26 research outputs found
Understanding the Correlation Gap For Matchings
Given a set of vertices V with |V| = n, a weight vector w in (R^+ cup {0})^{binom{V}{2}}, and a probability vector x In [0, 1]^{binom{V}{2}} in the matching polytope, we study the quantity (E_{G}[ nu_w(G)])/(sum_(u, v) in binom{V}{2} w_{u, v} x_{u, v}) where G is a random graph where each edge e with weight w_e appears with probability x_e independently, and let nu_w(G) denotes the weight of the maximum matching of G. This quantity is closely related to correlation gap and contention resolution schemes, which are important tools in the design of approximation algorithms, algorithmic game theory, and stochastic optimization.
We provide lower bounds for the above quantity for general and bipartite graphs, and for weighted and unweighted settings. The best known upper bound is 0.54 by Karp and Sipser, and the best lower bound is 0.4. We show that it is at least 0.47 for unweighted bipartite graphs, at least 0.45 for weighted bipartite graphs, and at least 0.43 for weighted general graphs. To achieve our results, we construct local distribution schemes on the dual which may be of independent interest
Maximizing Revenue in the Presence of Intermediaries
We study the mechanism design problem of selling items to unit-demand
buyers with private valuations for the items. A buyer either participates
directly in the auction or is represented by an intermediary, who represents a
subset of buyers. Our goal is to design robust mechanisms that are independent
of the demand structure (i.e. how the buyers are partitioned across
intermediaries), and perform well under a wide variety of possible contracts
between intermediaries and buyers.
We first study the case of identical items where each buyer draws its
private valuation for an item i.i.d. from a known -regular
distribution. We construct a robust mechanism that, independent of the demand
structure and under certain conditions on the contracts between intermediaries
and buyers, obtains a constant factor of the revenue that the mechanism
designer could obtain had she known the buyers' valuations. In other words, our
mechanism's expected revenue achieves a constant factor of the optimal welfare,
regardless of the demand structure. Our mechanism is a simple posted-price
mechanism that sets a take-it-or-leave-it per-item price that depends on
and the total number of buyers, but does not depend on the demand structure or
the downstream contracts.
Next we generalize our result to the case when the items are not identical.
We assume that the item valuations are separable. For this case, we design a
mechanism that obtains at least a constant fraction of the optimal welfare, by
using a menu of posted prices. This mechanism is also independent of the demand
structure, but makes a relatively stronger assumption on the contracts between
intermediaries and buyers, namely that each intermediary prefers outcomes with
a higher sum of utilities of the subset of buyers represented by it
Prior-Independent Auctions for Heterogeneous Bidders
We study the design of prior-independent auctions in a setting with
heterogeneous bidders. In particular, we consider the setting of selling to
bidders whose values are drawn from independent but not necessarily
identical distributions. We work in the robust auction design regime, where we
assume the seller has no knowledge of the bidders' value distributions and must
design a mechanism that is prior-independent. While there have been many strong
results on prior-independent auction design in the i.i.d. setting, not much is
known for the heterogeneous setting, even though the latter is of significant
practical importance. Unfortunately, no prior-independent mechanism can hope to
always guarantee any approximation to Myerson's revenue in the heterogeneous
setting; similarly, no prior-independent mechanism can consistently do better
than the second-price auction. In light of this, we design a family of
(parametrized) randomized auctions which approximates at least one of these
benchmarks: For heterogeneous bidders with regular value distributions, our
mechanisms either achieve a good approximation of the expected revenue of an
optimal mechanism (which knows the bidders' distributions) or exceeds that of
the second-price auction by a certain multiplicative factor. The factor in the
latter case naturally trades off with the approximation ratio of the former
case. We show that our mechanism is optimal for such a trade-off between the
two cases by establishing a matching lower bound. Our result extends to selling
identical items to heterogeneous bidders with an additional -factor in our trade-off between the two cases
On the Lovász theta function for independent sets in sparse graphs
We consider the maximum independent set problem on sparse graphs with maximum degree d. We show that the Lovász ϑ-function based semidefinite program (SDP) has an integrality gap of O(d/log3/2 d), improving on the previous best result of O(d/log d). This improvement is based on a new Ramsey-theoretic bound on the independence number of Kr-free graphs for large values of r. We also show that for stronger SDPs, namely, those obtained using polylog(d) levels of the SA+ semidefinite hierarchy, the integrality gap reduces to O(d/log2 d). This matches the best unique-games-based hardness result up to lower-order poly(log log d) factors. Finally, we give an algorithmic version of this SA+-based integrality gap result, albeit using d levels of SA+, via a coloring algorithm of Johansson