Improved Lower Bounds for Testing Triangle-freeness in Boolean Functions via Fast Matrix Multiplication

Understanding the query complexity for testing linear-invariant properties has been a central open problem in the study of algebraic property testing. Triangle-freeness in Boolean functions is a simple property whose testing complexity is unknown. Three Boolean functions $f_1$, $f_2$ and $f_3: \mathbb{F}_2^k \to \{0, 1\}$ are said to be triangle free if there is no $x, y \in \mathbb{F}_2^k$ such that $f_1(x) = f_2(y) = f_3(x + y) = 1$. This property is known to be strongly testable (Green 2005), but the number of queries needed is upper-bounded only by a tower of twos whose height is polynomial in 1 / \epsislon, where \epsislon is the distance between the tested function triple and triangle-freeness, i.e., the minimum fraction of function values that need to be modified to make the triple triangle free. A lower bound of $(1 / \epsilon)^{2.423}$ for any one-sided tester was given by Bhattacharyya and Xie (2010). In this work we improve this bound to $(1 / \epsilon)^{6.619}$. Interestingly, we prove this by way of a combinatorial construction called \emph{uniquely solvable puzzles} that was at the heart of Coppersmith and Winograd's renowned matrix multiplication algorithm

Optimal Trading Mechanisms with Ex Ante Unidentified Traders

We analyze optimal trading mechanisms in an exchange economy where each trader owns some units of a good to be traded and may be either a seller or a buyer, depending on the realization of the privately observed valuations. The concept of virtual valuation is extended to ex ante unidentified traders; contrary to the case where each trader is assigned a role as either a buyer or a seller, the traders' virtual valuations now depend on the choice of the trading mechanism and are generally non-monotonic even if the distribution of valuations is regular. We show that the trading mechanisms that maximize a broker's expected profit or expected total gains from trade are generalized double auctions which maximize the gains from trade measured in some modified monotonic virtual valuations for the traders. The bunching phenomena, which are here specific to ex ante unidentified traders, will be a general feature in these mechanisms. Furthermore, the randomization rule by which ties are broken is now part of the design of the optimal mechanisms. Finally, we show that the optimal mechanism converges toward a simple bid-ask mechanism as the number of participants in the market increases.

On the Complexity of Computing an Equilibrium in Combinatorial Auctions

We study combinatorial auctions where each item is sold separately but simultaneously via a second price auction. We ask whether it is possible to efficiently compute in this game a pure Nash equilibrium with social welfare close to the optimal one. We show that when the valuations of the bidders are submodular, in many interesting settings (e.g., constant number of bidders, budget additive bidders) computing an equilibrium with good welfare is essentially as easy as computing, completely ignoring incentives issues, an allocation with good welfare. On the other hand, for subadditive valuations, we show that computing an equilibrium requires exponential communication. Finally, for XOS (a.k.a. fractionally subadditive) valuations, we show that if there exists an efficient algorithm that finds an equilibrium, it must use techniques that are very different from our current ones

A behavioural model of the adoption and use of new telecommunications media: the effects of communication scenarios and media product/service attributes

Recent years have seen the dramatic growth of new modes of communication. Above and beyond using land line and mobile phone for voice real-time communication, people spend increasing amounts of time receiving and sending messages through social networks (e.g. Myspace or Facebook) and also through real-time communication software (e.g. Skype or MSN). As indicated by the significant decline on the amount of call volumes of land line and mobile phone during the period from 2000 to 2006 in UK and in Taiwan, we conjecture that consumers are transferring to these new forms of communication in order to satisfy their communication needs, diminishing the demand for established channels. The purpose of this research is to develop a behavioural model to analyse the perceived value and weight of the specific media attributes that drive people to adopt or use these new communication channels. Seven telecommunications media available in 2010 have been categorised in this research included land-line, mobile phone, short message service (SMS), E-mail, Internet telephony, instant messaging and social networking. Various media product/service attributes such as synchronicity, multi-tasking, price, quality, mobility, privacy and video which might affect the media choice of consumers were first identified. Importantly, this research has designed six types of communication scenarios in the online survey with 894 valid responses to clarify the effects of different communication aims, distinguish consumers' intended behaviours toward these telecommunications media. --Multi-attribute choice model,Telecommunications media,Communication scenario,New product adoption,Substitution effect,ICT forecasting

Optimal Auctions for Correlated Buyers with Sampling

Cr\'emer and McLean [1985] showed that, when buyers' valuations are drawn from a correlated distribution, an auction with full knowledge on the distribution can extract the full social surplus. We study whether this phenomenon persists when the auctioneer has only incomplete knowledge of the distribution, represented by a finite family of candidate distributions, and has sample access to the real distribution. We show that the naive approach which uses samples to distinguish candidate distributions may fail, whereas an extended version of the Cr\'emer-McLean auction simultaneously extracts full social surplus under each candidate distribution. With an algebraic argument, we give a tight bound on the number of samples needed by this auction, which is the difference between the number of candidate distributions and the dimension of the linear space they span

`Sinking' in a bed of grains activated by shearing

We show how a weak force, $f$, enables intruder motion through dense granular materials subject to external mechanical excitations, in the present case stepwise shearing. A force acts on a Teflon disc in a two dimensional system of photoelastic discs. This force is much smaller than the smallest force needed to move the disc without any external excitation. In a cycle, material + intruder are sheared quasi-statically from $\gamma = 0$ to $\gamma_{max}$, and then backwards to $\gamma = 0$. During various cycle phases, fragile and jammed states form. Net intruder motion, $\delta$, occurs during fragile periods generated by shear reversals. $\delta$ per cycle, e.g. the quasistatic rate $c$, is constant, linearly dependent on $\gamma_{max}$ and $f$. It vanishes as, $c \propto (\phi_c - \phi)^a$, with $a \simeq 3$ and $\phi_c \simeq \phi_J$, reflecting the stiffening of granular systems under shear as $\phi \rightarrow \phi_J$. The intruder motion induces large scale grain circulation. In the intruder frame, this motion is a granular analogue to fluid flow past a cylinder, where $f$ is the drag force exerted by the flow.Comment: 4 pages, 5 figures letter with supplementarie