Set Problem In Graphs Research Articles

A linear programming reformulation of the standard quadratic optimization problem

The problem of minimizing a quadratic form over the standard simplex is known as the standard quadratic optimization problem (SQO). It is NP-hard, and contains the maximum stable set problem in graphs as a special case. In this note, we show that the SQO problem may be reformulated as an (exponentially sized) linear program (LP). This reformulation also suggests a hierarchy of polynomial-time solvable LP's whose optimal values converge finitely to the optimal value of the SQO problem. The hierarchies of LP relaxations from the literature do not share this finite convergence property for SQO, and we review the relevant counterexamples.

A note on power domination in grid graphs

The problem of monitoring an electric power system by placing as few measurement devices in the system as possible is closely related to the well known vertex covering and dominating set problems in graphs (see [T.W. Haynes, S.M. Hedetniemi, S.T. Hedetniemi, M.A. Henning, Power domination in graphs applied to electrical power networks, SIAM J. Discrete Math. 15(4) (2002) 519–529]). A set S of vertices is defined to be a power dominating set of a graph if every vertex and every edge in the system is monitored by the set S (following a set of rules for power system monitoring). The minimum cardinality of a power dominating set of a graph is its power domination number. In this paper, we determine the power domination number of an n × m grid graph.

A Linear Programming Reformulation of the Standard Quadratic Optimization Problem

The problem of minimizing a quadratic form over the standard simplex is known as the standard quadratic optimization problem (SQO). It is NPhard, and contains the maximum stable set problem in graphs as a special case. In this note we show that the SQO problem may be reformulated as an (exponentially sized) linear program.

Online independent sets

We study the online version of the independent set problem in graphs. The vertices of an input graph are given one by one along with their edges to previous vertices, and the task is to decide whether to add each given vertex to an independent set solution. The goal is to maximize the size of the independent set, relative to the size of the optimal independent set. Since it is known that no online algorithm can attain competitive ratio better than n−1, where n denotes the number of vertices, we study here relaxations where the algorithm can hedge its bets by maintaining multiple alternative solutions. We introduce two models. In the first model, the algorithm can maintain a multiple number ( r(n)) of solutions (independent sets) and choose the largest one as the final solution. We show that the best competitive ratio for this model is θ(n/ log n) when r(n) is a polynomial and θ(n) when r(n) is a constant. In the second more powerful model, the algorithm can copy intermediate solutions and extend the copied solutions in different ways. We obtain an upper bound O(n/ log n) and a lower bound Ω(n/ log 3n) for the best possible competitive ratio when r(n) is a polynomial. Furthermore, we show a tight θ(n) bound when r(n) is a constant. Lower bound results of this paper hold also for randomized online algorithms against an oblivious adversary.

Domination in Graphs Applied to Electric Power Networks

The problem of monitoring an electric power system by placing as few measurement devices in the system as possible is closely related to the well-known vertex covering and dominating set problems in graphs. We consider the graph theoretical representation of this problem as a variation of the dominating set problem and define a set S to be a power dominating set of a graph if every vertex and every edge in the system is monitored by the set S (following a set of rules for power system monitoring). The minimum cardinality of a power dominating set of a graph G is the power domination number $\gamma_P(G)$. We show that the power dominating set (PDS) problem is NP-complete even when restricted to bipartite graphs or chordal graphs. On the other hand, we give a linear algorithm to solve the PDS for trees. In addition, we investigate theoretical properties of $\gamma_P(T)$ in trees T.

Approximation Algorithms for Connected Dominating Sets

The dominating set problem in graphs asks for a minimum size subset of vertices with the following property: each vertex is required to be either in the dominating set, or adjacent to some vertex i...

Stable sets and polynomials

Several applications of methods from nonlinear algebra to the stable set problem in graphs are surveyed. The most recent work cited was cowritten by A. Schrijver and involves nonlinear inequalities. These yield a procedure to generate facets of the stable set polytope. If a class of graphs has the property that all facets of the stable set polytope can be generated this way in a bounded number of setps, then the stable set problem is polynomially solvable for these graphs. Perfect, t-perfect and h-perfect graphs have this property.