The problem of finding a maximum independent set in an undirected graph is a well known NP-hard problem. On the other hand, the critical independent set problem is polynomially solvable. The relationship between these two problems is studied and a method that utilizes a nonempty critical independent set for solving the maximum independent set problem is developed. The efficiency of the proposed...

An independent set of a graph G = (V, E) with vertices V and edges E is a subset S ⊆ V, such that the subgraph induced by S does not contain any edges. The goal of the maximum independent set problem (MIS problem) is to find an independent set of maximum size. It is equivalent to the well-known vertex cover problem (VC problem) and maximum clique problem. This thesis consists of two main parts....

A bipartite graph G = (V;W; E) is called convex if the vertices in W can be ordered in such a way that the elements of W adjacent to any vertex v 2 V form an interval (i.e. a sequence consecutively numbered vertices). Such a graph can be represented in a compact form that requires O(n) space, where n = maxfjV j; jW jg. Given a convex bipartite graph G in the compact form Dekel and Sahni designe...

We consider the following map labelling problem: given distinct points p,p, . . . ,pn in the plane, and given σ, find a maximum cardinality set of pairwise disjoint axis-parallel σ× σ squares Q1, Q2, . . . , Qr. This problem reduces to that of finding a maximum cardinality independent set in an associated graph called the conflict graph. We describe several heuristics for the maximum cardinalit...

We study the maximum weighted independent-set problem on interval graphs with uncertainty on the vertex weights. We use the absolute robustness criterion and the min-max regret criterion to evaluate solutions. For a discrete scenario set, we find that the problem is NP-hard for each of the robustness criteria; we also provide pseudo-polynomial time algorithms when there is a constant number of ...

We study the computational complexity of finding a maximum independent set of vertices in a planar graph. In general, this problem is known to be NP-hard. However, under certain restrictions it becomes polynomial-time solvable. We identify a graph parameter to which the complexity of the problem is sensible and produce a number of both negative (intractable) and positive (solvable in polynomial...

A very general technique for converting approximation algorithms for the vertex coloring problem in a class of graphs into approximation algorithms for the maximum weight independent set problem (MWIS) in the same class of graphs is presented. The technique consists of solving an LP-relaxation of the MWIS problem with certain clique inequalities, constructing an instance of the vertex coloring ...

We consider the problem of finding a maximum independent set in a random graph. The random graphG is modelled as follows. Every edge is included independently with probability d n , where d is some sufficiently large constant. Thereafter, for some constant α, a subset I of αn vertices is chosen at random, and all edges within this subset are removed. In this model, the planted independent set I...

We study a natural extension of the Maximum Weight Independent Set Problem (MWIS), one of the most studied optimization problems in Graph algorithms. We are given a graph G = (V,E), a weight function w : V → R, a budget function b : V → Z, and a positive integer B. The weight (resp. budget) of a subset of vertices is the sum of weights (resp. budgets) of the vertices in the subset. A k-budgeted...

The maximum independent set problem is an NP-hard problem. In this paper, we consider Algorithm MAX, which is a polynomial time algorithm for finding a maximal independent set in a graph G. We present a set of forbidden induced subgraphs such that Algorithm MAX always results in finding a maximum independent set ofG. We also describe two modifications of Algorithm MAX and sets of forbidden indu...