Algorithmic and hardness results for some graph optimization problems

Abstract

Many real world optimization problems can be modeled as graph optimization problems. However, several graph optimization problems that are practically significant are NP-hard for general graphs, and hence, it is unlikely to find exact solutions in polynomial-time. One approach to tackle this is to study the problem for some restricted graph classes, as most of the time graphs obtained by modeling real world problems exhibit some special properties. Many of the researchers are working to design polynomial-time algorithms for NP-hard graph optimization problems for restricted graph classes. This thesis considers five important graph optimization problems, namely (i) MAXIMUM I NTERNAL SPANNING TREE Problem,(ii) MINIMUM EDGE TOTAL DOMINATING SET Problem, (iii) GRUNDY (DOUBLE) DOMINATION Problem, (iv) MAXIMUM WEIGHTED EDGE BICLIQUE Problem and (v) NEIGHBOR-LOCATING COLORING Problem. Aspanning tree of a graph G containing the maximum number of internal vertices among all spanning trees of G is called a maximum internal spanning tree (MIST) of G. The MAXIMUM INTERNAL SPANNING TREE problem is to find a MIST for a given graph G. For some special graph classes, we provide linear-time algorithms to compute a MIST and relate the number of internal vertices in a MIST with an important graph parameter. For a graph G = (V,E) without an isolated edge, a set D ⊆ E is called an edge total dominating set (ETD-set) of G if every edge e ∈ E is adjacent to at least one edge of D. For a given graph Gwith no isolated edges, the MINIMUM EDGE TOTAL DOMINATING SET Problem is to f ind an ETD-set of G with minimum cardinality. We study the problem in subclasses of bipartite graphs and chordal graphs. We prove some results from the approximation aspects. We also discuss the complexity difference between the MINIMUM EDGE DOMINATING SET problem and the MINIMUM EDGE TOTAL DOMINATING SET problem. Asequence S = (v1,v2,...,vk) is a dominating sequence if N[vi] \ i−1 j=1 N[vj] ̸ = ∅ holds for every i ∈ {2,...,k} and S = {v1,v2,...,vk} is a dominating set of G. The GRUNDY DOMINATION problem is to find a dominating sequence of maximum length for a given graph G. Moreover, a sequence S = (v1,v2,...,vk) is a double dominating sequence if (i) for each i ∈ [k], the vertex vi dominates at least one vertex of G that was dominated at most once by the previous vertices of S and (ii), S = {v1,v2,...,vk} is a double dominating set of G. The GRUNDY DOUBLE DOMINATION problem is to find a double dominating sequence of maximum length for a given graph G. We find some graph classes for which the problems are solvable in polynomial-time. Additionally, we identify certain graph classes for which these problems are NP-hard. Given a weighted graph G = (V,E,w), where each edge e ∈ E has a weight w(e) ∈ R, the MAXIMUM WEIGHTED EDGE BICLIQUE problemistofindabicliqueC ofGsuchthat the sum of the weights of edges of C is maximum. We obtain some algorithmic results when w(e) ∈ R+. Aproper coloring of a graph is called a neighbor-locating coloring if, for any two vertices of the same color, the sets of colors of their neighborhoods are different. Given a graph G, the NEIGHBOR LOCATING COLORING problem requires assigning a color to each vertex of G such that the coloring is a neighbor-locating coloring and the number of colors used is minimized. We provide some algorithmic and combinatorial results for this parameter in some special graph classes.