Monnot [3,33], where it was shown that the problem is strongly NP-complete on bipartite graphs with maximum vertex degree of three. The authors also presented a pseudo-polynomial algorithm for the problem in cographs, chordal graphs, and interval graphs.…”

“…The KPCC has been investigated by many researchers, and it is known under different names. Some authors call it the maximum independent set problem with a budget constraint [1][2][3], while some others call it the disjunctively constrained knapsack problem [4][5][6][7][8][9][10]. We use the name knapsack problem with conflict pair constraints since it belongs to the broader class of combinatorial optimization problems with conflict pair constraints [11][12][13][14][15][16][17][18][19][20].…”

Knapsack problem with objective value gaps

“…Leclerc [9] and Barahona and Pulleyblank [2] suggested pseudo-polynomial time algorithms for the exact spanning tree problem, the exact perfect matching problem on planar graph, the exact cycle sum problem on planar directed graph and the exact cut problem on planar or toroidal graph. Milanic and Monnot [12] proved that the exact weighted independent set problem and the exact weighted maximum independent set problem are strongly NP-complete for cubic bipartite graphs and that these problems are pseudo-polynomial solvable for mK 2 -free graphs, k-thin graphs (including interval graphs), chordal graphs, AT-free graphs, (claw,net)-free graphs, distance-hereditary graphs, circle graphs, graphs of bounded treewidth, graphs of bounded clique-width, and certain sub-classes of P 5 -free and fork-free graphs. The results of Milanic and Monnot [12] imply that the exact perfect matching problem is pseudo-polynomially solvable for graphs with treewidth bounded by a constant.…”

“…, U . Therefore, the NP-completeness results for exact problems mentioned in the previous paragraph apply for their counterparts with two gaps, the algorithm of Karzanov [6] can be employed to solve the assignment problem with gaps in polynomial time in the case of 0-1 costs, and the algorithms of Leclerc [9], Barahona and Pulleyblank [2] and Milanic and Monnot [12] can be employed to solve the relevant graph problems with gaps in pseudo-polynomial time. Computational complexity of the assignment problem with two gaps in unary encoding is an open question.…”

Simple paths with exact and forbidden lengths

“…Pseudo‐polynomial time algorithms for the exact value spanning tree problem, the exact value perfect matching problem on planar graph, the exact value cycle sum problem on planar directed graph and the exact value cut problem on planar or toroidal graphs have been presented by Leclerc () and Barahona and Pulleyblank (). A number of computational complexity and algorithmic results for the exact weight (maximum) independent set problem on various classes of graphs have been obtained by Milanic and Monnot (). Computational complexity of the exact weight subgraph problems, in which the number of vertices of the subgraph is a constant, has been studied by Vassilevska and Williams () and Abboud and Lewi ().…”