Problem In Graph Theory Research Articles

Cost-based filtering algorithms for a Capacitated Lot Sizing Problem and the Constrained Arborescence Problem

Constraint Programming (CP) is a paradigm derived from artificial intelligence, operational research, and algorithmics that can be used to solve combinatorial problems. CP solves problems by interleaving search (assign a value to an unassigned variable) and propagation. Constraint propagation aims at removing/filtering inconsistent values from the domains of the variables in order to reduce the search space of the problem. In this thesis, we develop filtering algorithms for two complex combinatorial optimization problems: a Capacitated Lot Sizing Problem (CLSP) and the Constrained Arborescence Problem (CAP). Each of these problems has many variants and practical applications. The CLSP is the problem of finding an optimal production plan for single or multiple items while satisfying demands of clients and respecting resource restrictions. The CLSP finds important applications in production planning. In this thesis, we introduce a CLSP in CP. In many lot sizing and scheduling problems, in particular when the planning horizon is discrete and finite, there are stocking costs to be minimized. These costs depend on the time spent between the production of an order and its delivery. We focus on developing specialized filtering algorithms to handle the stocking cost part of a class of the CLSP. We propose the global optimization constraint StockingCost when the perperiod stocking cost is the same for all orders; and its generalized version, the IDStockingCost constraint (ID stands for Item Dependent). In this thesis, we also deal with a well-known problem in graph theory: the Minimum Weight Arborescence (MWA) problem. Consider a weighted directed graph in which we distinguish one vertex r as the root. An MWA rooted at r is a directed spanning tree rooted at r with minimum total weight. We focus on the CAP that requires one to find an arborescence that satisfies some side constraints (for example resource, degree, or diameter constraints) and that has minimum weight. The CAP has many real life applications in telecommunication networks, computer networks, transportation problems, scheduling problems, etc. After sensitivity analysis of the MWA, we introduce the CAP in CP. We propose a dedicated global optimization constraint to handle any known variant of the CAP in CP: the MinArborescence constraint. All the proposed filtering algorithms are analyzed theoretically and evaluated experimentally. The different experimental evaluations of these propagators against the state-of-the-art propagators show their respective efficiencies.

Martingales and the fixation probability of high-dimensional evolutionary graphs

A principal problem of evolutionary graph theory is to find the probability that an initial mutant population will fix on a graph, i.e. that the mutants will eventually replace the indigenous population. This problem is particularly difficult when the dimensionality of a graph is high. Martingales can yield compact and exact expressions for the fixation probability of an evolutionary graph. Crucially, the tractability of martingales does not necessarily depend on the dimensionality of a graph. We will use martingales to obtain the exact fixation probability of graphs with high dimensionality, specifically k-partite graphs (or ‘circular flows’) and megastars (or ‘superstars’). To do so, we require that the edges of the graph permit mutants to reproduce in one direction and indigenous in the other. The resultant expressions for fixation probabilities explicitly show their dependence on the parameters that describe the graph structure, and on the starting position(s) of the initial mutant population. In particular, we will investigate the effect of funneling on the fixation probability of k-partite graphs, as well as the effect of placing an initial mutant in different partitions. These are the first exact and explicit results reported for the fixation probability of evolutionary graphs with dimensionality greater than 2, that are valid over all parameter space. It might be possible to extend these results to obtain fixation probabilities of high-dimensional evolutionary graphs with undirected or directed connections. Martingales are a formidable theoretical tool that can solve fundamental problems in evolutionary graph theory, often within a few lines of straightforward mathematics.

English

Finding and discovering any class of graphs which are determined by their spectra is always an important and interesting problem in the spectral graph theory. The main aim of this study is to characterize two classes of multicone graphs which are determined by both their adjacency and Laplacian spectra. A multicone graph is defined to be the join of a clique and a regular graph. Let Kw denote a complete graph on w vertices, and let m be a positive integer number. In A.Z.Abdian (2016) it has been shown that multicone graphs Kw ▽ P17 and Kw ▽ S are determined by both their adjacency and Laplacian spectra, where P17 and S denote the Paley graph of order 17 and the Schlafli graph, respectively. In this paper, we generalize these results and we prove that multicone graphs Kw ▽mP17 and Kw▽mS are determined by their adjacency spectra as well as their Laplacian spectra.

Equivalence between Clar covering polynomials of single zigzag chains and tiling polynomials of [formula omitted] rectangles

This paper offers a formal explanation of a rather puzzling and surprising equivalence between the Clar covering polynomials of single zigzag chains and the tiling polynomials of 2×n rectangles for tilings using 1 × 2, 2 × 1 and 2 × 2 tiles. It is demonstrated that the set of Clar covers of single zigzag chains N(n−1) is isomorphic to the set of tilings of a 2×n rectangle. In particular, this isomorphism maps Clar covers of N(n−1) with k aromatic sextets to tilings of a 2×n rectangle using k square 2 × 2 tiles. The proof of this fact is an application of the recently introduced interface theory of Clar covers. The existence of a similar relationship between the Clar covers of more general benzenoid structures and more general tilings of rectangles remains an interesting open problem in chemical graph theory.

Supersaturated Sparse Graphs and Hypergraphs

AbstractA central problem in extremal graph theory is to estimate, for a given graph H, the number of H-free graphs on a given set of n vertices. In the case when H is not bipartite, Erd̋s, Frankl, and Rödl proved that there are 2(1+o(1))ex(n, H) such graphs. In the bipartite case, however, bounds of the form 2O(ex(n, H)) have been proven only for relatively few special graphs H. As a 1st attempt at addressing this problem in full generality, we show that such a bound follows merely from a rather natural assumption on the growth rate of n ↦ ex(n, H); an analogous statement remains true when H is a uniform hypergraph. Subsequently, we derive several new results, along with most previously known estimates, as simple corollaries of our theorem. At the heart of our proof lies a general supersaturation statement that extends the seminal work of Erd̋s and Simonovits. The bounds on the number of H-free hypergraphs are derived from it using the method of hypergraph containers.

Band connectivity for topological quantum chemistry: Band structures as a graph theory problem

The conventional theory of solids is well suited to describing band structures locally near isolated points in momentum space, but struggles to capture the full, global picture necessary for understanding topological phenomena. In part of a recent paper [B. Bradlyn et al., Nature 547, 298 (2017)], we have introduced the way to overcome this difficulty by formulating the problem of sewing together many disconnected local "k-dot-p" band structures across the Brillouin zone in terms of graph theory. In the current manuscript we give the details of our full theoretical construction. We show that crystal symmetries strongly constrain the allowed connectivities of energy bands, and we employ graph-theoretic techniques such as graph connectivity to enumerate all the solutions to these constraints. The tools of graph theory allow us to identify disconnected groups of bands in these solutions, and so identify topologically distinct insulating phases.

Building blocks of topological quantum chemistry: Elementary band representations

The link between chemical orbitals described by local degrees of freedom and band theory, which is defined in momentum space, was proposed by Zak several decades ago for spinless systems with and without time-reversal in his theory of "elementary" band representations. In Nature 547, 298-305 (2017), we introduced the generalization of this theory to the experimentally relevant situation of spin-orbit coupled systems with time-reversal symmetry and proved that all bands that do not transform as band representations are topological. Here, we give the full details of this construction. We prove that elementary band representations are either connected as bands in the Brillouin zone and are described by localized Wannier orbitals respecting the symmetries of the lattice (including time-reversal when applicable), or, if disconnected, describe topological insulators. We then show how to generate a band representation from a particular Wyckoff position and determine which Wyckoff positions generate elementary band representations for all space groups. This theory applies to spinful and spinless systems, in all dimensions, with and without time reversal. We introduce a homotopic notion of equivalence and show that it results in a finer classification of topological phases than approaches based only on the symmetry of wavefunctions at special points in the Brillouin zone. Utilizing a mapping of the band connectivity into a graph theory problem, which we introduced in Nature 547, 298-305 (2017), we show in companion papers which Wyckoff positions can generate disconnected elementary band representations, furnishing a natural avenue for a systematic materials search.

On the analysis of independent sets via multilevel splitting

Counting the number of independent sets is an important problem in graph theory, combinatorics, optimization, and social sciences. However, a polynomial‐time exact calculation, or even a reasonably close approximation, is widely believed to be impossible, since their existence implies an efficient solution to various problems in the non‐deterministic polynomial‐time complexity class. To cope with the approximation challenge, we express the independent set counting problem as a rare‐event estimation problem. We develop a multilevel splitting algorithm which is generally capable of delivering accurate results, while using a manageable computational effort, even when applied to large graphs. We apply the algorithm to both counting and optimization (finding a maximum independent set) problems, and show that it compares favorably with the existing state of the art.

Optical NP problem solver on laser-written waveguide platform.

Cognitive photonic networks are researched to efficiently solve computationally hard problems. Flexible fabrication techniques for the implementation of such networks into compact and scalable chips are desirable for the study of new optical computing schemes and algorithm optimization. Here we demonstrate a femtosecond laser-written optical oracle based on cascaded directional couplers in glass, for the solution of the Hamiltonian path problem. By interrogating the integrated photonic chip with ultrashort laser pulses, we were able to distinguish the different paths traveled by light pulses, and thus infer the existence or the absence of the Hamiltonian path in the network by using an optical correlator. This work proves that graph theory problems may be easily implemented in integrated photonic networks, down scaling the net size and speeding up execution times.

An Upper Bound on the Number of Edges of Graphs containing No r Vertex-Disjoint Odd Cycles

In [5], we found an upper bound on the number of edges, $\mathcal{E}(G)$, of a graph $G$ containing no $r$ vertex-disjoint cycles of length 3. In this paper we generalize this result to graphs containing no $r$ vertex-disjoint cycles of length $2k+1$. We showed that $\mathcal{E}(G)\les \lfloor \frac{(n-r+1)^2}{4} \rfloor +(r-1)(n-r+1)$ for every $G\in\mathcal{G}(n,V_{r,2k+1})$, the class of all graphs on $n$ vertices containing no $r$ vertex-disjoint cycles of length $2k+1$. Determination of the maximum number of edges in a given graph that contains no specific subgraphs is one of the important problems in graph theory. Solving such problems has attracted the attention of many researchers in graph theory.

A Novel Index for Assessing the Robustness of Integrated Electrical Network and a Natural Gas Network

The robustness of an integrated power grid and natural gas network can be transformed into a graph theory problem theoretically. With the concept of tree-coritivity, a novel assessment to analyze the robustness of the interdependent network is proposed in this paper. This method contains a two-stage optimization method and the calculation framework to reduce the computation burden and examine the results, respectively. Compared with existing robustness indices, the proposed tree-coritivity shows the advanced performance in terms of the connected and unconnected network. Furthermore, this index is able to measure the vulnerability of graphs with the same coritivity. Case studies on the standard and modified IEEE New England 39-bus systems integrated with the natural gas network show the effectiveness of the proposed algorithm and index.

An Energy Efficient Schedule for IEEE 802.15.4/ZigBee Cluster Tree WSN with Multiple Collision Domains and Period Crossing Constraint

Cluster scheduling respecting collision avoidance is a complex problem in cluster-tree wireless sensor networks (WSNs). The difficulty of the problem also increases significantly when the traffic is organized as time-constrained data flows with opposite directions. Thus, in this paper, we seek a collision-free cluster schedule that meets all the data flow deadlines as given in time units. In this context, we have found an elegant approach that expresses the deadline of each flow as an integer number of the length of the schedule period (i.e., period crossing constraints). Consequently, the data flow timeliness requirements become easier to be tackled. Due to the scarce resources of the WSNs, the minimization of the energy consumption of the nodes is a problem of paramount importance. Therefore, the objective is to maximize the lifetime of the network by maximizing the time when the nodes stay in low-power mode. In this paper, we present a novel heuristic scheduling algorithm to obtain the desired schedule. The algorithm is based on very interesting formulations of graph theory problems. Thus, it is efficient in both computational time (instances with thousands of devices are solved in a short time) and solution quality (evaluated over smaller size instances while comparing it with optimal solutions obtained by integer linear programming).

Two-point resistances in an Apollonian network.

The computation of resistance between two nodes in a resistor network is a classical problem in electric theory and graph theory. Based on the Apollonian packing, Andrade etal. introduced a deterministic growing type of networks A(k) [Phys. Rev. Lett. 94, 018702 (2005)PRLTAO0031-900710.1103/PhysRevLett.94.018702]. In this paper, first, a recursive algorithm for computing resistance between any two nodes in A(k) is given. Then as explanations, using the algorithm, explicit expressions for some resistances in A(k) are obtained.

Resistance Distances in Vertex-Face Graphs

Abstract The computation of two-point resistances in networks is a classical problem in electric circuit theory and graph theory. Let G be a triangulation graph with n vertices embedded on an orientable surface. Define K(G) to be the graph obtained from G by inserting a new vertex vϕ to each face ϕ of G and adding three new edges (u, vϕ ), (v, vϕ ) and (w, vϕ ), where u, v and w are three vertices on the boundary of ϕ. In this paper, using star-triangle transformation and resistance local-sum rules, explicit relations between resistance distances in K(G) and those in G are obtained. These relations enable us to compute resistance distance between any two points of Kk (G) recursively. As explanation examples, some resistances in several networks are computed, including the modified Apollonian network and networks constructed from tetrahedron, octahedron and icosahedron, respectively.

Edge-Maximal Graphs Containing No $r$ Vertex-Disjoint Triangles

An important problem in graph theory is that of determining the maximum number of edges in a given graph $G$ that contains no specific subgraphs. This problem has attracted the attention of many researchers. An example of such a problem is the determination of an upper bound on the number of edges of a graph that has no triangles. In this paper, we let $\mathcal{G}(n,V_{r,3})$ denote the class of graphs on $n$ vertices containing no $r$-vertex-disjoint cycles of length $3$. We show that for large $n$, $\mathcal{E}(G)\les \lfloor \frac{(n-r+1)^2}{4} \rfloor +(r-1)(n-r+1)$ for every $G\in\mathcal{G}(n,V_{r,3})$. Furthermore, equality holds if and only if $G=\Omega(n,r)=K_{r-1,\lfloor \frac{n-r+1}2\rfloor,\lceil \frac{n-r+1}2\rceil}$ where $\Omega(n,r)$ is a tripartite graph on $n$ vertices.

Pre-treatment of path problems with required lengths

Path problem is a core problem of graph theory, and its representative one is the shortest path problem. But in many cases the most reasonable solution of path problem is some paths with a required length rather than the shortest one. For the path problem with required length, we propose a pretreatment technology – simplification. We found the shortest path model, and reveal and summary properties of the model as a theorem which reflects relationships between parameters of the model and path lengths. Base on the path model, we design a simple algorithm to simplify the graph for reducing difficulty of the path problem.

A note on the thickness of some complete bipartite graphs

The thickness of a graph is the minimum number of planar subgraphs into which the graph can be decomposed. Determining the thickness for the complete bipartite graph is an unsolved problem in graph theory for over fifty years. Using a new planar decomposition for K 4 k − 4, 4 k ( k ≥ 4) , we obtain the thickness of the complete bipartite graph K n , n + 4 , for n ≥ 1.

A note on the Ramsey numbers for theta graphs versus the wheel of order [formula omitted

The study of exact values and bounds on the Ramsey numbers of graphs forms an important family of problems in the extremal graph theory. For a set of graphs S and a graph F, the Ramsey number R(S,F) is the smallest positive integer r such that for every graph G on r vertices, G contains a graph in S as a subgraph or the complement of G contains F as a subgraph. Ramsey numbers of various graphs including theta graphs and wheels have been extensively studied. We extend known results in the area by presenting exact values of the Ramsey numbers R(θn,W5) for n≥7, where θn is the set of theta graphs of order n and W5 is the wheel graph of order 5.

Logical and inequality implications for reducing the size and difficulty of quadratic unconstrained binary optimization problems

The quadratic unconstrained binary optimization (QUBO) problem arises in diverse optimization applications ranging from Ising spin problems to classical problems in graph theory and binary discrete optimization. The use of preprocessing to transform the graph representing the QUBO problem into a smaller equivalent graph is important for improving solution quality and time for both exact and metaheuristic algorithms and is a step towards mapping large scale QUBO to hardware graphs used in quantum annealing computers. In an earlier paper a set of rules was introduced that achieved significant QUBO reductions as verified through computational testing. Here this work is extended with additional rules that provide further reductions that succeed in exactly solving 10% of the benchmark QUBO problems. An algorithm and associated data structures to efficiently implement the entire set of rules is detailed and computational experiments are reported that demonstrate their efficacy.

Unicyclic and bicyclic graphs with exactly three Q-main eigenvalues

Finding all the graphs with a certain number of Q-main eigenvalues is an algebraic graph theory problem that scientists have sought to answer it for many years. The purpose of this research is finding relationships between the algebraic properties of a signless Laplacian matrix of a graph and the other properties of that graph. In order to achieve this, we choose to characterize all the unicyclic and bicyclic graphs with exactly three distinct Q-main eigenvalues, one of which is zero.