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340
A fully combinatorial algorithm for submodular function minimization
 J. COMBIN. THEORY
"... This paper presents a new simple algorithm for minimizing submodular functions. For integer valued submodular functions, the algorithm runs in O(n6EO log nM) time, where n is the cardinality of the ground set, M is the maximum absolute value of the function value, and EO is the time for function eva ..."
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Cited by 64 (7 self)
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This paper presents a new simple algorithm for minimizing submodular functions. For integer valued submodular functions, the algorithm runs in O(n6EO log nM) time, where n is the cardinality of the ground set, M is the maximum absolute value of the function value, and EO is the time for function evaluation. The algorithm can be improved to run in O((n4EO+n 5) log nM) time. The strongly polynomial version of this faster algorithm runs in O((n5EO + n6) log n) time for real valued general submodular functions. These are comparable to the best known running time bounds for submodular function minimization. The algorithm can also be implemented in strongly polynomial time using only additions, subtractions, comparisons, and the oracle calls for function evaluation. This is the first fully combinatorial submodular function minimization algorithm that does not rely on the scaling method.
Convexity and Steinitz’s Exchange Property
 ADVANCES IN MATHEMATICS, 124 (1996), 272–311
, 1997
"... “Convex analysis” is developed for functions defined on integer lattice points. We investigate the class of functions which enjoy a variant of Steinitz’s exchange property. It includes linear functions on matroids, valuations on matroids (in the sense of Dress and Wenzel), and separable concave func ..."
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Cited by 63 (27 self)
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“Convex analysis” is developed for functions defined on integer lattice points. We investigate the class of functions which enjoy a variant of Steinitz’s exchange property. It includes linear functions on matroids, valuations on matroids (in the sense of Dress and Wenzel), and separable concave functions on the integral base polytope of submodular systems. It is shown that a function ω has the Steinitz exchange property if and only if it can be extended to a concave function ω such that the maximizers of (ω+any linear function) form an integral base polytope. A Fencheltype minmax theorem and discrete separation theorems are established which imply, as immediate consequences, Frank’s discrete separation theorem for submodular functions, Edmonds’ intersection theorem, Fujishige’s Fencheltype minmax theorem for submodular functions, and also Frank’s weight splitting theorem for weighted matroid intersection.
Minimizing Energy Consumption in Largescale Sensor Networks through Distributed Data Compression and Hierarchical Aggregation
, 2004
"... In this paper we study how to reduce energy consumption in largescale sensor networks which systematically sample a spatiotemporal field. We begin by formulating a distributed compression problem subject to aggregation (energy) costs to a single sink. We show that the optimal solution is greedy an ..."
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Cited by 59 (3 self)
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In this paper we study how to reduce energy consumption in largescale sensor networks which systematically sample a spatiotemporal field. We begin by formulating a distributed compression problem subject to aggregation (energy) costs to a single sink. We show that the optimal solution is greedy and based on ordering sensors according to their aggregation costs typically related to proximity and, perhaps surprisingly, it is independent of the distribution of data sources. Next we consider a simplified hierarchical model for a sensor network including multiple sinks, compressors/aggregation nodes and sensors. Using a reasonable metric for energy cost, we show that the optimal organization of devices is associated with a JohnsonMehl tessellation induced by their locations. Drawing on techniques from stochastic geometry, we analyze the energy savings that optimal hierarchies provide relative to previously proposed organizations based on proximity, i.e., associated Voronoi tessellations. Our analysis and simulations show that an optimal organization of aggregation/compression can yield 828% energy savings depending on the compression ratio.
The convex hull of two core capacitated network design problems
 MATHEMATICAL PROGRAMMING
, 1993
"... The network loading problem (NLP) is a specialized capacitated network design problem in which prescribed pointtopoint demand between various pairs of nodes of a network must be met by installing (loading) a capacitated facility. We can load any number of units of the facility on each of the arcs ..."
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Cited by 58 (1 self)
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The network loading problem (NLP) is a specialized capacitated network design problem in which prescribed pointtopoint demand between various pairs of nodes of a network must be met by installing (loading) a capacitated facility. We can load any number of units of the facility on each of the arcs at a specified arc dependent cost. The problem is to determine the number of facilities to be loaded on the arcs that will satisfy the given demand at minimum cost. This paper studies two core subproblems of the NLP. The first problem, motivated by a Lagrangian relaxation approach for solving the problem, considers a multiple commodity, single arc capacitated network design problem. The second problem is a three node network; this specialized network arises in larger networks if we aggregate nodes. In both cases, we develop families of facets and completely characterize the convex hull of feasible solutions to the integer programming formulation of the problems. These results in turn strengthen the formulation of the NLP.
Structured sparsityinducing norms through submodular functions
 IN ADVANCES IN NEURAL INFORMATION PROCESSING SYSTEMS
, 2010
"... Sparse methods for supervised learning aim at finding good linear predictors from as few variables as possible, i.e., with small cardinality of their supports. This combinatorial selection problem is often turnedinto a convex optimization problem byreplacing the cardinality function by its convex en ..."
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Cited by 58 (10 self)
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Sparse methods for supervised learning aim at finding good linear predictors from as few variables as possible, i.e., with small cardinality of their supports. This combinatorial selection problem is often turnedinto a convex optimization problem byreplacing the cardinality function by its convex envelope (tightest convex lower bound), in this case the ℓ1norm. In this paper, we investigate more general setfunctions than the cardinality, that may incorporate prior knowledge or structural constraints which are common in many applications: namely, we show that for nonincreasing submodular setfunctions, the corresponding convex envelope can be obtained from its Lovász extension, a common tool in submodular analysis. This defines a family of polyhedral norms, for which we provide generic algorithmic tools (subgradients and proximal operators) and theoretical results (conditions for support recovery or highdimensional inference). By selecting specific submodular functions, we can give a new interpretation to known norms, such as those based on rankstatistics or grouped norms with potentially overlapping groups; we also define new norms, in particular ones that can be used as nonfactorial priors for supervised learning.
Deltamatroids, Jump Systems and Bisubmodular Polyhedra
, 1993
"... We relate an axiomatic generalization of matroids, called a jump system, to polyhedra arising from bisubmodular functions. Unlike the case for usual submodularity, the points of interest are not all the integral points in the relevant polyhedron, but form a subset of them. However, we do show that t ..."
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Cited by 54 (0 self)
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We relate an axiomatic generalization of matroids, called a jump system, to polyhedra arising from bisubmodular functions. Unlike the case for usual submodularity, the points of interest are not all the integral points in the relevant polyhedron, but form a subset of them. However, we do show that the convex hull of the set of points of a jump system is a bisubmodular polyhedron, and that the integral points of an integral bisubmodular polyhedron determine a (special) jump system. We also prove addition and composition theorems for jump systems, which have several applications for deltamatroids and matroids. Copyright (C) by the Society for Industrial and Applied Mathematics, in SIAM Journal on Discrete Mathematics, 8 (1995) pp. 1732. y Partially supported by an N.S.E.R.C. International Scientific Exchange Award at Carleton University z Partially supported by an N.S.E.R.C. of Canada operating grant 1 Introduction Matroids are important as a unifying structure in pure combin...
Survivable networks, linear programming relaxations and the parsimonious property
, 1993
"... We consider the survivable network design problem the problem of designing, at minimum cost, a network with edgeconnectivity requirements. As special cases, this problem encompasses the Steiner tree problem, the traveling salesman problem and the kedgeconnected network design problem. We establ ..."
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Cited by 51 (11 self)
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We consider the survivable network design problem the problem of designing, at minimum cost, a network with edgeconnectivity requirements. As special cases, this problem encompasses the Steiner tree problem, the traveling salesman problem and the kedgeconnected network design problem. We establish a property, referred to as the parsimonious property, of the linear programming (LP) relaxation of a classical formulation for the problem. The parsimonious property has numerous consequences. For example, we derive various structural properties of these LP relaxations, we present some algorithmic improvements and we perform tight worstcase analyses of two heuristics for the survivable network design problem.
Ratesplitting multiple access for discrete memoryless channels
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 2001
"... It is shown that the encoding/decoding problem for any asynchronoususer discrete memoryless multipleaccess channel can be reduced to corresponding problems for at most 2M  1 singleuser discrete memoryless channels. This result, which extends a similar result for Gaussian channels, reduces the se ..."
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Cited by 51 (1 self)
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It is shown that the encoding/decoding problem for any asynchronoususer discrete memoryless multipleaccess channel can be reduced to corresponding problems for at most 2M  1 singleuser discrete memoryless channels. This result, which extends a similar result for Gaussian channels, reduces the seemingly hard task of finding good multipleaccess codes to the much better understood task of finding good codes for singleuser channels. As a byproduct, some interesting properties of the capacity region ofuser asynchronous discrete memoryless channels are derived.
Submodular Function Minimization under Covering Constraints
, 2009
"... This paper addresses the problems of minimizing nonnegative submodular functions under covering constraints, which generalize the vertex cover, edge cover, and set cover problems. We give approximation algorithms for these problems exploiting the discrete convexity of submodular functions. We first ..."
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Cited by 47 (1 self)
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This paper addresses the problems of minimizing nonnegative submodular functions under covering constraints, which generalize the vertex cover, edge cover, and set cover problems. We give approximation algorithms for these problems exploiting the discrete convexity of submodular functions. We first present a rounding 2approximation algorithm for the submodular vertex cover problem based on the halfintegrality of the continuous relaxation problem, and show that the rounding algorithm can be performed by one application of submodular function minimization on a ring family. We also show that a rounding algorithm and a primaldual algorithm for the submodular cost set cover problem are both constant factor approximation algorithms if the maximum frequency is fixed. In addition, we give an essentially tight lower bound on the approximability of the submodular edge cover problem.
SINGLE MACHINE SCHEDULING WITH RELEASE DATES
, 2002
"... We consider the scheduling problem of minimizing the average weighted completion time of n jobs with release dates on a single machine. We first study two linear programming relaxations of the problem, one based on a timeindexed formulation, the other on a completiontime formulation. We show their ..."
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Cited by 45 (13 self)
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We consider the scheduling problem of minimizing the average weighted completion time of n jobs with release dates on a single machine. We first study two linear programming relaxations of the problem, one based on a timeindexed formulation, the other on a completiontime formulation. We show their equivalence by proving that a O(n log n) greedy algorithm leads to optimal solutions to both relaxations. The proof relies on the notion of mean busy times of jobs, a concept which enhances our understanding of these LP relaxations. Based on the greedy solution, we describe two simple randomized approximation algorithms, which are guaranteed to deliver feasible schedules with expected objective function value within factors of 1.7451 and 1.6853, respectively, of the optimum. They are based on the concept of common and independent αpoints, respectively. The analysis implies in particular that the worstcase relative error of the LP relaxations is at most 1.6853, and we provide instances showing thatitis atleaste/(e − 1) ≈ 1.5819. Both algorithms may be derandomized; their deterministic versions run in O(n²) time. The randomized algorithms also apply to the online setting, in which jobs arrive dynamically over time and one must decide which job to process without knowledge of jobs that will be released afterwards.