153 research outputs found
On a generalization of iterated and randomized rounding
We give a general method for rounding linear programs that combines the
commonly used iterated rounding and randomized rounding techniques. In
particular, we show that whenever iterated rounding can be applied to a problem
with some slack, there is a randomized procedure that returns an integral
solution that satisfies the guarantees of iterated rounding and also has
concentration properties. We use this to give new results for several classic
problems where iterated rounding has been useful
The Price of Information in Combinatorial Optimization
Consider a network design application where we wish to lay down a
minimum-cost spanning tree in a given graph; however, we only have stochastic
information about the edge costs. To learn the precise cost of any edge, we
have to conduct a study that incurs a price. Our goal is to find a spanning
tree while minimizing the disutility, which is the sum of the tree cost and the
total price that we spend on the studies. In a different application, each edge
gives a stochastic reward value. Our goal is to find a spanning tree while
maximizing the utility, which is the tree reward minus the prices that we pay.
Situations such as the above two often arise in practice where we wish to
find a good solution to an optimization problem, but we start with only some
partial knowledge about the parameters of the problem. The missing information
can be found only after paying a probing price, which we call the price of
information. What strategy should we adopt to optimize our expected
utility/disutility?
A classical example of the above setting is Weitzman's "Pandora's box"
problem where we are given probability distributions on values of
independent random variables. The goal is to choose a single variable with a
large value, but we can find the actual outcomes only after paying a price. Our
work is a generalization of this model to other combinatorial optimization
problems such as matching, set cover, facility location, and prize-collecting
Steiner tree. We give a technique that reduces such problems to their non-price
counterparts, and use it to design exact/approximation algorithms to optimize
our utility/disutility. Our techniques extend to situations where there are
additional constraints on what parameters can be probed or when we can
simultaneously probe a subset of the parameters.Comment: SODA 201
On minimum -claw deletion in split graphs
For , is called -claw. In minimum -claw deletion
problem (\texttt{Min--Claw-Del}), given a graph , it is required
to find a vertex set of minimum size such that is
-claw free. In a split graph, the vertex set is partitioned into two sets
such that one forms a clique and the other forms an independent set. Every
-claw in a split graph has a center vertex in the clique partition. This
observation motivates us to consider the minimum one-sided bipartite -claw
deletion problem (\texttt{Min--OSBCD}). Given a bipartite graph , in \texttt{Min--OSBCD} it is asked to find a vertex set of
minimum size such that has no -claw with the center
vertex in . A primal-dual algorithm approximates \texttt{Min--OSBCD}
within a factor of . We prove that it is \UGC-hard to approximate with a
factor better than . We also prove it is approximable within a factor of 2
for dense bipartite graphs. By using these results on \texttt{Min--OSBCD},
we prove that \texttt{Min--Claw-Del} is \UGC-hard to approximate within a
factor better than , for split graphs. We also consider their complementary
maximization problems and prove that they are \APX-complete.Comment: 11 pages and 1 figur
On a generalization of iterated and randomized rounding
We give a general method for rounding linear programs that combines the commonly used iterated rounding and randomized rounding techniques. In particular, we show that whenever iterated rounding can be applied to a problem with some slack, there is a randomized procedure that returns an integral solution that satisfies the guarantees of iterated rounding and also has concentration properties. We use this to give new results for several classic problems such as rounding column-sparse LPs, makespan minimization on unrelated machines, degree-bounded spanning trees and multi-budgeted matchings
Nash welfare, valuated matroids, and gross substitutes
We study computational aspects of equilibria and fair division problems with a focus on demand and valuation functions that satisfy the (weak) gross substitutes property. We study the Arrow-Debreu exchange market model with divisible goods where agents’ demands satisfy the weak gross substitutes (WGS) property. We give an auction algorithm that obtains an approximate market equilibrium for WGS demands. Previously, such algorithms were known only for restricted classes of WGS demands. We also derive the implications of our technique for spending-restricted market equilibrium for budget-separable piecewise linear concave (budget-SPLC) utilities. Spending-restricted equilibrium was introduced as a continuous relaxation of the Nash SocialWelfare (NSW) problem. Next, we present the first polynomial-time constant-factor approximation algorithm for the NSW problem under Rado valuations. Rado valuations form a general class of valuation functions that arise from maximum cost independent matching problems. They include as special cases assignment (OXS) valuations and weighted matroid rank functions. Our approach also gives the first polynomial-time constant-factor approximation algorithm for the asymmetric NSW problem under Rado valuations, provided that the maximum ratio between the weights is bounded by a constant. We examine the Matroid Based Valuation (MBV) conjecture by Ostrovsky and Paes Leme (Theoretical Economics 2015). It asserts that every (discrete) gross substitute valuation is a matroid based valuation—a valuation obtained from weighted matroid rank functions by repeated applications of merge and endowment operations. Each matroid based valuation turns out to be an endowment of some Rado valuation. By introducing complete classes of valuated matroids, we exhibit a family of valuations that are gross substitutes but not endowed Rado valuations. This refutes the MBV conjecture. The family is defined via sparse paving matroids
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