Examples of max-flow and min-cut problems with duality gaps in continuous networks

Strang (Mathematical Programming 26, 1983) gave a method to establish a max-flow min-cut theorem in a domain of a Euclidean space. The method can be applied also to max-flow min-cut problems defined by Iri (Survey of Mathematical Programming, North-Holland, 1979) whenever the capacity functions of max-flow problems are bounded and continuous. This paper deals with max-flow min-cut problems of Strang and Iri with unbounded or noncontinuous capacity functions. It is proved that, in such problems, max-flow min-cut theorems may fail to hold.     

Reduced graphs for min-cut/max-flow approaches in image segmentation

In few years, min-cut/max-flow approach has become a leading method for solving a wide range of problems in computer vision. However, min-cut/max-flow approaches involve the construction of huge graphs which sometimes do not fit in memory. Currently, most of the max-flow algorithms are impracticable to solve such large scale problems. In this paper, we introduce a new strategy for reducing exactly graphs in the image segmentation context. During the creation of the graph, we test if the node is really useful to the max-flow computation. Numerical experiments validate the relevance of this technique to segment large scale images.     

Improved bounds on the max-flow min-cut ratio for multicommodity flows

In this paper we consider the worst case ratio between the capacity of min-cuts and the value of max-flow for multicommodity flow problems. We improve the best known bounds for the min-cut max-flow ratio for multicommodity flows in undirected graphs, by replacing theO(logD) in the bound byO(logk), whereD denotes the sum of all demands, andk demotes the number of commodities. In essence we prove that up to constant factors the worst min-cut max-flow ratios appear in problems where demands are integral and polynomial in the number of commodities.Klein, Rao, Agrawal, and Ravi have previously proved that if the demands and the capacities are integral, then the min-cut max-flow ratio in general undirected graphs is bounded byO(logClogD), whereC denotes the sum of all the capacities. Tragoudas has improved this bound toO(lognlogD), wheren is the number of nodes in the network. Garg, Vazirani and Yannakakis further improved this toO(logklogD). Klein, Plotkin and Rao have proved that for planar networks, the ratio isO(logD).Our result improves the bound for general networks toO(log² k) and the bound for planar networks toO(logk). In both cases our result implies the first non-trivial bound that is independent of the magnitude of the numbers involved. The method presented in this paper can be used to give polynomial time approximation algorithms to the minimum cuts in the network up to the above factors.Preliminary version appeared in Proceedings of the 25th Annual ACM Symposium on the Theory of Computing, 1993, 691-697.Research supported by U.S. Army Research Office Grant DAAL-03-91-G-0102, and by a grant from Mitsubishi Electric Laboratories.Research supported in part by a Packard Fellowship, an NSF PYI award, a Sloan Fellowship, and by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS-8920550.     

An approximate max-flow min-cut relation for undirected multicommodity flow,with applications

In this paper, we prove the first approximate max-flow min-cut theorem for undirected multicommodity flow. We show that for a feasible flow to exist in a multicommodity problem, it is sufficient that every cut's capacity exceeds its demand by a factor ofO(logClogD), whereC is the sum of all finite capacities andD is the sum of demands. Moreover, our theorem yields an algorithm for finding a cut that is approximately minimumrelative to the flow that must cross it. We use this result to obtain an approximation algorithm for T. C. Hu's generalization of the multiway-cut problem. This algorithm can in turn be applied to obtain approximation algorithms for minimum deletion of clauses of a 2-CNF formula, via minimization, and other problems. We also generalize the theorem to hypergraph networks; using this generalization, we can handle CNF clauses with an arbitrary number of literals per clause.Most of the results in this paper were presented in preliminary form in Approximation through multicommodity flow,Proceedings, 31th Annual Symposium on Foundations of Computer Science (1990), pp. 726–737.Research supported by the National Science Foundation under NSF grant CDA 8722809, by the Office of Naval and the Defense Advanced Research Projects Agency under contract N00014-83-K-0146, and ARPA Order No. 6320, Amendament 1.Research supported by NSF grant CCR-9012357 and by an NSF Presidential Young Investigator Award.     

Improved approximations for two-stage min-cut and shortest path problems under uncertainty

Mathematical Programming - In this paper, we study the robust and stochastic versions of the two-stage min-cut and shortest path problems introduced in Dhamdhere et al. (in How to pay, come what...     

A continuous framework for open pit mine planning

This paper proposes a new mathematical framework for the open pit mine planning problem, based on continuous functional analysis. The main challenge for engineers is to determine a sequence of nested profiles maximizing the net present value of the mining operation. The traditional models for this problem have been constructed by using binary decision variables, giving rise to large-scale combinatorial and Mixed Integer Programming problems. Instead, we use a continuous approach which allows for a refined imposition of slope constraints associated with geotechnical stability. The framework introduced here is posed in a suitable functional space, essentially the real-valued functions that are Lipschitz continuous on a given two dimensional bounded region. We derive existence results and investigate qualitative properties of the solutions.     

IIPS: A framework for specifying inductive-inference problems

We propose the IIPS framework for specifying inductive inference problems. Unlike the specification outline given in [1], the IIPS framework is formally defined and can specify problems where the inducer influences the presentation order of the examples. The framework is suited to specifying machine-discovery problems.     

A framework for decision support systems for scheduling problems

Considering a decision support system as a tool where executive's judgment can be included along with the mathematical tool kit of the management scientist, this paper shows the need to include problem management as an integral component of the decision support system for scheduling problems. A methodology based on the resolution of conflicts among various constraints in scheduling problems is proposed to implement the problem management system in a decision support system for these problems. The paper concludes with some guidelines to create a workable framework for providing effective decision support to solve scheduling problems and the identification of some fruitful directions for future research.     

A computational framework for spatially explicit agroecosystem modeling: Application to regional simulation

Site-based agroecosystem model has been applied at regional and state level to enable comprehensive analyses of environmental sustainability of food and biofuel production. However, spatially explicit ecosystem simulations over large landscape present computational challenges. This paper presents a framework to support spatially explicit agroecosystem modeling and data analysis over large landscape, which includes four major phases of agroecosystem simulation: simulation data preparation, site-based simulation on high performance computers, data management and data analysis. Then, a case study on a regional intensive modeling area (RIMA) was presented as an application to demonstrate the system implementation and capability.     

A new local search for continuous location problems

This paper presents a new local search approach for solving continuous location problems. The main idea is to exploit the relation between the continuous model and its discrete counterpart. A local search is first conducted in the continuous space until a local optimum is reached. It then switches to a discrete space that represents a discretisation of the continuous model to find an improved solution from there. The process continues switching between the two problem formulations until no further improvement can be found in either. Thus, we may view the procedure as a new adaption of formulation space search. The local search is applied to the multi-source Weber problem where encouraging results are obtained. This local search is also embedded within Variable Neighbourhood Search producing excellent results.     

A hyper-solution framework for classification problems via metaheuristic approaches

This is a summary of the author’s PhD thesis, supervised by Prof. Domenico Conforti and defended on 26-02-2010 at the Universitá della Calabria, Cosenza. The thesis is written in Italian and a copy is available from the author upon request. This work deals with the development of a high-level classification framework which combines parameters optimization of a single classifier with classifiers ensemble optimization, through meta-heuristics. Support Vector Machines (SVM) is used for learning while the meta-heuristics adopted and compared are Genetic-Algorithms (GA), Tabu-Search (TS) and Ant Colony Optimization (ACO). Single SVM optimization usually concerns two approaches: searching for optimal set up of a SVM with fixed kernel (Model Selection) or with linear combination of basic kernels (Multiple Kernel Learning), both issues were considered. Meta-heuristics were used in order to avoid time consuming grid-approach for testing several classifiers configurations and some ad-hoc variations to GA were proposed. Finally, different frameworks were developed and then tested on 8 datasets providing reliable solutions.     

A unified solution framework for multi-attribute vehicle routing problems

Vehicle routing attributes are extra characteristics and decisions that complement the academic problem formulations and aim to properly account for real-life application needs. Hundreds of methods have been introduced in recent years for specific attributes, but the development of a single, general-purpose algorithm, which is both efficient and applicable to a wide family of variants remains a considerable challenge. Yet, such a development is critical for understanding the proper impact of attributes on resolution approaches, and to answer the needs of actual applications. This paper contributes towards addressing these challenges with a component-based design for heuristics, targeting multi-attribute vehicle routing problems, and an efficient general-purpose solver. The proposed Unified Hybrid Genetic Search metaheuristic relies on problem-independent unified local search, genetic operators, and advanced diversity management methods. Problem specifics are confined to a limited part of the method and are addressed by means of assignment, sequencing, and route-evaluation components, which are automatically selected and adapted and provide the fundamental operators to manage attribute specificities. Extensive computational experiments on 29 prominent vehicle routing variants, 42 benchmark instance sets and overall 1099 instances, demonstrate the remarkable performance of the method which matches or outperforms the current state-of-the-art problem-tailored algorithms. Thus, generality does not necessarily go against efficiency for these problem classes.     

A multiobjective framework for heavily constrained examination timetabling problems

University examination timetabling is a challenging set partitioning problem that comes in many variations, and real world applications usually carry multiple constraints and require the simultaneous optimization of several (often conflicting) objectives. This paper presents a multiobjective framework capable of solving heavily constrained timetabling problems. In this prototype study, we focus on the two objectives: minimizing timetable length while simultaneously optimizing the spread of examinations for individual students. Candidate solutions are presented to a multiobjective memetic algorithm as orderings of examinations, and a greedy algorithm is used to construct violation free timetables from permutation sequences of exams. The role of the multiobjective algorithm is to iteratively improve a population of orderings, with respect to the given objectives, using various mutation and reordering heuristics.     

Compact linearization for binary quadratic problems

We show that a well-known linearization technique initially proposed for quadratic assignment problems can be generalized to a broader class of quadratic 0–1 mixed-integer problems subject to assignment constraints. The resulting linearized formulation is more compact and tighter than that obtained with a more usual linearization technique. We discuss the application of the compact linearization to three classes of problems in the literature, among which the graph partitioning problem.      

A global continuation algorithm for solving binary quadratic programming problems

In this paper, we propose a new continuous approach for the unconstrained binary quadratic programming (BQP) problems based on the Fischer-Burmeister NCP function. Unlike existing relaxation methods, the approach reformulates a BQP problem as an equivalent continuous optimization problem, and then seeks its global minimizer via a global continuation algorithm which is developed by a sequence of unconstrained minimization for a global smoothing function. This smoothing function is shown to be strictly convex in the whole domain or in a subset of its domain if the involved barrier or penalty parameter is set to be sufficiently large, and consequently a global optimal solution can be expected. Numerical results are reported for 0-1 quadratic programming problems from the OR-Library, and the optimal values generated are made comparisons with those given by the well-known SBB and BARON solvers. The comparison results indicate that the continuous approach is extremely promising by the quality of the optimal values generated and the computational work involved, if the initial barrier parameter is chosen appropriately. This work is partially supported by the Doctoral Starting-up Foundation (B13B6050640) of GuangDong Province.     

A generalized approximation framework for fractional network flow and packing problems

We generalize the fractional packing framework of Garg and Koenemann (SIAM J Comput 37(2):630–652, 2007) to the case of linear fractional packing problems over polyhedral cones. More precisely, we provide approximation algorithms for problems of the form \(\max \{c^T x : Ax \le b, x \in C \}\), where the matrix A contains no negative entries and C is a cone that is generated by a finite set S of non-negative vectors. While the cone is allowed to require an exponential-sized representation, we assume that we can access it via one of three types of oracles. For each of these oracles, we present positive results for the approximability of the packing problem. In contrast to other frameworks, the presented one allows the use of arbitrary linear objective functions and can be applied to a large class of packing problems without much effort. In particular, our framework instantly allows to derive fast and simple fully polynomial-time approximation algorithms (FPTASs) for a large set of network flow problems, such as budget-constrained versions of traditional network flows, multicommodity flows, or generalized flows. Some of these FPTASs represent the first ones of their kind, while others match existing results but offer a much simpler proof.     

A library for continuous convex separable quadratic knapsack problems

The Continuous Convex Separable Quadratic Knapsack problem (CQKnP) is an easy but useful model that has very many different applications. Although the problem can be solved quickly, it must typically be solved very many times within approaches to (much) more difficult models; hence an efficient solution approach is required. We present and discuss a small open-source library for its solution that we have recently developed and distributed.