16 research outputs found
Strongly polynomial algorithm for a class of minimum-cost flow problems with separable convex objectives
A well-studied nonlinear extension of the minimum-cost flow problem is to
minimize the objective over feasible flows ,
where on every arc of the network, is a convex function. We give
a strongly polynomial algorithm for the case when all 's are convex
quadratic functions, settling an open problem raised e.g. by Hochbaum [1994].
We also give strongly polynomial algorithms for computing market equilibria in
Fisher markets with linear utilities and with spending constraint utilities,
that can be formulated in this framework (see Shmyrev [2009], Devanur et al.
[2011]). For the latter class this resolves an open question raised by Vazirani
[2010]. The running time is for quadratic costs,
for Fisher's markets with linear utilities and
for spending constraint utilities.
All these algorithms are presented in a common framework that addresses the
general problem setting. Whereas it is impossible to give a strongly polynomial
algorithm for the general problem even in an approximate sense (see Hochbaum
[1994]), we show that assuming the existence of certain black-box oracles, one
can give an algorithm using a strongly polynomial number of arithmetic
operations and oracle calls only. The particular algorithms can be derived by
implementing these oracles in the respective settings
Approximating Minimum-Cost k-Node Connected Subgraphs via Independence-Free Graphs
We present a 6-approximation algorithm for the minimum-cost -node
connected spanning subgraph problem, assuming that the number of nodes is at
least . We apply a combinatorial preprocessing, based on the
Frank-Tardos algorithm for -outconnectivity, to transform any input into an
instance such that the iterative rounding method gives a 2-approximation
guarantee. This is the first constant-factor approximation algorithm even in
the asymptotic setting of the problem, that is, the restriction to instances
where the number of nodes is lower bounded by a function of .Comment: 20 pages, 1 figure, 28 reference
Concave Generalized Flows with Applications to Market Equilibria
We consider a nonlinear extension of the generalized network flow model, with
the flow leaving an arc being an increasing concave function of the flow
entering it, as proposed by Truemper and Shigeno. We give a polynomial time
combinatorial algorithm for solving corresponding flow maximization problems,
finding an epsilon-approximate solution in O(m(m+log n)log(MUm/epsilon))
arithmetic operations and value oracle queries, where M and U are upper bounds
on simple parameters. This also gives a new algorithm for linear generalized
flows, an efficient, purely scaling variant of the Fat-Path algorithm by
Goldberg, Plotkin and Tardos, not using any cycle cancellations.
We show that this general convex programming model serves as a common
framework for several market equilibrium problems, including the linear Fisher
market model and its various extensions. Our result immediately extends these
market models to more general settings. We also obtain a combinatorial
algorithm for nonsymmetric Arrow-Debreu Nash bargaining, settling an open
question by Vazirani.Comment: Major revision. Instead of highest gain augmenting paths, we employ
the Fat-Path framework. Many parts simplified, running time for the linear
case improve
The Cutting Plane Method is Polynomial for Perfect Matchings
The cutting plane approach to optimal matchings has been discussed by several
authors over the past decades (e.g., Padberg and Rao '82, Grotschel and Holland
'85, Lovasz and Plummer '86, Trick '87, Fischetti and Lodi '07) and its
convergence has been an open question. We give a cutting plane algorithm that
converges in polynomial-time using only Edmonds' blossom inequalities; it
maintains half-integral intermediate LP solutions supported by a disjoint union
of odd cycles and edges. Our main insight is a method to retain only a subset
of the previously added cutting planes based on their dual values. This allows
us to quickly find violated blossom inequalities and argue convergence by
tracking the number of odd cycles in the support of intermediate solutions
Concave Generalized Flows with Applications to Market Equilibria
We consider a nonlinear extension of the generalized network flow model, with the flow leaving an arc being an increasing concave function of the flow entering it, as proposed by Truemper and Shigeno. We give a polynomial time combinatorial algorithm for solving corresponding flow maximization problems, finding an epsilon-approximate solution in O(m(m+log n)log(MUm/epsilon)) arithmetic operations and value oracle queries, where M and U are upper bounds on simple parameters. This also gives a new algorithm for linear generalized flows, an efficient, purely scaling variant of the Fat-Path algorithm by Goldberg, Plotkin and Tardos, not using any cycle cancellations. We show that this general convex programming model serves as a common framework for several market equilibrium problems, including the linear Fisher market model and its various extensions. Our result immediately extends these market models to more general settings. We also obtain a combinatorial algorithm for nonsymmetric Arrow-Debreu Nash bargaining, settling an open question by Vazirani.