Graph Optimization Problem Research Articles

A graph approach for knowledge reduction in formal contexts

Knowledge reduction in formal concept analysis (FCA) is an important procedure for knowledge processing and data analysis. Currently, the knowledge reduction in FCA based on granular computing (GrC) provides another way to analyze and represent the structure of concept lattices. However, the granular reduction method is based on Boolean reasoning and thus is an NP-hard problem. It is therefore natural to develop some heuristic methods to deal with this problem especially for the large data. In this paper, a new framework based on graph theory is used to study the granular reduction in FCA. A graph representation for the granular reduction is first investigated. The results in this paper show that the granular reduction computation in FCA can be translated into a graph optimization problem. Two new heuristic graph-based algorithms for the granular reduction in formal contexts and formal decision contexts are then respectively presented. Furthermore, numerical experiments are conducted to evaluate the effectiveness of the proposed methods.

Supermodularity in Unweighted Graph Optimization I: Branchings and Matchings

The main result of this paper is motivated by the following two apparently unrelated graph optimization problems: (A) As an extension of Edmonds’ disjoint branchings theorem, characterize digraphs comprising k disjoint branchings Bi each having a specified number μi of arcs. (B) As an extension of Ryser’s maximum term rank formula, determine the largest possible matching number of simple bipartite graphs complying with degree-constraints. The solutions to these problems and to their generalizations will be obtained from a new min-max theorem on covering a supermodular function by a simple degree-constrained bipartite graph. A specific feature of the result is that its minimum cost extension is already NP-hard. Therefore classic polyhedral tools themselves definitely cannot be sufficient for solving the problem, even though they make some good service in our approach.

SDN-Based Traffic Aware Placement of NFV Middleboxes

Network function virtualization (NFV) enables flexible deployment of middleboxes as virtual machines running on general hardware. Since different middleboxes may change the volume of processed traffic in different ways, improper deployment of NFV middleboxes will result in hot spots and congestion. In this paper, we study the traffic changing effects of middleboxes, and propose software-defined networking based middlebox placement solutions to achieve optimal load balancing. We formulate the traffic aware middlebox placement (TAMP) problem as a graph optimization problem with the objective to minimize the maximum link load ratio. First, we solve the TAMP problem when the flow paths are predetermined, such as the case in a tree. For a single flow, we propose the least-first-greatest-last (LFGL) rule and prove its optimality; for multiple flows, we first show the NP-hardness of the problem, and then propose an efficient heuristic. Next, for the general TAMP problem without predetermined flow paths, we prove that it is NP-hard even for a single flow, and propose the LFGL based MinMax routing algorithm by integrating LFGL with MinMax routing. We use a joint emulation and simulation approach to evaluate the proposed solutions, and present extensive experimental and simulation results to demonstrate the effectiveness of our design.

The weighted independent domination problem: Integer linear programming models and metaheuristic approaches

This work deals with the so-called weighted independent domination problem, which is an NP-hard combinatorial optimization problem in graphs. In contrast to previous work, this paper considers the problem from a non-theoretical perspective. The first contribution consists in the development of three integer linear programming models. Second, two greedy heuristics are proposed. Finally, the last contribution is a population-based iterated greedy metaheuristic which is applied in two different ways: (1) the metaheuristic is applied directly to each problem instance, and (2) the metaheuristic is applied at each iteration of a higher-level framework – known as construct, merge, solve and adapt – to sub-instances of the tackled problem instances. The results of the considered algorithmic approaches show that integer linear programming approaches can only compete with the developed metaheuristics in the context of graphs with up to 100 nodes. When larger graphs are concerned, the application of the populated-based iterated greedy algorithm within the higher-level framework works generally best. The experimental evaluation considers graphs of different types, sizes, densities, and ways of generating the node and edge weights.

Trajectory optimization in radiotherapy using sectioning (TORUS).

A challenging problem in trajectory optimization for radiotherapy is properly handling the synchronization of the medical accelerators dynamic delivery. The initial coarse sampling of control points implemented in a Progressive Resolution Optimization type approach (VMAT) routinely results in MLC aperture forming contention issues as the sampling resolution increases. This work presents an approach to optimize continuous, beam-on radiation trajectories through exploration of the anatomical topology present in the patient and formation of a novel dual-metric graph optimization problem. This work presents a new perspective on trajectory optimization in radiotherapy using the concept of sectioning (TORUS). TORUS avoids degradation of 3D dose optimization quality by mapping the connectedness of target regions from the BEV perspective throughout the space of deliverable coordinates. This connectedness information is then incorporated into a graph optimization problem to define ideal trajectories. The unique usage of two distance functions in this graph optimization permits the TORUS algorithm to generate efficient dynamic trajectories for delivery while maximizing the angular flux through all PTV voxels. 3D dose optimization is performed for trajectories using a commercial TPS progressive resolution optimizer. The TORUS algorithm is applied to three example treatments: chest wall, scalp, and the TG-119 C-shape phantom. When static collimator coplanar trajectories are generated for the chest wall and scalp cases, the TORUS trajectories are found to outperform both 7 field IMRT and 2 arc VMAT plans in delivery time, organ at risk sparing, conformality, and homogeneity. For the TG-119 phantom, when static couch and collimator non-coplanar trajectories are optimized, TORUS trajectories have superior sparing of the central core avoidance with shorter delivery times, with similar dose conformality and homogeneity. The TORUS algorithm is able to automatically generate trajectories having improved plan quality and delivery time over standard IMRT and VMAT treatments. TORUS offers an exciting and promising avenue forward toward increasing dynamic capabilities in radiation delivery.

Self-weighted spectral clustering with parameter-free constraint

The constrained spectral clustering (or known as the semi-supervised spectral clustering) focuses on enhancing the clustering capability by utilizing the side information. In this paper, a novel constrained spectral clustering method is proposed based on deriving a sparse parameter-free similarity. Different from other works, the proposed method transforms the given pairwise constraints into the intrinsic graph similarity and the penalty graph similarity respectively instead of incorporating them into one single similarity. Besides, the optimal weight can be automatically achieved to balance the graph optimization problems between the intrinsic graph and the penalty graph. Equipped with a general framework of efficiently unraveling the bi-objective optimization, the proposed method could obtain both ratio cut and normalized cut clusterings via updating the weighted Laplacian matrix until convergence. Moreover, the proposed method is equivalent to the spectral clustering, when no side information is provided. Consequently, the effectiveness and the superiority of the proposed method are further verified both analytically and empirically.

Randomized OBDD-based graph algorithms

Implicit graph algorithms deal with the characteristic function χE of the edge set E of a graph G=(V,E). Encoding the nodes by binary vectors, χE can be represented by an Ordered Binary Decision Diagram (OBDD) which is a well known data structure for Boolean functions. OBDD-based graph algorithms solve graph optimization problems by mainly using functional operations and are a heuristic approach to cope with massive graphs. These algorithms heavily rely on a compact representation of the underlying Boolean functions which is why all previously known OBDD-based algorithms are deterministic since random functions are not compressible in general. Here, the first randomized OBDD-based algorithms are presented where random functions with limited independence are used to overcome the large representation size. On the theoretical part, we investigate the size of OBDDs representing k-wise independent random functions. In addition, we give a construction of almost k-wise independent random functions by means of a random OBDD generation and show an almost matching lower bound of the OBDD size representing such random functions. On the algorithmic part, randomization often facilitates the design of simple algorithms which in the context of OBDD-based algorithms means a small number of functional operations and as few input variables of the used Boolean functions as possible. We present two randomized implicit algorithms: A randomized minimum spanning tree algorithm and a maximal matching algorithm. The matching algorithm needs O(log3⁡|V|) functional operations in expectation and uses functions with at most 3log⁡|V| variables which is both better than the best known deterministic algorithms w.r.t. functional operations and variables. The algorithm may be of independent interest. The experimental evaluation of the algorithms shows that they outperform known OBDD-based algorithms for the maximal matching problem and the minimum spanning tree problem.

A constant-time SLAM back-end in the continuum between global mapping and submapping: application to visual stereo SLAM

This work addresses the development and application of a novel approach, called sparser relative bundle adjustment (SRBA), which exploits the inherent flexibility of the relative bundle adjustment (RBA) framework to devise a continuum of strategies, ranging from RBA with linear graphs to classic bundle adjustment (BA) in global coordinates, where submapping with local maps emerges as a natural intermediate solution. This method leads to graphs that can be optimized in bounded time even at loop closures, regardless of the loop length. Furthermore, it is shown that the pattern in which relative coordinate variables are defined among keyframes has a significant impact on the graph optimization problem. By using the proposed scheme, optimization can be done more efficiently than in standard RBA, allowing the optimization of larger local maps for any given maximum computational cost. The main algorithms involved in the graph management, along with their complexity analyses, are presented to prove their bounded-time nature. One key advance of the present work is the demonstration that, under mild assumptions, the spanning trees for every single keyframe in the map can be incrementally built by a constant-time algorithm, even for arbitrary graph topologies. We validate our proposal within the scope of visual stereo simultaneous localization and mapping (SLAM) by developing a complete system that includes a front-end that seamlessly integrates several state-of-the-art computer vision techniques such as ORB features and bag-of-words, along with a decision scheme for keyframe insertion and a SRBA-based back-end that operates as graph optimizer. Finally, a set of experiments in both indoor and outdoor conditions is presented to test the capabilities of this approach. Open-source implementations of the SRBA back-end and the stereo front-end have been released online.

An algorithm for the maximum weight independent set problem on outerstring graphs

Outerstring graphs are the intersection graphs of curves that lie inside a disk such that each curve intersects the boundary of the disk. Outerstring graphs are among the most general classes of intersection graphs studied. To date, no polynomial time algorithm is known for any of the classical graph optimization problems on outerstring graphs; in fact, most are NP-hard. It is known that there is an intersection model for any outerstring graph that consists of polygonal arcs attached to a circle. However, this representation may require an exponential number of segments relative to the size of the graph.Given an outerstring graph and an intersection model consisting of polygonal arcs with a total of N segments, we develop an algorithm that solves the Maximum Weight Independent Set problem in O(N3) time. If the polygonal arcs are restricted to single segments, then outersegment graphs result. For outersegment graphs, we solve the Maximum Weight Independent Set problem in O(n3) time where n is the number of vertices in the graph.

All-Optical Implementation of the Ant Colony Optimization Algorithm.

We report all-optical implementation of the optimization algorithm for the famous “ant colony” problem. Ant colonies progressively optimize pathway to food discovered by one of the ants through identifying the discovered route with volatile chemicals (pheromones) secreted on the way back from the food deposit. Mathematically this is an important example of graph optimization problem with dynamically changing parameters. Using an optical network with nonlinear waveguides to represent the graph and a feedback loop, we experimentally show that photons traveling through the network behave like ants that dynamically modify the environment to find the shortest pathway to any chosen point in the graph. This proof-of-principle demonstration illustrates how transient nonlinearity in the optical system can be exploited to tackle complex optimization problems directly, on the hardware level, which may be used for self-routing of optical signals in transparent communication networks and energy flow in photonic systems.

A Unified Framework for Integer Programming Formulation of Graph Matching Problems

Graph theory has been a powerful tool in solving difficult and complex problems arising in all disciplines. In particular, graph matching is a classical problem in pattern analysis with enormous applications. Many graph problems have been formulated as a mathematical program then solved using exact, heuristic and/or approximated-guaranteed procedures. On the other hand, graph theory has been a powerful tool in visualizing and understanding of complex mathematical programming problems, especially integer programs. Formulating a graph problem as a natural integer program (IP) is often a challenging task. However, an IP formulation of the problem has many advantages. Several researchers have noted the need for natural IP formulation of graph theoretic problems. The aim of the present study is to provide a unified framework for IP formulation of graph matching problems. Although there are many surveys on graph matching problems, however, none is concerned with IP formulation. This paper is the first to provide a comprehensive IP formulation for such problems. The framework includes variety of graph optimization problems in the literature. While these problems have been studied by different research communities, however, the framework presented here helps to bring efforts from different disciplines to tackle such diverse and complex problems. We hope the present study can significantly help to simplify some of difficult problems arising in practice, especially in pattern analysis.

Semidefinite relaxations for partitioning, assignment and ordering problems

Semidefinite optimization is a strong tool in the study of NP-hard combinatorial optimization problems. On the one hand, semidefinite optimization problems are in principle solvable in polynomial time (with fixed precision), on the other hand, their modeling power allows to naturally handle quadratic constraints. Contrary to linear optimization with the efficiency of the Simplex method, the algorithmic treatment of semidefinite problems is much more subtle and also practically quite expensive. This survey-type article is meant as an introduction for a non-expert to this exciting area. The basic concepts are explained on a mostly intuitive level, and pointers to advanced topics are given. We provide a variety of semidefinite optimization models on a selection of graph optimization problems and give a flavour of their practical impact.

Convex relaxations of penalties for sparse correlated variables with bounded total variation

We study the problem of statistical estimation with a signal known to be sparse, spatially contiguous, and containing many highly correlated variables. We take inspiration from the recently introduced k-support norm, which has been successfully applied to sparse prediction problems with correlated features, but lacks any explicit structural constraints commonly found in machine learning and image processing. We address this problem by incorporating a total variation penalty in the k-support framework. We introduce the (k, s) support total variation norm as the tightest convex relaxation of the intersection of a set of sparsity and total variation constraints. We show that this norm leads to an intractable combinatorial graph optimization problem, which we prove to be NP-hard. We then introduce a tractable relaxation with approximation guarantees that scale well for grid structured graphs. We devise several first-order optimization strategies for statistical parameter estimation with the described penalty. We demonstrate the effectiveness of this penalty on classification in the low-sample regime, classification with M/EEG neuroimaging data, and image recovery with synthetic and real data background subtracted image recovery tasks. We extensively analyse the application of our penalty on the complex task of identifying predictive regions from low-sample high-dimensional fMRI brain data, we show that our method is particularly useful compared to existing methods in terms of accuracy, interpretability, and stability.

Physarum-Inspired Applications in Graph-Optimization Problems

In the past decade, we have witnessed the rapid development of a increasing body of research focused on Physarum polycephalum, which has been employed to solve many complicated problems in real-world applications. In this paper, we give an overview towards the developments of Physarum-inspired models for graph-optimization problems. Specifically, we reported the applications of Physarum in the following graph-optimization problems: influential nodes identification, shortest path tree problem, biobjective shortest path problem, improved Physarum algorithm and approximating the transport network.

A Hierarchy of Subgraph Projection-Based Semidefinite Relaxations for Some NP-Hard Graph Optimization Problems

Many important NP-hard combinatorial problems can be efficiently approximated using semidefinite programming relaxations. We propose a new hierarchy of semidefinite relaxations for classes of such problems that are based on graphs and for which the projection of the problem onto a subgraph shares the same structure as the original problem. This includes the well-studied max-cut and stable-set problems. Each level k of the proposed hierarchy consists of the basic semidefinite relaxation of the problem augmented by the constraints enforcing the structural projection condition on every k-node subgraph of the problem. This hierarchy has the distinguishing feature that all the relaxations are formulated in the space of the original semidefinite relaxation. Because the size of the relaxations increases rapidly with the number of subgraphs, we explore the possibility of adding the projection constraints only for selected subgraphs. Preliminary computational results show that the proposed hierarchy yields improved bounds when compared to the initial relaxation for benchmark instances of the max-cut and stable-set problems, and that the improved bounds result in significantly smaller enumeration trees when the relaxation is used in a branch-and-bound scheme.

A survey of transformation based Website Structure Optimization models

The unabated growth of the World Wide Web in the last decade and the increasing size of the websites have resulted in significant amount of research activity to improve the link structure of the websites. A website can be considered as a directed graph with webpages as nodes and hyperlinks as edges referred to as the Webgraph. The researchers have applied various heuristic, metaheuristic and mathematical programming models to optimize the website link structure. Website structure optimization or reorganization is also considered as a graph optimization problem. This paper presents a pioneering comprehensive survey of the recent and the most successful models that are based on transformation and are developed for the newly emerged website structure optimization (WSO) problem. Transformation based models alter the link structure according to the browsing behavior of the users to ease the navigation for all the users. In this article, the models have been classified and discussed. The origins, formulation and comparison of models have also been presented.

Fast Algorithms for the Undirected Negative Cost Cycle Detection Problem

The undirected negative cost cycle detection (UNCCD) problem is concerned with checking whether an undirected, weighted graph contains a negative cost cycle. Known approaches for solving this problem involve reducing it to either the minimum weight$$b$$b-matching problem or the minimum weight$$T$$T-join problem. In this paper, we formally describe these two approaches and provide the tightest known analysis of the $$b$$b-matching approach. Our analysis establishes that the $$b$$b-matching approach runs in $$O((n+m)^{2}\cdot \log (n+m))$$O((n+m)2·log(n+m)) time on a graph with $$n$$n nodes and $$m$$m edges. We also explore the case where the edge costs are integers in the range $$\{-K,\ldots , K\}$${-K,?,K}. For $$K = O(1)$$K=O(1), we provide improved time bounds for both the above-mentioned approaches by exploiting the existence of specialized procedures for optimization problems in graphs with integral positive edge costs. We show that while the $$T$$T-join algorithm has a better time bound for general graphs, the $$b$$b-matching algorithm is more efficient for sparse graphs. Finally, we present one of the first, extensive empirical studies for the UNCCD problem. We examine both approaches on several different families of undirected graphs. Our findings provide insight into the actual efficiency of each approach and reinforce the asymptotic analysis of our algorithms.

Website Structure Optimization Model Based on Ant Colony System and Local Search

The unabated growth of the World Wide Web in the last decade and the increasing size of the websites have resulted in significant amount of research activity to improve the link structure of the websites. A website can be considered as a directed graph with webpages as nodes and hyperlinks as edges referred to as the Webgraph. Website structure optimization or reorganization is also considered as a graph optimization problem. The researchers have developed few models to optimize the website link structure. It is observed that the heuristic and mathematical models cannot optimize webgraphs of large size and are also time consuming. This paper presents an Ant Colony and local search based hybrid metaheuristic model for the newly emerged website structure optimization (WSO) problem. The developed hybrid model is also compared with Ant colony method and it is observed that the model performs better than Ant Colony System based approach.

Node-Weighted Steiner Tree and Group Steiner Tree in Planar Graphs

We improve the approximation ratios for two optimization problems in planar graphs. For node-weighted Steiner tree, a classical network-optimization problem, the best achievable approximation ratio in general graphs is Θ (log n ), and nothing better was previously known for planar graphs. We give a constant-factor approximation for planar graphs. Our algorithm generalizes to allow as input any nontrivial minor-closed graph family, and also generalizes to address other optimization problems such as Steiner forest, prize-collecting Steiner tree, and network-formation games. The second problem we address is group Steiner tree: given a graph with edge weights and a collection of groups (subsets of nodes), find a minimum-weight connected subgraph that includes at least one node from each group. The best approximation ratio known in general graphs is O (log 3 n ), or O (log 2 n ) when the host graph is a tree. We obtain an O (log n polyloglog n ) approximation algorithm for the special case where the graph is planar embedded and each group is the set of nodes on a face. We obtain the same approximation ratio for the minimum-weight tour that must visit each group.

Optimization problems in dotted interval graphs

The class of D-dotted interval (d-DI) graphs is the class of intersection graphs of arithmetic progressions with jump (common difference) at most d. We consider various classical graph-theoretic optimization problems in d-DI graphs of arbitrarily, but fixed, d.We show that Maximum Independent Set, Minimum Vertex Cover, and Minimum Dominating Set can be solved in polynomial time in this graph class, answering an open question posed by Jiang (2006). We also show that Minimum Vertex Cover can be approximated within a factor of (1+ε), for any ε>0, in linear time. This algorithm generalizes to a wide class of deletion problems including the classical Minimum Feedback Vertex Set and Minimum Planar Deletion problems.Our algorithms are based on classical results in algorithmic graph theory and new structural properties of d-DI graphs that may be of independent interest.