Genetic Algorithms for the Tree T-Spanner Problem

A tree t-spanner, in a given connected graph, is a spanning tree T in which the distance between every two vertices in T is at most t times their shortest distance in the graph, where the parameter t is called stretch factor of T. This paper studies the tree t-spanner problem on a given connected, undirected and edge-weighted graph G that seeks to find a spanning tree in G whose stretch factor is minimum among all tree spanners in G. This problem is \(\mathcal {NP}\)-Hard for any fixed \(t > 1\). In the domain of metaheuristic techniques, only genetic algorithm (GA) based on generational population model is proposed in the literature. This paper presents a steady-state genetic algorithm (SSGA) for this problem, which is quite different from the existing GA in the literature, not only in the population management strategy, but also in genetic operators. Genetic operators in SSGA use problem-specific knowledge for generating offspring, making SSGA highly effective in finding high-quality solutions in comparison to the existing GA. On a set of randomly generated instances, computational results justify the effectiveness of SSGA over the existing GA.

A spanning tree T of a graph G is said to be a tree t-spanner if the distance between any two vertices in T is at most t times their distance in G. A graph that has a tree t-spanner is called a tree t -spanner admissible graph. The complexity of recognizing tree 3-spanner admissible graphs is still unknown. In this paper, a characterization of tree 3-spanner admissible 2-trees is presented. Linear time algorithms for recognizing tree 3-spanner admissible 2-trees and for constructing tree 3-spanners in such 2-trees are also proposed.

Spanners in sparse graphs

The t-spanner problem is a popular combinatorial optimization problem and has different applications in communication networks and distributed systems. This chapter considers the problem of constructing a t-spanner subgraph H in a given undirected edge-weighted graph G in the sense that the distance between every pair of vertices in H is at most t times the shortest distance between the two vertices in G. The value of t, called the stretch factor, quantifies the quality of the distance approximation of the corresponding t-spanner subgraph. This chapter studies two variations of the problem, the Minimum t-Spanner Subgraph (MtSS) and the Minimum Maximum Stretch Spanning Tree(MMST). Given a value for the stretch factor t, the MtSS problem asks to find the t-spanner subgraph of the minimum total weight in G. The MMST problem looks for a tree T in G that minimizes the maximum distance between all pairs of vertices in V (i.e., minimizing the stretch factor of the constructed tree). It is easy to conclude from the literatures that the above problems are NP-hard. This chapter presents genetic algorithms that returns a high quality solution for those two problems.

The Minimum Spanning Tree (MST) is of crucial importance for Communication Networks (CNs), which can solve problems of unconstrained CNs effectively. However, in practical CNs, the degree of the spanning tree is constrained. In such a case, the problem is very difficult, which has been proved to be NP-hard. In this paper, an improved genetic algorithm (I-GA) is proposed for solving the problem of Degree Constrained Minimum Spanning Tree (DCMST). We use Prufer number as the chromosome, and improve the crossover and mutation processes of the existing genetic algorithm for obtaining high locality, heritability, and self-adaptation. Simulation results show that our mechanism of I-GA can get relatively better results.

A graph is denoted by $G$ with the vertex set $V(G)$ and the edge set $E(G)$.A path $P=\langle v_0,v_1,\cdots,v_m\rangle$ is a sequence of adjacent vertices. Two paths with equal length $P_1=\langle u_1,u_2,\ldots,u_m\rangle$ and $P_2=\langle v_1,v_2,\ldots,v_m\rangle$ from $a$ to $b$ are {\it independent} if $u_1=v_1=a$, $u_m=v_m=b$, and $u_i\neq v_i$ for $2\leq i\leq m-1$.Paths with equal length $\{P_i\}^{n}_{i=1}$ from $a$ to $b$ are {\it mutually independent} if they are pairwisely independent. Let $u$ and $v$ be two distinct vertices of a bipartite graph $G$, and let $l$ be a positive integer length, $d_G(u,v)\le l \le |V(G)-1|$ with $(l-d_G(u,v))$ being even. We say that the pair of vertices $u$, $v$ is {\it $(m,l)$-mutually independent bipanconnected} if there exist $m$ mutually independent paths $\{P_i^l\}_{i=1}^m$ with length $l$ from $u$ to $v$. In this paper, we explore yet another strong property of the hypercubes. We prove that every pair of vertices $u$ and $v$ in the $n$-dimensional hypercube, with $d_{Q_n}(u,v) \ge n-1$, is $(n-1,l)$-mutually independent bipanconnected for every $l$, $d_{Q_n}(u,v) \le l \le |V(Q_n)-1|$ with $(l-d_{Q_n}(u,v))$ being even. As for $d_{Q_n}(u,v) \le n-2$, it is also $(n-1,l)$-mutually independent bipanconnected if $l \ge d_{Q_n}(u,v)+2$, and is only $(l,l)$-mutually independent bipanconnected if $l=d_{Q_n}(u,v)$.

This paper studies the minimum spanning tree problem on a graph with uncertain edge weights, which are formulated as uncertain variables. The concept of ideal uncertain minimum spanning tree (ideal UMST) is initiated by extending the definition of the uncertain [Formula: see text]-minimum spanning tree to reect the overall properties of the α-minimum spanning tree weights at any confidence level [Formula: see text]. On the basis of this new concept, the definition of uncertain distribution-minimum spanning tree is proposed in three ways. Particularly, by considering the tail value at risk from the perspective of risk management, the notion of uncertain [Formula: see text]-distribution-minimum spanning tree ([Formula: see text]-distribution-UMST) is suggested. It is shown that the [Formula: see text]-distribution-UMST is just the uncertain expected minimum spanning tree when [Formula: see text] = 0. For any [Formula: see text], this problem can be effectively solved via the proposed deterministic graph transformation-based approach with the aid of the [Formula: see text]-distribution-path optimality condition. Furthermore, the proposed definitions and solutions are illustrated by some numerical examples.

Given an undirected weighted connected graph \(G=(V,E)\) with vertex set V, edge set E and a designated vertex \(r \in V\), this chapter studies the following constrained tree problems in G. The first problem, called Constrained Minimum Spanning Tree Problem (CMST), asks for a rooted tree T in G that minimizes the total weight of T such that the distance between the r and any vertex v in T is at most a given constant C times the shortest distance between the two vertices in G. The second problem, Constrained Shortest Path Tree Problem (CSPT) requires a rooted tree T in G that minimizes the maximum distance between r and all vertices in V such that the total weight of T is at most a given constant C times the minimum tree weight in G. It is easy to conclude from the literatures that the above problems are NP-hard. This chapter presents efficient genetic algorithms that return (as shown by our experimental results) high quality solutions for those two problems.

Let G = (V, E) be a connected and simple graphs with vertex set V and edge set E. A coloring of graph G is rainbow connected if there is a rainbow path that connects each two vertices of graph G. The minimum k such that G has a rainbow-connected using k colors of the edges of G is the rainbow connection number rc(G) of G. A graph with a bijective mapping f : E → {1, 2, …, |E|}. The sums of each paired vertex has distinct value, defined as ∑ e ∈E(v)f(e). Thus, the function of G clearly an antimagic labeling if the sums of each paired vertex has distinct value. It is clear that rainbow antimagic connection number is the smallest number of colors which are needed to make G rainbow connected, denoted by rcA (G). A bijection function f : E → {1, 2, …, |E|} is called a rainbow antimagic labeling if there is a rainbow path between every pair of vertices and for each edge e = uv ∈ E(G), the weight w(e) = f(u) + f(v). A graph G is rainbow antimagic if G has a rainbow antimagic labeling. In this paper, we will analyze the rainbow antimagic coloring of related book graph.

Motivated by the increasing need for fast distributed processing of large-scale graphs such as the Web graph and various social networks, we study a number of fundamental graph problems in the message-passing model, where we have k machines that jointly perform computation on an arbitrary n-node (typically, n > k) input graph. The graph is assumed to be randomly partitioned among the k ≥ 2 machines (a common implementation in many real world systems). The communication is point-to-point, and the goal is to minimize the time complexity i.e., the number of communication rounds, of solving various fundamental graph problems.We present lower bounds that quantify the fundamental time limitations of distributively solving graph problems. We first show a lower bound of Ω(n/k) rounds for computing a spanning tree (ST) of the input graph. This result also implies the same bound for other fundamental problems such as computing a minimum spanning tree (MST), breadth-first tree (BFS), and shortest paths tree (SPT). We also show an Ω(n/k2) lower bound for connectivity ST verification and other related problems. Our lower bounds develop and use new bounds in random-partition communication complexity.To complement our lower bounds, we also give algorithms for various fundamental graph problems, e.g., PageRank, MST, connectivity, ST verification, shortest paths, cuts, spanners, covering problems, densest subgraph, subgraph isomorphism, finding triangles, etc. We show that problems such as PageRank, MST, connectivity, and graph covering can be solved in O(n/k) time (the notation O hides polylog(n) factors and an additive polylog(n) term); this shows that one can achieve almost linear (in k) speedup, whereas for shortest paths, we present algorithms that run in O(n/Ok) time (for (1 + e)-factor approximation) and in O(n/k) time (for O(log n)-factor approximation) respectively.Our results are a step towards understanding the complexity of distributively solving large-scale graph problems.

In this chapter we elaborate some of the algebraic concepts introduced in Chapter 4 to provide foundations for a deeper understanding of signed networks and balance algorithms discussed in Chapter 10.

A method to design resilient protocols for spanning tree problems is proposed. The method consists of a generic resilient protocol that can be used for any spanning tree problem. A resilient protocol for a particular spanning tree problem can be obtained by adjusting various run-time parameters of the generic protocol. The generic protocol is designed by using the concepts of replacement sets that contain the links that can be used to replace a link that fails and replacement values that measure the change in the spanning tree when a link in a replacement set is used to recreate a spanning tree. The approach is as general as the common approach (for the case of spanning trees) because the generic protocol can solve a new spanning tree problem without an individual design. The protocols obtained with this method are efficient and easy to implement and verify because they are all derived from the generic protocol. >

In a previous article, using underlying graph theoretical properties, Gouveia and Magnanti (2003) described several network flow‐based formulations for diameter‐constrained tree problems. Their computational results showed that, even with several enhancements, models for situations when the tree diameter D is odd proved to be more difficult to solve than those when D is even. In this article we provide an alternative modeling approach for the situation when D is odd. The approach views the diameter‐constrained minimum spanning tree as being composed of a variant of a directed spanning tree (from an artificial root node) together with two constrained paths, a shortest and a longest path, from the root node to any node in the tree. We also show how to view the feasible set of the linear programming relaxation of the new formulation as the intersection of two integer polyhedra, a so‐called triangle‐tree polyhedron and a constrained path polyhedron. This characterization improves upon a model of Gouveia and Magnanti (2003) whose linear programming relaxation feasible set is the intersection of three rather than two integer polyhedra. The linear programming gaps for the tightened model are very small, typically less than 0.5%, and are usually one third to one tenth of the gaps of the best previous model described in Gouveia and Magnanti (2003). Moreover, using the new model, we have been able to solve large Euclidean problem instances that are not solvable by the previous approaches. © 2004 Wiley Periodicals, Inc.

Degree-bounded minimum spanning trees

We present a new bicriteria approximation algorithm for the degree-bounded minimum-cost spanning tree (MST) problem: Given an undirected graph with nonnegative edge weights and a degree bound B, find a spanning tree of maximum node-degree B and minimum total edge-cost. Our algorithm outputs a tree of maximum degree at most a constant times B and total edge-cost at most a constant times that of a minimum-cost degree-B-bounded spanning tree. While our new algorithm is based on ideas from Lagrangian relaxation, as is our previous work [SIAM J. Comput., 31 (2002), pp. 1783--1793], it does not rely on computing a solution to a linear program. Instead, it uses a repeated application of Kruskal's MST algorithm interleaved with a combinatorial update of approximate Lagrangian node-multipliers maintained by the algorithm. These updates cause subsequent repetitions of the spanning tree algorithm to run for longer and longer times, leading to overall progress and a proof of the performance guarantee. A second useful feature of our algorithm is that it can handle nonuniform degree bounds on the nodes: Given distinct bounds Bv for every node $v \in V$, the output tree has degree at most O(Bv + log|V|) for every $v \in V$. As before, the cost of the tree is at most a constant times that of a minimum-cost tree obeying all degree bounds.