Weighted Spanning Tree Constraint with Explanations

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.

The Minimum Conflict Weighted Spanning Tree Problem is a variant of the Minimum Spanning Tree Problem in which, given a list of conflicting edges modelled as a conflict graph, we want to find a weighted spanning tree with the minimum number of conflicts as main objective function and minimize the total weight of spanning trees as secondary objective function. The problem is proved to be NP-Hard in its general form and finds applications in several real-case scenarios such as the modelling of road networks in which some movements are prohibited. We propose a genetic algorithm designed to minimize the number of conflict edge pairs and the total weight of the spanning tree. We tested our approach on benchmark instances, the results of our GA showed that we outperform the other approaches proposed in the literature.

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 message-passing distributed computing model for graph processing and present lower bounds and algorithms for several graph problems. This work is inspired by recent large-scale graph processing systems (e.g., Pregel and Giraph) which are designed based on the message-passing model of distributed computing. Our model consists of a point-to-point communication network of $k$ machines interconnected by bandwidth-restricted links. Communicating data between the machines is the costly operation (as opposed to local computation). The network is used to process an arbitrary $n$-node input graph (typically $n \gg k > 1$) that is randomly partitioned among the $k$ machines (a common implementation in many real world systems). Our goal is to study fundamental complexity bounds for solving graph problems in this model. We present techniques for obtaining lower bounds on the distributed time complexity. Our lower bounds develop and use new bounds in random-partition communication complexity. We first show a lower bound of $\Omega(n/k)$ 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). We also show an $\Omega(n/k^2)$ lower bound for connectivity, ST verification and other related problems. We give algorithms for various fundamental graph problems in our model. We show that problems such as PageRank, MST, connectivity, and graph covering can be solved in $\tilde{O}(n/k)$ time, whereas for shortest paths, we present algorithms that run in $\tilde{O}(n/\sqrt{k})$ time (for $(1+\epsilon)$-factor approx.) and in $\tilde{O}(n/k)$ time (for $O(\log n)$-factor approx.) respectively.

An effective genetic algorithm approach to the quadratic minimum spanning tree problem

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.

This paper considers the problem of minimum spanning trees in uncertain networks in which the edge weights are random variables. We propose the concept of expected minimum spanning tree and formulate the model according to expected value. In order to solve the model, a hybrid intelligent algorithm combined genetic algorithm and stochastic simulation is given, and the Prufer encoding schemes represented spanning trees are adopted. The algorithm has been proved to be useful for solving practical problems by a numerical example.

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. >

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.

In this paper, we study undirected multiple graphs of any natural multiplicity $$k > 1$$ . There are edges of three types: ordinary edges, multiple edges and multi-edges. Each edge of the last two types is a union of $$k$$ linked edges, which connect 2 or $$k + 1$$ vertices, correspondingly. The linked edges should be used simultaneously. If a vertex is incident to a multiple edge, it can be also incident to other multiple edges, and it can be the common ending vertex to $$k$$ linked edges of a multi-edge. If a vertex is the common end of some multi-edge, it cannot be the common end of any other multi-edge. Special attention is paid to the class of divisible multiple graphs. The main peculiarity of them is a possibility to divide the graph into $$k$$ parts, which are adjusted on the linked edges and which have no common edges. Each part is an ordinary graph. The definition of a multiple tree is stated and the basic properties of such trees are studied. Unlike ordinary trees, the number of edges in a multiple tree is not fixed. In the article, the estimation of the minimum and maximum number of edges in the divisible tree is stated and proved. Next, the definitions of the spanning tree and the complete spanning tree of a multiple graph are given. The criterion of completeness of the spanning tree is proved for divisible graphs. It is also proved that a complete spanning tree exists in any divisible graph. If the multiple graph is weighted, the minimum spanning tree problem and the minimum complete spanning tree problem can be set. In the article, we suggest a heuristic algorithm for the minimum complete spanning tree problem for a divisible graph.

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.

Complexities of some interesting problems on spanning trees

In a complete graphwith independent uniform(or exponential) edge weights, letbe the minimum‐weight spanning tree (MST), andthe MST after deleting the edges of all previous trees. We show that each tree's weightconverges in probability to a constant, with, and we conjecture that. The problem is distinct from Frieze and Johansson's minimum combined weightofkedge‐disjoint spanning trees; indeed,. With an edge of weightw“arriving” at time, Kruskal's algorithm defines forests, initially empty and eventually equal to, each edge added to the first possible. Using tools of inhomogeneous random graphs we obtain structural results including that the fraction of vertices in the largest component ofconverges to some. We conjecture that the functionstend to time translations of a single function.

This study addresses constructing Minimum Spanning Trees (MST) in stochastic weighted distribution networks, where edge costs have inherent uncertainties with known means and variances. Traditional deterministic methods often fail, and existing stochastic approaches are frequently unstable or computationally complex under high uncertainty. A novel variance-based deterministic transformation algorithm is proposed. Its core feature is transforming stochastic edge costs into robust deterministic equivalents by computing an aggregate variance term from the largest (n – 1) edge variances, enabling MST construction via classical algorithms. This method fundamentally enhances stability and ensures feasibility, particularly in high-variance scenarios, improving upon traditional confidence interval-based techniques. The algorithm’s efficacy was rigorously validated. Its performance was compared against a probabilistic Qij-based method under moderate variance, demonstrating consistent and accurate MSTs. It was then applied to a complex 21-edge distribution network with high variance parameters. Results confirm the algorithm’s broad applicability, precision, and capability to construct reliable spanning trees under both moderate and substantial uncertainty. The algorithm demonstrates significant computational efficiency (O(r log r)), ensuring practicality and scalability across varying uncertainty levels. Unlike iterative or constraint-heavy models, this algorithm simplifies optimization while preserving uncertainty representation. This makes it well-suited for large-scale networks and real-world systems where cost variability is critical. Future research includes expanding this approach to multi-objective optimization or dynamic networks

A general formulation is presented for continuum scaling limits of stochastic spanning trees. A spanning tree is expressed in this limit through a consistent collection of subtrees, which includes a tree for every finite set of endpoints in $\R^d$. Tightness of the distribution, as $\delta \to 0$, is established for the following two-dimensional examples: the uniformly random spanning tree on $\delta \Z^2$, the minimal spanning tree on $\delta \Z^2$ (with random edge lengths), and the Euclidean minimal spanning tree on a Poisson process of points in $\R^2$ with density $\delta^{-2}$. In each case, sample trees are proven to have the following properties, with probability one with respect to any of the limiting measures: i) there is a single route to infinity (as was known for $\delta > 0$), ii) the tree branches are given by curves which are regular in the sense of H\"older continuity, iii) the branches are also rough, in the sense that their Hausdorff dimension exceeds one, iv) there is a random dense subset of $\R^2$, of dimension strictly between one and two, on the complement of which (and only there) the spanning subtrees are unique with continuous dependence on the endpoints, v) branching occurs at countably many points in $\R^2$, and vi) the branching numbers are uniformly bounded. The results include tightness for the loop erased random walk (LERW) in two dimensions. The proofs proceed through the derivation of scale-invariant power bounds on the probabilities of repeated crossings of annuli.