Total Weight Of Edges Research Articles

The ultrametric backbone is the union of all minimum spanning forests

Minimum spanning trees and forests are powerful sparsification techniques that remove cycles from weighted graphs to minimize total edge weight while preserving node reachability, with applications in computer science, network science, and graph theory. Despite their utility and ubiquity, they have several limitations, including that they are only defined for undirected networks, they significantly alter dynamics on networks, and they do not generally preserve important network features such as shortest distances, shortest path distribution, and community structure. In contrast, distance backbones, which are subgraphs formed by all edges that obey a generalized triangle inequality, are well defined in directed and undirected graphs and preserve those and other important network features. The backbone of a graph is defined with respect to a specified path-length operator that aggregates weights along a path to define its length, thereby associating a cost to indirect connections. The backbone is the union of all shortest paths between each pair of nodes according to the specified operator. One such operator, the max function, computes the length of a path as the largest weight of the edges that compose it (a weakest link criterion). It is the only operator that yields an algebraic structure for computing shortest paths that is consistent with De Morgan's laws. Applying this operator yields the ultrametric backbone of a graph in that (semi-triangular) edges whose weights are larger than the length of an indirect path connecting the same nodes (i.e. those that break the generalized triangle inequality based on max as a path-length operator) are removed. We show that the ultrametric backbone is the union of minimum spanning forests in undirected graphs and provides a new generalization of minimum spanning trees to directed graphs that, unlike minimum equivalent graphs and minimum spanning arborescences, preserves all shortest paths and De Morgan's law consistency.

Implementation of the Shortest Path Method with Excel Solver to Optimize Goods Delivery Routes

The shortest path solver is a program that aims to find the route with the lowest total edge weight between two points in a graph. Commonly used algorithms include Dijkstra for graphs with non-negative edge weights, Bellman-Ford for graphs with negative edge weights, and Floyd-Warshall for finding the shortest path between all point pairs. Its application is wide, ranging from navigation systems, computer networks, to logistics and games. The process of using it involves creating a graph model , selecting the appropriate algorithm, running a solver, and analyzing the results. A practical example shows how the Dijkstra algorithm can be used to determine the shortest route between cities in a road network , with effective and accurate results . The shortest path solver proves to be a versatile and essential tool for solving a wide range of problems in a variety of fields. This research uses quantitative methods with an experimental approach to test the effectiveness and efficiency of using the shortest path method with Solver Excel in optimizing goods delivery routes. The research object is the delivery route from the warehouse to several delivery destinations, with a sample of 10 routes that are most frequently used and have the highest delivery volume. Primary data was obtained through direct observation and interviews with company logistics managers. The results of this study show that the use of shortest path method with Excel Solver is effective in optimising the route of delivery of goods, reducing the cost and time of delivery by 15% and 10%. Although there are limitations for large networks, this tool remains useful and flexible. This implementation can be a reference for other companies to improve their logistics efficiency.

Tight Lower Bounds for Directed Cut Sparsification and Distributed Min-Cut

In this paper, we consider two fundamental cut approximation problems on large graphs. We prove new lower bounds for both problems that are optimal up to logarithmic factors. The first problem is approximating cuts in balanced directed graphs. In this problem, we want to build a data structure that can provide (1 ± ε)-approximation of cut values on a graph with n vertices. For arbitrary directed graphs, such a data structure requires Ω(n 2 ) bits even for constant ε. To circumvent this, recent works study β-balanced graphs, meaning that for every directed cut, the total weight of edges in one direction is at most β times the total weight in the other direction. We consider the for-each model, where the goal is to approximate each cut with constant probability, and the for-all model, where all cuts must be preserved simultaneously. We improve the previous Ømega(n √β/ε) lower bound in the for-each model to ~Ω (n √β /ε) and we improve the previous Ω(n β/ε) lower bound in the for-all model to Ω(n β/ε 2 ). This resolves the main open questions of (Cen et al., ICALP, 2021). The second problem is approximating the global minimum cut in a local query model, where we can only access the graph via degree, edge, and adjacency queries. We prove an ΩL(min m, m/ε 2 k R) lower bound for this problem, which improves the previous ΩL(m/k R) lower bound, where m is the number of edges, k is the minimum cut size, and we seek a (1+ε)-approximation. In addition, we show that existing upper bounds with minor modifications match our lower bound up to logarithmic factors.

An approximation algorithm for high-dimensional table compression on balanced [formula omitted]-partite graph

This paper considers the high-dimensional table compression problem on balanced k-partite graph. The objective is to choose half of the vertices from each of the k-partite to maximize the total weight of edges connecting the chosen vertices. Our main contribution is a 0.8785-approximation algorithm for the high-dimensional table compression problem by introducing the α-independent solutions through Lasserre semidefinite programming. This new algorithm improved two previous low-dimensional results, namely the 0.8731-approximation algorithm of Wu et al. for the one-dimensional case and the 0.6708-approximation of Xu and Du for the two-dimensional case.

Brain Tumor Localization Using N-Cut

A brain tumor is an abnormal collection of tissue in the brain. When tumors form, they are classified as either malignant or benign. It is critical to notice and identify the existence of tumors in brain images since they can be life threatening. This paper illustrates a novel segmentation method in which threshold technique is combined with normalized cut (Ncut) for the segregation of the tumors from brain magnetic resonance (MR) images. Image segmentation is a technique for grouping images. It is a method of splitting an image into sections with comparable attributes such as intensity, texture, colour, and so on. In thresholding, an object is distinguished from the background, and for the proposed segmentation methodology, the threshold value is determined by normalized graph cut. A weighted graph is divided into disjointed sets (groups) in which the similarity within a group is high and the similarity across groups is low. A graph-cut is a grouping approach in which the total weight of edges eliminated between these two pieces is used to calculate the degree of dissimilarity between these two groups. The normalized cut criterion is used to calculate the total likeness within the groups as well as the dissimilarity between the different groups.

The comparison of three MST algorithms

The minimum spanning tree, connecting all the nodes of a connected, weighted graph with the minimum possible total edge weight, is a fundamental concept of graph theory. Thus, how to find the minimum spanning tree (MST) with higher efficiency appears to be crucial. Three kinds of algorithms for this problem have already been developed: Kruskal, Prim, Boruvka. In order to make readers have a better understanding of the three algorithms in the process of studying and teaching, this paper uses the method of comparing to indicate the differences and commonalities between these three algorithms, draws a conclusion afterwards, and points out the most suitable situations to apply them in the end: Kruskals algorithm is suitable in sparse graphs while Prims algorithm and Boruvkas algorithm are suitable in dense graphs.

A 3/2-approximation algorithm for the multiple Hamiltonian path problem with no prefixed endpoints

We study the multiple Hamiltonian path problem (MHPP) defined on a complete undirected graph G with n vertices. The edge weights of G are non-negative and satisfy the triangle inequality. The MHPP seeks to find a collection of k paths with exactly one visit to each vertex of G with the minimum total edge weight, where endpoints of the paths are not prefixed. We present a 3/2-approximation algorithm for MHPP with time complexity O(n3) for arbitrary k≥1.

New algorithms for a simple measure of network partitioning

Partitioning a network into k pieces is a fundamental problem in network science. A simple measure of partitioning a network is provided by the Maxk-Uncut problem. Given an n-vertex undirected graph G with nonnegative weights defined on edges, and a positive integer k, the Maxk-Uncut problem asks to find a k-partition of the vertices of G to maximize the total weight of edges that are not in the cut. This problem is the complement of the classic Mink-Cut problem, and has close relation to many combinatorial optimization problems, including the famous Densestk-Subgraph problem. In this paper, we propose a greedy approximation algorithm for the Maxk-Uncut problem with performance ratio 1−2(k−1)n. The algorithm is very simple, which consists of only k−1 min cut computations. The algorithm has fast running time O(kn2) and is hence implementable. The experimental results show that the algorithm has excellent practical performance.

Improved Merlin–Arthur Protocols for Central Problems in Fine-Grained Complexity

In a Merlin–Arthur proof system, the proof verifier (Arthur) accepts valid proofs (from Merlin) with probability 1, and rejects invalid proofs with probability arbitrarily close to 1. The running time of such a system is defined to be the length of Merlin’s proof plus the running time of Arthur. We provide new Merlin–Arthur proof systems for some key problems in fine-grained complexity. In several cases our proof systems have optimal running time. Our main results include:Certifying that a list of n integers has no 3-SUM solution can be done in Merlin–Arthur time tilde{O}(n). Previously, Carmosino et al. [ITCS 2016] showed that the problem has a nondeterministic algorithm running in tilde{O}(n^{1.5}) time (that is, there is a proof system with proofs of length tilde{O}(n^{1.5}) and a deterministic verifier running in tilde{O}(n^{1.5}) time).Counting the number of k-cliques with total edge weight equal to zero in an n-node graph can be done in Merlin–Arthur time {tilde{O}}(n^{lceil k/2rceil }) (where kge 3). For odd k, this bound can be further improved for sparse graphs: for example, counting the number of zero-weight triangles in an m-edge graph can be done in Merlin–Arthur time {tilde{O}}(m). Previous Merlin–Arthur protocols by Williams [CCC’16] and Björklund and Kaski [PODC’16] could only count k-cliques in unweighted graphs, and had worse running times for small k.Computing the All-Pairs Shortest Distances matrix for an n-node graph can be done in Merlin–Arthur time tilde{O}(n^2). Note this is optimal, as the matrix can have Omega (n^2) nonzero entries in general. Previously, Carmosino et al. [ITCS 2016] showed that this problem has an tilde{O}(n^{2.94}) nondeterministic time algorithm.Certifying that an n-variable k-CNF is unsatisfiable can be done in Merlin–Arthur time 2^{n/2 - n/O(k)}. We also observe an algebrization barrier for the previous 2^{n/2}cdot textrm{poly}(n)-time Merlin–Arthur protocol of R. Williams [CCC’16] for #SAT: in particular, his protocol algebrizes, and we observe there is no algebrizing protocol for k-UNSAT running in 2^{n/2}/n^{omega (1)} time. Therefore we have to exploit non-algebrizing properties to obtain our new protocol.Certifying a Quantified Boolean Formula is true can be done in Merlin–Arthur time 2^{4n/5}cdot textrm{poly}(n). Previously, the only nontrivial result known along these lines was an Arthur–Merlin–Arthur protocol (where Merlin’s proof depends on some of Arthur’s coins) running in 2^{2n/3}cdot textrm{poly}(n) time. Due to the centrality of these problems in fine-grained complexity, our results have consequences for many other problems of interest. For example, our work implies that certifying there is no Subset Sum solution to n integers can be done in Merlin–Arthur time 2^{n/3}cdot textrm{poly}(n), improving on the previous best protocol by Nederlof [IPL 2017] which took 2^{0.49991n}cdot textrm{poly}(n) time.

Graph Algorithm Based Submodular Function for Sparsest Cut Problem

Sparsest cut problems are very important graph partitions, which have been widely applied in expander graphs, Markov chains, and image segmentation. In this paper, we study the edge-weighted version of the Sparse Cut Problem, which minimizes the ratio of the total weight of edges between blocks and the total weight of edges incident to vertices in one block. We first prove that the problem is even NP-hard for an edge-weighted graph with bridges. Then, we combine and generalize submodular functions and principal partition to design a graph algorithm to improve the initial bipartition, which runs in polynomial time by using network flow as its subroutines.

Approximating probabilistic group steiner trees in graphs

Consider an edge-weighted graph, and a number of properties of interests (PoIs). Each vertex has a probability of exhibiting each PoI. The joint probability that a set of vertices exhibits a PoI is the probability that this set contains at least one vertex that exhibits this PoI. The probabilistic group Steiner tree problem is to find a tree such that (i) for each PoI, the joint probability that the set of vertices in this tree exhibits this PoI is no smaller than a threshold value, e.g. , 0.97; and (ii) the total weight of edges in this tree is the minimum. Solving this problem is useful for mining various graphs with uncertain vertex properties, but is NP-hard. The existing work focuses on certain cases, and cannot perform this task. To meet this challenge, we propose 3 approximation algorithms for solving the above problem. Let |Γ| be the number of PoIs, and ξ be an upper bound of the number of vertices for satisfying the threshold value of exhibiting each PoI. Algorithms 1 and 2 have tight approximation guarantees proportional to |Γ| and ξ, and exponential time complexities with respect to ξ and |Γ|, respectively. In comparison, Algorithm 3 has a looser approximation guarantee proportional to, and a polynomial time complexity with respect to, both |Γ| and ξ. Experiments on real and large datasets show that the proposed algorithms considerably outperform the state-of-the-art related work for finding probabilistic group Steiner trees in various cases.

An accurate star identification approach based on spectral graph matching for attitude measurement of spacecraft

Star identification is the foundation of star trackers, which are used to precisely determine the attitude of spacecraft. In this paper, we propose a novel star identification approach based on spectral graph matching. In the proposed approach, we construct a feature called the neighbor graph for each main star, transforming the star identification to the problem of finding the most similar neighbor graph. Then the rough search and graph matching are cooperated to form a dynamic search framework to solve the problem. In the rough search stage, the total edge weight in the minimum spanning tree of the neighbor graph is selected as an indicator, then the k-vector range search is applied for reducing the search scale. Spectral graph matching is utilized to achieve global matching, identifying all stars in the neighbor circle with good noise-tolerance ability. Extensive simulation experiments under the position noise, lost-star noise, and fake-star noise show that our approach achieves higher accuracy (mostly over 99%) and better robustness results compared with other baseline algorithms in most cases.

Efficient Maximum Edge-Weighted Biclique Search on Large Bipartite Graphs

Given a bipartite graph, the maximum edge biclique problem (MEB) aims to find a biclique with the largest number of edges. MEB is a fundamental problem with many real applications, such as community analysis, E-commerce services and bioinformatics. However, in some scenarios, the weight of an edge reflects valuable and important information on the relationship between two entities. Motivated by this, in this paper, we investigate the problem of maximum edge-weighted biclique search (MEWB), which finds a biclique with the largest total weight of edges in a weighted bipartite graph. MEWB has many real applications, including item recommendation, fraud detection, gene clustering, etc. Although we show that MEWB can be resolved by adapting the search algorithm designed for MEB, the performance of this method is yet unsatisfactory. To improve the computation efficiency, two optimizations in terms of upper bound and search order are proposed. For the upper bound, we consider the degree distribution for vertices in the candidate set, and thus have a chance to discard a few edges to tighten the upper bound. For search order, we theoretically show that a vertex order generating the most similar search depth on vertices can achieve the least time cost for MEWB. Guided by this fact, we propose the global summation vertex order. To further accelerate the computation, we extend our approach to a parallel environment, and develop a heuristic approach to deal with large-scale graphs by slightly sacrificing the answer quality. Extensive performance studies conducted on real datasets demonstrate that our proposals can significantly outperform the baseline method by up to two orders of magnitude. Besides, our heuristic approach gives the optimal result on 8 out of 10 real datasets, while achieving more than an order of magnitude of speed-up.

A combinatorial Schur expansion of triangle-free horizontal-strip LLT polynomials

In recent years, Alexandersson and others proved combinatorial formulas for the Schur function expansion of the horizontal-strip LLT polynomial \(G_{\boldsymbol\lambda}(\boldsymbol x;q)\) in some special cases. We associate a weighted graph \(\Pi\) to \(\boldsymbol\lambda\) and we use it to express a linear relation among LLT polynomials. We apply this relation to prove an explicit combinatorial Schur-positive expansion of \(G_{\boldsymbol\lambda}(\boldsymbol x;q)\) whenever \(\Pi\) is triangle-free. We also prove that the largest power of \(q\) in the LLT polynomial is the total edge weight of our graph.Keywords: Charge, chromatic symmetric function, cocharge, Hall--Littlewood polynomial, jeu de taquin, LLT polynomial, interval graph, Schur function, Schur-positive, symmetric function.Mathematics Subject Classifications: 05E05, 05E10, 05C15

Optimal realizations and the block decomposition of a finite metric space

Finite metric spaces are an essential tool in discrete mathematics and have applications in several areas including computational biology, image analysis, speech recognition, and information retrieval. Given any such metric D on a finite set X, an important problem is to find appropriate ways to realize D by weighting the edges in some graph G containing X in its vertex set such that D(x,y) equals the length of a shortest path from x to y in G for all x,y∈X. Here we focus on realizations with minimum total edge weight, called optimal realizations. By considering the 2-connected components and bridges in any optimal realization G of D we obtain an additive decomposition of D into simpler metrics. We show that this decomposition, called the block decomposition, is canonical in that it only depends on D and not on G, and that the decomposition can be computed in O(|X|3) time. As well as providing a fundamental new way to decompose any finite metric space, we expect that the block decomposition will provide a useful preprocessing tool for deriving metric realizations.

A (probably) optimal algorithm for Bisection on bounded-treewidth graphs

The maximum/minimum bisection problems are, given an edge-weighted graph, to find a bipartition of the vertex set into two sets whose sizes differ by at most one, such that the total weight of edges between the two sets is maximized/minimized. Although these two problems are known to be NP-hard, there is an efficient algorithm for bounded-treewidth graphs. In particular, Jansen et al. (2005) [5] gave an O(2tn3)-time algorithm when given a tree decomposition of width t of the input graph, where n is the number of vertices of the input graph. Eiben et al. (2021) [10] improved the dependency of n in the running time by giving an O(8tt5n2log⁡n)-time algorithm. Moreover, they showed that there is no O(n2−ε)-time algorithm for trees under some reasonable complexity assumption.In this paper, we show an O(2t(tn)2)-time algorithm for both problems, which is asymptotically tight to their conditional lower bound. We also show that the exponential dependency of the treewidth is asymptotically optimal under the Strong Exponential Time Hypothesis. Finally, we discuss the (in)tractability of both problems with respect to special graph classes.

Approximating Max k-Uncut via LP-rounding plus greed, with applications to Densest k-Subgraph

The Maxk-Uncut problem arose from the study of homophily of large-scale networks. Given an n-vertex undirected graph G=(V,E) with nonnegative weights defined on edges and a positive integer k, the Maxk-Uncut problem asks to find a partition {V1,V2,⋯,Vk} of V such that the total weight of edges that are not cut is maximized. This problem is the complement of the classic Mink-Cut problem, and was proved to have surprisingly rich connection to the Densestk-Subgraph problem. In this paper, we give an approximation algorithm for Maxk-Uncut using a non-uniform approach combining LP-rounding and the greedy strategy. The algorithm partitions the vertices of G into at least (1−1e)k parts in expectation, and achieves a good expected approximation ratio 12(1+(n−kn)2).

Optimizing Wiener and Randić Indices of Graphs

Wiener and Randić indices have long been studied in chemical graph theory as connection strength measures of graphs. Later, these indices were used in different fields such as network analysis. We consider two optimization problems related to these indices, with potential applications to network theory, in particular to epidemiological networks. Given a connected graph and a fixed total edge weight, we investigate how individual weights must be assigned to edges, minimizing the connection strength of the graph. In order to measure the connection strength, we use the weighted Wiener index and a modified version of the ordinary Randić index. Wiener index optimization is linear, while Randić index optimization turns out to be both nonlinear and nonconvex. Hence, we adopt the technique of separable programming to generate solutions. We present our experimental results by applying relevant algorithms to several graphs.

A Modified Decomposition Algorithm for Maximum Weight Bipartite Matching and Its Experimental Evaluation

Let G be an undirected bipartite graph with positive integer weights on the edges. We refine the existing decomposition theorem originally proposed by Kao et al., for computing maximum weight bipartite matching. We apply it to design an efficient version of the decomposition algorithm to compute the weight of a maximum weight bipartite matching of G in O(|V |W /k(|V |, W /N))-time by employing an algorithm designed by Feder and Motwani as a subroutine, where |V | and N denote the number of nodes and the maximum edge weight of G, respectively and k(x, y) = log x/ log(x 2 /y). The parameter W is smaller than the total edge weight W, essentially when the largest edge weight differs by more than one from the second largest edge weight in the current working graph in any decomposition step of the algorithm. In best case W = O(|E|) where |E| be the number of edges of G and in worst case W = W, that is, |E| ≤ W ≤ W. In addition, we talk about a scaling property of the algorithm and research a better bound of the parameter W. An experimental evaluation on randomly generated data shows that the proposed improvement is significant in general.

Heuristic and exact algorithms for minimum-weight non-spanning arborescences

We address the problem of finding an arborescence of minimum total edge weight rooted at a given vertex in a directed, edge-weighted graph. If the arborescence must span all vertices the problem is solvable in polynomial time, but the non-spanning version is NP-hard. We propose reduction rules which determine vertices that are required or can be excluded from optimal solutions, a modification of Edmonds algorithm to construct arborescences that span a given set of selected vertices, and embed this procedure into an iterated local search for good vertex selections. Moreover, we propose a cutset-based integer linear programming formulation, provide different linear relaxations to reduce the number of variables in the model and solve the reduced model using a branch-and-cut approach. We give extensive computational results showing that both the heuristic and the exact methods are effective and obtain better solutions on instances from the literature than existing approaches, often in much less time.