1,624 research outputs found
Polynomial sequences of binomial-type arising in graph theory
In this paper, we show that the solution to a large class of "tiling"
problems is given by a polynomial sequence of binomial type. More specifically,
we show that the number of ways to place a fixed set of polyominos on an
toroidal chessboard such that no two polyominos overlap is
eventually a polynomial in , and that certain sets of these polynomials
satisfy binomial-type recurrences. We exhibit generalizations of this theorem
to higher dimensions and other lattices. Finally, we apply the techniques
developed in this paper to resolve an open question about the structure of
coefficients of chromatic polynomials of certain grid graphs (namely that they
also satisfy a binomial-type recurrence).Comment: 15 page
Information complexity is computable
The information complexity of a function is the minimum amount of
information Alice and Bob need to exchange to compute the function . In this
paper we provide an algorithm for approximating the information complexity of
an arbitrary function to within any additive error , thus
resolving an open question as to whether information complexity is computable.
In the process, we give the first explicit upper bound on the rate of
convergence of the information complexity of when restricted to -bit
protocols to the (unrestricted) information complexity of .Comment: 30 page
Load limiting energy absorbing lightweight debris catcher
In the representative embodiment of the invention disclosed, a load limiting, energy absorbing net is arranged to overlay a normally-covered vent opening in the rear bulkhead of the space orbiter vehicle. Spatially-disposed flexible retainer straps are extended from the net and respectively secured to bulkhead brackets spaced around the vent opening. The intermediate portions of the straps are doubled over and stitched together in a pattern enabling the doubled-over portions to progressively separate at a predicable load designed to be well below the tensile capability of the straps as the stitches are successively torn apart by the forces imposed on the retainer members whenever the cover plate is explosively separated from the bulkhead and propelled into the net. By arranging these stitches to be successively torn away at a load below the strap strength in response to forces acting on the retainers that are less than the combined strength of the retainers, this tearing action serves as a predictable compact energy absorber for safely halting the cover plate as the retainers are extended as the net is deployed. The invention further includes a block of an energy-absorbing material positioned in the net for receiving loose debris produced by the explosive release of the cover plate
Fast Dynamic Pointer Following via Link-Cut Trees
In this paper, we study the problem of fast dynamic pointer following: given
a directed graph where each vertex has outdegree , efficiently support
the operations of i) changing the outgoing edge of any vertex, and ii) find the
vertex vertices `after' a given vertex. We exhibit a solution to this
problem based on link-cut trees that requires time per operation,
and prove that this is optimal in the cell-probe complexity model.Comment: 7 page
Condorcet-Consistent and Approximately Strategyproof Tournament Rules
We consider the manipulability of tournament rules for round-robin
tournaments of competitors. Specifically, competitors are competing for
a prize, and a tournament rule maps the result of all
pairwise matches (called a tournament, ) to a distribution over winners.
Rule is Condorcet-consistent if whenever wins all of her matches,
selects with probability .
We consider strategic manipulation of tournaments where player might
throw their match to player in order to increase the likelihood that one of
them wins the tournament. Regardless of the reason why chooses to do this,
the potential for manipulation exists as long as increases by
more than decreases. Unfortunately, it is known that every
Condorcet-consistent rule is manipulable (Altman and Kleinberg). In this work,
we address the question of how manipulable Condorcet-consistent rules must
necessarily be - by trying to minimize the difference between the increase in
and decrease in for any potential manipulating
pair.
We show that every Condorcet-consistent rule is in fact -manipulable,
and that selecting a winner according to a random single elimination bracket is
not -manipulable for any . We also show that many
previously studied tournament formats are all -manipulable, and the
popular class of Copeland rules (any rule that selects a player with the most
wins) are all in fact -manipulable, the worst possible. Finally, we consider
extensions to match-fixing among sets of more than two players.Comment: 20 page
Selling to a No-Regret Buyer
We consider the problem of a single seller repeatedly selling a single item
to a single buyer (specifically, the buyer has a value drawn fresh from known
distribution in every round). Prior work assumes that the buyer is fully
rational and will perfectly reason about how their bids today affect the
seller's decisions tomorrow. In this work we initiate a different direction:
the buyer simply runs a no-regret learning algorithm over possible bids. We
provide a fairly complete characterization of optimal auctions for the seller
in this domain. Specifically:
- If the buyer bids according to EXP3 (or any "mean-based" learning
algorithm), then the seller can extract expected revenue arbitrarily close to
the expected welfare. This auction is independent of the buyer's valuation ,
but somewhat unnatural as it is sometimes in the buyer's interest to overbid. -
There exists a learning algorithm such that if the buyer bids
according to then the optimal strategy for the seller is simply
to post the Myerson reserve for every round. - If the buyer bids according
to EXP3 (or any "mean-based" learning algorithm), but the seller is restricted
to "natural" auction formats where overbidding is dominated (e.g. Generalized
First-Price or Generalized Second-Price), then the optimal strategy for the
seller is a pay-your-bid format with decreasing reserves over time. Moreover,
the seller's optimal achievable revenue is characterized by a linear program,
and can be unboundedly better than the best truthful auction yet simultaneously
unboundedly worse than the expected welfare
- …