/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

Models for the processes by which ideas and influence propagate through a social network have been studied in a number of domains, including the diffusion of medical and technological innovations, the sudden and widespread adoption of various strategies in game-theoretic settings, and the effects of "word of mouth" in the promotion of new products. Recently, motivated by the design of viral marketing strategies, Domingos and Richardson posed a fundamental algorithmic problem for such social network processes: if we can try to convince a subset of individuals to adopt a new product or innovation, and the goal is to trigger a large cascade of further adoptions, which set of individuals should we target?We consider this problem in several of the most widely studied models in social network analysis. The optimization problem of selecting the most influential nodes is NP-hard here, and we provide the first provable approximation guarantees for efficient algorithms. Using an analysis framework based on submodular functions, we show that a natural greedy strategy obtains a solution that is provably within 63% of optimal for several classes of models; our framework suggests a general approach for reasoning about the performance guarantees of algorithms for these types of influence problems in social networks.We also provide computational experiments on large collaboration networks, showing that in addition to their provable guarantees, our approximation algorithms significantly out-perform node-selection heuristics based on the well-studied notions of degree centrality and distance centrality from the field of social networks.

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Maximizing the spread of influence through a social network

LetN be a finite set andz be a real-valued function defined on the set of subsets ofN that satisfies z(S)+z(T)źz(SźT)+z(SźT) for allS, T inN. Such a function is called submodular. We consider the problem maxSźN{a(S):|S|≤K,z(S) submodular}.

Several hard combinatorial optimization problems can be posed in this framework. For example, the problem of finding a maximum weight independent set in a matroid, when the elements of the matroid are colored and the elements of the independent set can have no more thanK colors, is in this class. The uncapacitated location problem is a special case of this matroid optimization problem.

We analyze greedy and local improvement heuristics and a linear programming relaxation for this problem. Our results are worst case bounds on the quality of the approximations. For example, whenz(S) is nondecreasing andz(0) = 0, we show that a "greedy" heuristic always produces a solution whose value is at least 1 ź[(K ź 1)/K]K times the optimal value. This bound can be achieved for eachK and has a limiting value of (e ź 1)/e, where e is the base of the natural logarithm.

An analysis of approximations for maximizing submodular set functions--I

LetN be a finite set andz be a real-valued function defined on the set of subsets ofN that satisfies z(S)+z(T)gez(SxcupT)+z(SxcapT) for allS, T inN. Such a function is called submodular. We consider the problem maxSsubN{a(S):|S|leK,z(S) submodular}. Several hard combinatorial optimization problems can be posed in this framework. For example, the problem of finding a maximum weight independent set in a matroid, when the elements of the matroid are colored and the elements of the independent set can have no more thanK colors, is in this class. The uncapacitated location problem is a special case of this matroid optimization problem. We analyze greedy and local improvement heuristics and a linear programming relaxation for this problem. Our results are worst case bounds on the quality of the approximations. For example, whenz(S) is nondecreasing andz(0) = 0, we show that a ldquogreedyrdquo heuristic always produces a solution whose value is at least 1 –[(K – 1)/K] K times the optimal value. This bound can be achieved for eachK and has a limiting value of (e – 1)/e, where e is the base of the natural logarithm.

An analysis of approximations for maximizing submodular set functions II

A unified view of metaheuristics This book provides a complete background on metaheuristics and shows readers how to design and implement efficient algorithms to solve complex optimization problems across a diverse range of applications, from networking and bioinformatics to engineering design, routing, and scheduling. It presents the main design questions for all families of metaheuristics and clearly illustrates how to implement the algorithms under a software framework to reuse both the design and code. Throughout the book, the key search components of metaheuristics are considered as a toolbox for: Designing efficient metaheuristics (e.g. local search, tabu search, simulated annealing, evolutionary algorithms, particle swarm optimization, scatter search, ant colonies, bee colonies, artificial immune systems) for optimization problems Designing efficient metaheuristics for multi-objective optimization problems Designing hybrid, parallel, and distributed metaheuristics Implementing metaheuristics on sequential and parallel machines Using many case studies and treating design and implementation independently, this book gives readers the skills necessary to solve large-scale optimization problems quickly and efficiently. It is a valuable reference for practicing engineers and researchers from diverse areas dealing with optimization or machine learning; and graduate students in computer science, operations research, control, engineering, business and management, and applied mathematics.

/pdf/metaheuristics-from-design-to-implementation-1vqdfv8nk9.pdf

Metaheuristics: From Design to Implementation

http://egon.cheme.cmu.edu/Papers/OaBB.pdf

An algorithmic framework for convex mixed integer nonlinear programs

The number of days required to clear a check drawn on a bank in city j depends on the city i in which the check is cashed. Thus, to maximize its available funds, a company that pays bills to numero...

Exceptional Paper—Location of Bank Accounts to Optimize Float: An Analytic Study of Exact and Approximate Algorithms

We propose a cutting plane algorithm for mixed 0---1 programs based on a family of polyhedra which strengthen the usual LP relaxation. We show how to generate a facet of a polyhedron in this family which is most violated by the current fractional point. This cut is found through the solution of a linear program that has about twice the size of the usual LP relaxation. A lifting step is used to reduce the size of the LP's needed to generate the cuts. An additional strengthening step suggested by Balas and Jeroslow is then applied. We report our computational experience with a preliminary version of the algorithm. This approach is related to the work of Balas on disjunctive programming, the matrix cone relaxations of Lovasz and Schrijver and the hierarchy of relaxations of Sherali and Adams.

/pdf/a-lift-and-project-cutting-plane-algorithm-for-mixed-0-1-mf8s1bsfjx.pdf

A lift-and-project cutting plane algorithm for mixed 0-1 programs

: An economic problem of great practical importance is to choose the location of facilities, such as industrial plants or warehouses, in order to minimize the cost (or maximize the profit) of satisfying the demand for some commodity. In general there are fixed costs for locating the facilities and transportation costs for distributing the commodities between the facilities and the clients. This problem has been extensively studied in the literature and is often referred to as the plant, warehouse or facility location problem. When each potential facility has a capacity, which is the maximum demand that it can supply, the problem is known as the capacitated facility location problem. When the capacity hypothesis is not needed, we have the simple or uncapacitated facility location problem, which the authors abbreviate by UL. The mathematical formulation of these problems as integer programs has proven very fruitful in the derivation of solution methods.

The uncapacitated facility location problem

In the course of the deliberations of the 1977 Lanchester Prize Committee, Alan J. Goldman brought to our attention an error in the proof of Lemma 1 of our paper (Cornuejols, G., M. L. Fisher, G. L. Nemhauser. 1977. Location of bank accounts to optimize float: an analytic study of exact and approximate algorithms. Management Sci. 23 789–810.). The lemma, however, is true and the original correct, but long and intricate, proof was provided to the Committee, see (Cornuejols, G., M. L. Fisher, G. L. Nemhauser. 1977. On the uncapacitated location problem. Ann. Discrete Math. 1 163–178.) for details.

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Gérard Cornuéjols

Papers

An algorithmic framework for convex mixed integer nonlinear programs

Exceptional Paper—Location of Bank Accounts to Optimize Float: An Analytic Study of Exact and Approximate Algorithms

A lift-and-project cutting plane algorithm for mixed 0-1 programs

The uncapacitated facility location problem

Note—On “Location of Bank Accounts to Optimize Float: An Analytic Study of Exact and Approximate Algorithms”