The geodesic cover problem for butterfly networks

Fuzzy minimum weight edge covering problem

We provide efficient quadratic unconstrained binary optimization (QUBO) formulations for the Dominating Set and Edge Cover combinatorial problems suitable for adiabatic quantum computers, which are viewed as a real-world enhanced model of simulated annealing (e.g. a type of genetic algorithm with quantum tunneling). The number of qubits (dimension of QUBO matrices) required to solve these set cover problems are O(n + n lg n) and O(m + n lg n) respectively, where n is the number of vertices and m is the number of edges. We also extend our formulations for the Minimum Vertex-Weighted Dominating Set problem and the Minimum Edge-Weighted Edge Cover problem. Experimental results for the Dominating Set and Edge Cover problems using a D-Wave Systems quantum computer with 1098 active qubit-coupled processors are also provided for a selection of known common graphs.

One significant problem in exchange networks is finding the equilibrium. To solve this problem, the concept of stable outcome has been developed. However, there are few effective methods to solve it from the point of graph theory. In this paper, we propose a minimum cost stable outcome (MCSO) problem, which is to find a stable outcome whose total transaction cost is minimized. Two algorithms have been designed to solve this problem on unit and general profit networks respectively. For unit profit networks, we use minimum cost edge cover based method to give the optimal solution. For general profit networks, we develop an approximate algorithm and prove that performance ratio is no more than twice the optimal value. Moreover, we provide the probabilistic analysis. At last, extensive experiments have been conducted on synthetic and real-life datasets. Experimental results validate the performance of the proposed algorithms.

An edge covering of G is a subset S ⊆ E (G) such that each vertex of G is end of some edge in S. The number of edges in a minimum edge covering of G, denoted by β’ (G) is the edge covering number of G. A subset T ⊆ S is called a forcing subset for S if S is the unique minimum edge covering containing T. A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing edge covering number of S, denoted by fβ’ (S), is the cardinality of a minimum forcing subset of S. The forcing edge covering number of G, denoted by fβ’ (G), is fβ’ (G) = min {fβ’ (S)}, where the minimum is taken over all minimum edge coverings S in G. Some general properties satisfied by this concept is studied. The forcing edge covering number of certain classes of graphs are determined. It is shown that for every pair a, b of integer with 0 ≤ a < b and b ≥ 2, there exists a connected graph G such that fβ’ (G) = a and β’ (G) = b.

In this work, the geodesic equations and their numerical solution in Cartesian coordinates on an oblate spheroid, presented by Panou and Korakitis (2017), are generalized on a triaxial ellipsoid. A new exact analytical method and a new numerical method of converting Cartesian to ellipsoidal coordinates of a point on a triaxial ellipsoid are presented. An extensive test set for the coordinate conversion is used, in order to evaluate the performance of the two methods. The direct geodesic problem on a triaxial ellipsoid is described as an initial value problem and is solved numerically in Cartesian coordinates. The solution provides the Cartesian coordinates and the angle between the line of constant λ and the geodesic, at any point along the geodesic. Also, the Liouville constant is computed at any point along the geodesic, allowing to check the precision of the method. An extensive data set of geodesics is used, in order to demonstrate the validity of the numerical method for the geodesic problem. We conclude that a complete, stable and precise solution of the problem is accomplished.

Let G be a finite undirected simple graph with chromatic index \chi ' (G). For a minimal proper edge coloring \psi: E(G) \to \{1,2,..., \chi ' (G)\} in G, an edge covering set S is an edge covering coloring set if each color class in \{1,2,...,\chi ' (G)\} appears in some edge in S. The edge cover coloring number of G, denoted by \chi_{\alpha} ' (G), is the minimum cardinality taken over all possible edge cover coloring sets and all minimal proper edge colorings in G. This paper discusses the problem of finding the edge cover coloring number in graphs, particularly for paths, cycles, complete graphs, wheels, and (n,k)-tadpole graphs.

We study two problems related to the Small Set Expansion Conjecture [14]: the Maximum weight \(m'\) -edge cover (MWEC) problem and the Fixed cost minimum edge cover (FCEC) problem. In the MWEC problem, we are given an undirected simple graph \(G=(V,E)\) with integral vertex weights. The goal is to select a set \(U\subseteq V\) of maximum weight so that the number of edges with at least one endpoint in \(U\) is at most \(m'\). Goldschmidt and Hochbaum [8] show that the problem is NP-hard and they give a \(3\)-approximation algorithm for the problem. The approximation guarantee was improved to \(2+\epsilon \), for any fixed \(\epsilon > 0\) [12]. We present an approximation algorithm that achieves a guarantee of \(2\). Interestingly, we also show that for any constant \(\epsilon > 0\), a \((2-\epsilon )\)-ratio for MWEC implies that the Small Set Expansion Conjecture [14] does not hold. Thus, assuming the Small Set Expansion Conjecture, the bound of 2 is tight. In the FCEC problem, we are given a vertex weighted graph, a bound \(k\), and our goal is to find a subset of vertices \(U\) of total weight at least \(k\) such that the number of edges with at least one edges in \(U\) is minimized. A \(2(1+\epsilon )\)-approximation for the problem follows from the work of Carnes and Shmoys [3]. We improve the approximation ratio by giving a \(2\)-approximation algorithm for the problem and show a \((2-\epsilon )\)-inapproximability under Small Set Expansion Conjecture conjecture. Only the NP-hardness result was known for this problem [8]. We show that a natural linear program for FCEC has an integrality gap of \(2-o(1)\). We also show that for any constant \(\rho >1\), an approximation guarantee of \(\rho \) for the FCEC problem implies a \(\rho (1+o(1))\) approximation for MWEC. Finally, we define the Degrees density augmentation problem which is the density version of the FCEC problem. In this problem we are given an undirected graph \(G=(V,E)\) and a set \(U\subseteq V\). The objective is to find a set \(W\) so that \((e(W)+e(U,W))/deg(W)\) is maximum. This problem admits an LP-based exact solution [4]. We give a combinatorial algorithm for this problem.KeywordsVertex WeightsApproximation GuaranteeMinimum Edge CoverNatural Integer Linear ProgramMaximum WeightThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In an $n$ by $n$ complete bipartite graph with independent exponentially distributed edge costs, we ask for the minimum total cost of a set of edges of which each vertex is incident to at least one. This so-called minimum edge cover problem is a relaxation of perfect matching. We show that the large $n$ limit cost of the minimum edge cover is $W(1)^2+2W(1)\approx 1.456$, where $W$ is the Lambert $W$-function. In particular this means that the minimum edge cover is essentially cheaper than the minimum perfect matching, whose limit cost is $\pi^2/6\approx 1.645$. We obtain this result through a generalization of the perfect matching problem to a setting where we impose a (poly-)matroid structure on the two vertex-sets of the graph, and ask for an edge set of prescribed size connecting independent sets.

For an undirected graph G = (V, E), an edge cover is defined as a set of edges that covers all vertices of V. It is known that a minimum edge cover can be found in polynomial time and forms a collection of star graphs. In this paper, we consider the problem of finding a balanced edge cover where the degrees of star center vertices are balanced, which can be applied to optimize sensor network structures, for example. To this end, we formulate the problem as a minimization of the summation of strictly monotone increasing convex costs associated with degrees for covered vertices, and show that the optimality can be characterized as the non-existence of certain alternating paths. By using this characterization, we show that the optimal covers are also minimum edge covers, have the lexicographically smallest degree sequence of the covered vertices, and minimize the maximum degree of covered vertices. Based on the characterization we also present an O(|V||E|) time algorithm.

We describe a paradigm for designing parallel algorithms via approximation, and illustrate it on the b-Edge Cover problem. A b-Edge Cover of minimum weight in a graph is a subset C of its edges such that at least a specified number b(v) of edges in C is incident on each vertex v, and the sum of the edge weights in C is minimum. The Greedy algorithm and a variant, the LSE algorithm, provide 3/2-approximation guarantees in the worst-case for this problem, but these algorithms have limited parallelism. Hence we design two new 2-approximation algorithms with greater concurrency. The MCE algorithm reduces the computation of a b-Edge Cover to that of finding a b'-Matching, by exploiting the relationship between these subgraphs in an approximation context. The LSENW is derived from the LSE algorithm using static edge weights rather than dynamically computing effective edge weights. This relaxation gives S-LSE a worse approximation guarantee but makes it more amenable to parallelization. We prove that both the MCE and S-LSE algorithms compute the same b-EDGE COVER with at most twice the weight of the minimum weight edge cover. In practice, the 2-approximation and 3/2-approximation algorithms compute edge covers of weight within 10% the optimal. We implement three of the approximation algorithms, MCE, LSE, and S-LSE, on shared memory multi-core machines, including an Intel Xeon and an IBM Power8 machine with 8 TB memory. The MCE algorithm is the fastest of these by an order of magnitude or more. It computes an edge cover in a graph with billions of edges in 20 seconds using two hundred threads on the IBM Power8. We also show that the parallel depth and work can be bounded for the Suitor and b-Suitor algorithms when edge weights are random.

Dedication

On set expansion problems and the small set expansion conjecture

We consider the problem of computing edge covers that are tolerant to a certain number of edge deletions. We call the problem of finding a minimum such cover r-Tolerant Edge Cover (r-EC) and the problem of finding a maximum minimal such cover Upper r-EC. We present several NP-hardness and inapproximability results for Upper r-EC and for some of its special cases.

We discuss an efficient primal‐dual algorithm for finding a minimum cost edge cover in an undirected network G with n nodes and m edges, which is based on the blossom algorithm for 1‐matching problems. We establish that its worst case computational complexity is O(n3), the same as that for blossom or primal algorithms for 1‐matching problems.

Constrained weighted matchings and edge coverings in graphs