174 research outputs found
Ranking with Submodular Valuations
We study the problem of ranking with submodular valuations. An instance of
this problem consists of a ground set , and a collection of monotone
submodular set functions , where each .
An additional ingredient of the input is a weight vector . The
objective is to find a linear ordering of the ground set elements that
minimizes the weighted cover time of the functions. The cover time of a
function is the minimal number of elements in the prefix of the linear ordering
that form a set whose corresponding function value is greater than a unit
threshold value.
Our main contribution is an -approximation algorithm
for the problem, where is the smallest non-zero marginal value that
any function may gain from some element. Our algorithm orders the elements
using an adaptive residual updates scheme, which may be of independent
interest. We also prove that the problem is -hard to
approximate, unless P = NP. This implies that the outcome of our algorithm is
optimal up to constant factors.Comment: 16 pages, 3 figure
Ranking with submodular functions on a budget
Submodular maximization has been the backbone of many important machine-learning problems, and has applications to viral marketing, diversification, sensor placement, and more. However, the study of maximizing submodular functions has mainly been restricted in the context of selecting a set of items. On the other hand, many real-world applications require a solution that is a ranking over a set of items. The problem of ranking in the context of submodular function maximization has been considered before, but to a much lesser extent than item-selection formulations. In this paper, we explore a novel formulation for ranking items with submodular valuations and budget constraints. We refer to this problem as max-submodular ranking (MSR). In more detail, given a set of items and a set of non-decreasing submodular functions, where each function is associated with a budget, we aim to find a ranking of the set of items that maximizes the sum of values achieved by all functions under the budget constraints. For the MSR problem with cardinality- and knapsack-type budget constraints we propose practical algorithms with approximation guarantees. In addition, we perform an empirical evaluation, which demonstrates the superior performance of the proposed algorithms against strong baselines.Peer reviewe
Matrix bids in combinatorial auctions: expressiveness and micro-economic properties
A combinatorial auction is an auction where multiple items are for sale simultaneously to a set of buyers. Furthermore, buyers are allowed to place bids on subsets of the available items. This paper focuses on a combinatorial auction where a bidder can express his preferences by means of a so-called ordered matrix bid. Ordered matrix bids are a bidding language that allows a compact representation of a bidder''s preferences, and was developed by Day (2004). We give an overview of how a combinatorial auction with matrix bids works. We elaborate on the relevance of the matrix bid auction and we develop methods to verify whether a given matrix bid satisfies properties related to micro-economic theory as free disposal, subadditivity, submodularity and the gross substitutes property. Finally, we investigate how a collection of arbitrary bids can be represented as a matrix bid.microeconomics ;
Composable and Efficient Mechanisms
We initiate the study of efficient mechanism design with guaranteed good
properties even when players participate in multiple different mechanisms
simultaneously or sequentially. We define the class of smooth mechanisms,
related to smooth games defined by Roughgarden, that can be thought of as
mechanisms that generate approximately market clearing prices. We show that
smooth mechanisms result in high quality outcome in equilibrium both in the
full information setting and in the Bayesian setting with uncertainty about
participants, as well as in learning outcomes. Our main result is to show that
such mechanisms compose well: smoothness locally at each mechanism implies
efficiency globally.
For mechanisms where good performance requires that bidders do not bid above
their value, we identify the notion of a weakly smooth mechanism. Weakly smooth
mechanisms, such as the Vickrey auction, are approximately efficient under the
no-overbidding assumption. Similar to smooth mechanisms, weakly smooth
mechanisms behave well in composition, and have high quality outcome in
equilibrium (assuming no overbidding) both in the full information setting and
in the Bayesian setting, as well as in learning outcomes.
In most of the paper we assume participants have quasi-linear valuations. We
also extend some of our results to settings where participants have budget
constraints
Credible Comparisons in Multi-Issue Bargaining
We find that bargaining over multiple issues as a bundle allows for credible cheap talk on the relative importance of the issues even when interests are directly opposed on each issue. This communication increases the chance of agreement by allowing the offerer to concede more on the issue of greater importance to the offeree. In contrast, if the issues are bargained over separately we find that the offeree will lie about which issue is of greater importance. The communication gains from bundling are in addition to the gains from bundling previously identified in the monopoly pricing literature. Applied to negotiations between an executive and a legislature, our results imply that bundling multiple issues in a single bill with no line-item veto increases communication and can benefit both the legislature and, in many cases, the executive as well.bundling; bargaining; cheap talk; veto
Draft Auctions
We introduce draft auctions, which is a sequential auction format where at
each iteration players bid for the right to buy items at a fixed price. We show
that draft auctions offer an exponential improvement in social welfare at
equilibrium over sequential item auctions where predetermined items are
auctioned at each time step. Specifically, we show that for any subadditive
valuation the social welfare at equilibrium is an -approximation
to the optimal social welfare, where is the number of items. We also
provide tighter approximation results for several subclasses. Our welfare
guarantees hold for Bayes-Nash equilibria and for no-regret learning outcomes,
via the smooth-mechanism framework. Of independent interest, our techniques
show that in a combinatorial auction setting, efficiency guarantees of a
mechanism via smoothness for a very restricted class of cardinality valuations,
extend with a small degradation, to subadditive valuations, the largest
complement-free class of valuations. Variants of draft auctions have been used
in practice and have been experimentally shown to outperform other auctions.
Our results provide a theoretical justification
Precoder Design for Physical Layer Multicasting
This paper studies the instantaneous rate maximization and the weighted sum
delay minimization problems over a K-user multicast channel, where multiple
antennas are available at the transmitter as well as at all the receivers.
Motivated by the degree of freedom optimality and the simplicity offered by
linear precoding schemes, we consider the design of linear precoders using the
aforementioned two criteria. We first consider the scenario wherein the linear
precoder can be any complex-valued matrix subject to rank and power
constraints. We propose cyclic alternating ascent based precoder design
algorithms and establish their convergence to respective stationary points.
Simulation results reveal that our proposed algorithms considerably outperform
known competing solutions. We then consider a scenario in which the linear
precoder can be formed by selecting and concatenating precoders from a given
finite codebook of precoding matrices, subject to rank and power constraints.
We show that under this scenario, the instantaneous rate maximization problem
is equivalent to a robust submodular maximization problem which is strongly NP
hard. We propose a deterministic approximation algorithm and show that it
yields a bicriteria approximation. For the weighted sum delay minimization
problem we propose a simple deterministic greedy algorithm, which at each step
entails approximately maximizing a submodular set function subject to multiple
knapsack constraints, and establish its performance guarantee.Comment: 37 pages, 8 figures, submitted to IEEE Trans. Signal Pro
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