Weight Independent Set Problem Research Articles

Delay-Optimal Joint Processing in Computation-Constrained Fog Radio Access Networks

With fog radio access networks (F-RANs), the computation capability is provided in the physical proximity of the users, which can significantly lower the delay and mitigate the heavy traffic over backhaul links. As the number of remote radio head (RRH) increases, the computational complexity becomes a severe issue and it is necessary to group the RRHs into multiple clusters. In this paper, we optimize the joint processing strategy, including the RRH clustering and the RRH-server matching to minimize the delay for F-RANs. We model the delay-optimal joint processing problem in computation-constrained F-RANs as a Markov decision process (MDP) problem. By deriving the optimality condition of this MDP, we obtain a per-slot weighted sum rate maximization problem, in which the RRH clustering and the RRH-server matching are solved jointly. Specifically, we transform the weighted sum rate maximization problem into a combinatorial auction problem (CAP). Since the CAP is NP-hard, we greedily obtain an initial solution of CAP and gradually improve the performance by adopting local α-good improvement algorithm based on the weighted independent set problem (WISP). Furthermore, we theoretically derive the bound of the performance of the proposed algorithm.

Independent sets in some classes of $$S_{i, j, k}$$ S i , j , k -free graphs

The maximum weight independent set (MWIS) problem on graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum total weight. In 1982, Alekseev (Comb Algebraic Methods Appl Math 132:3---13, 1982) showed that the M(W)IS problem remains NP-complete on H-free graphs, whenever H is connected, but neither a path nor a subdivision of the claw. We will focus on graphs without a subdivision of a claw. For integers $$i, j, k \ge 1$$i,j,kź1, let $$S_{i, j, k}$$Si,j,k denote a tree with exactly three vertices of degree one, being at distance i, j and k from the unique vertex of degree three. Note that $$S_{i,j, k}$$Si,j,k is a subdivision of a claw. The computational complexity of the MWIS problem for the class of $$S_{1, 2, 2}$$S1,2,2-free graphs, and for the class of $$S_{1, 1, 3}$$S1,1,3-free graphs are open. In this paper, we show that the MWIS problem can be solved in polynomial time for ($$S_{1, 2, 2}, S_{1, 1, 3}$$S1,2,2,S1,1,3, co-chair)-free graphs, by analyzing the structure of the subclasses of this class of graphs. This also extends some known results in the literature.

A Generalization of Join and an Algorithmic Recognition Problem

We consider a new graph operation c-join which generalizes join and co-join. We show that odd hole-free graphs ( odd antihole-free graphs) are closed under c-join and describe a polynomial time algorithm to recognize graphs that admit a c-join. The time complexity of the (a) recognition problem, (b) maximum weight independent set (MWIS) problem, and (c) minimum coloring (MC) problem for odd hole-free graphs are still unknown. Let H be an odd hole-free graph that contains an odd antihole as an induced subgraph and GH be the class of all graphs generated from the induced subgraphs of H by using c-join recursively. Then GH is odd hole-free, contains all P4-free graphs, complement of all bipartite graphs, and some imperfect graphs. We show that the MWIS problem, maximum weight clique (MWC) problem, MC problem, and minimum clique cover (MCC) problem can be solved efficiently for GH .