17 research outputs found
Revenue Maximization and Ex-Post Budget Constraints
We consider the problem of a revenue-maximizing seller with m items for sale
to n additive bidders with hard budget constraints, assuming that the seller
has some prior distribution over bidder values and budgets. The prior may be
correlated across items and budgets of the same bidder, but is assumed
independent across bidders. We target mechanisms that are Bayesian Incentive
Compatible, but that are ex-post Individually Rational and ex-post budget
respecting. Virtually no such mechanisms are known that satisfy all these
conditions and guarantee any revenue approximation, even with just a single
item. We provide a computationally efficient mechanism that is a
-approximation with respect to all BIC, ex-post IR, and ex-post budget
respecting mechanisms. Note that the problem is NP-hard to approximate better
than a factor of 16/15, even in the case where the prior is a point mass
\cite{ChakrabartyGoel}. We further characterize the optimal mechanism in this
setting, showing that it can be interpreted as a distribution over virtual
welfare maximizers.
We prove our results by making use of a black-box reduction from mechanism to
algorithm design developed by \cite{CaiDW13b}. Our main technical contribution
is a computationally efficient -approximation algorithm for the algorithmic
problem that results by an application of their framework to this problem. The
algorithmic problem has a mixed-sign objective and is NP-hard to optimize
exactly, so it is surprising that a computationally efficient approximation is
possible at all. In the case of a single item (), the algorithmic problem
can be solved exactly via exhaustive search, leading to a computationally
efficient exact algorithm and a stronger characterization of the optimal
mechanism as a distribution over virtual value maximizers
The unsplittable stable marriage problem
The Gale-Shapley "propose/reject" algorithm is a wellknown procedure for solving the classical stable marriage problem. In this paper we study this algorithm in the context of the many-to-many stable marriage problem, also known as the stable allocation or ordinal transportation problem. We present an integral variant of the Gale- Shapley algorithm that provides a direct analog, in the context of "ordinal" assignment problems, of a well-known bicriteria approximation algorithm of Shmoys and Tardos for scheduling on unrelated parallel machines with costs. If we are assigning, say, jobs to machines, our algorithm nds an unsplit (non-preemptive) stable assignment where every job is assigned at least as well as it could be in any fractional stable assignment, and where each machine is congested by at most the processing time of the largest job.4th IFIP International Conference on Theoretical Computer ScienceRed de Universidades con Carreras en Informática (RedUNCI
An Improved Randomized Truthful Mechanism for Scheduling Unrelated Machines
We study the scheduling problem on unrelated machines in the mechanism design
setting. This problem was proposed and studied in the seminal paper (Nisan and
Ronen 1999), where they gave a 1.75-approximation randomized truthful mechanism
for the case of two machines. We improve this result by a 1.6737-approximation
randomized truthful mechanism. We also generalize our result to a
-approximation mechanism for task scheduling with machines, which
improve the previous best upper bound of $0.875m(Mu'alem and Schapira 2007)
Energy-Efficient Multiprocessor Scheduling for Flow Time and Makespan
We consider energy-efficient scheduling on multiprocessors, where the speed
of each processor can be individually scaled, and a processor consumes power
when running at speed , for . A scheduling algorithm
needs to decide at any time both processor allocations and processor speeds for
a set of parallel jobs with time-varying parallelism. The objective is to
minimize the sum of the total energy consumption and certain performance
metric, which in this paper includes total flow time and makespan. For both
objectives, we present instantaneous parallelism clairvoyant (IP-clairvoyant)
algorithms that are aware of the instantaneous parallelism of the jobs at any
time but not their future characteristics, such as remaining parallelism and
work. For total flow time plus energy, we present an -competitive
algorithm, which significantly improves upon the best known non-clairvoyant
algorithm and is the first constant competitive result on multiprocessor speed
scaling for parallel jobs. In the case of makespan plus energy, which is
considered for the first time in the literature, we present an
-competitive algorithm, where is the total number of
processors. We show that this algorithm is asymptotically optimal by providing
a matching lower bound. In addition, we also study non-clairvoyant scheduling
for total flow time plus energy, and present an algorithm that achieves -competitive for jobs with arbitrary release time and
-competitive for jobs with identical release time. Finally,
we prove an lower bound on the competitive ratio of
any non-clairvoyant algorithm, matching the upper bound of our algorithm for
jobs with identical release time
Approximating Star Cover Problems
Given a metric space , we consider star covers of with
balanced loads. A star is a pair where and , and the load of a star is . In minimum load
-star cover problem , one tries to cover the set of
clients using stars that minimize the maximum load of a star, and in
minimum size star cover one aims to find the minimum number
of stars of load at most needed to cover , where is a given
parameter.
We obtain new bicriteria approximations for the two problems using novel
rounding algorithms for their standard LP relaxations. For , we
find a star cover with stars and
load where
is the optimum load. For , we find
a star cover with stars of load
at most where is the optimal
number of stars for the problem. Previously, non-trivial bicriteria
approximations were known only when
FASTER ALGORITHMS FOR STABLE ALLOCATION PROBLEMS
We consider a high-multiplicity generalization of the classical stable matching problem known as the stable allocation problem, introduced by Baiou and Balinski in 2002. By leveraging new structural properties and sophisticated data structures, we show how to solve this problem in O(mlog n) time on an bipartite instance with n nodes and m edges, improving the best known running time of O(mn). Our approach simplifies the algorithmic landscape for this problem by providing a common generalization of two different approaches from the literature -- the classical Gale-Shapley algorithm, and a recent algorithm of Baiou and Balinski. Building on this algorithm, we provide an O(m log n) algorithm for the non-bipartite stable allocation problem that introduces a new and useful transformation from non-bipartite to bipartite instances. We also give a polynomial-time algorithm for solving the \u27optimal\u27 variant of the bipartite stable allocation problem, as well as a 2-approximation algorithm for the NP-hard \u27optimal\u27 variant of the non-bipartite stable allocation problem. Finally, we highlight some important connections between the stable allocation problem and the maximum flow problem
The unsplittable stable marriage problem
The Gale-Shapley "propose/reject" algorithm is a wellknown procedure for solving the classical stable marriage problem. In this paper we study this algorithm in the context of the many-to-many stable marriage problem, also known as the stable allocation or ordinal transportation problem. We present an integral variant of the Gale- Shapley algorithm that provides a direct analog, in the context of "ordinal" assignment problems, of a well-known bicriteria approximation algorithm of Shmoys and Tardos for scheduling on unrelated parallel machines with costs. If we are assigning, say, jobs to machines, our algorithm nds an unsplit (non-preemptive) stable assignment where every job is assigned at least as well as it could be in any fractional stable assignment, and where each machine is congested by at most the processing time of the largest job.4th IFIP International Conference on Theoretical Computer ScienceRed de Universidades con Carreras en Informática (RedUNCI