arXiv subjects

Jon Schneider

Publications and source records attributed to Jon Schneider.

61 records · Page 4Linked to original sources

arXiv2016

Condorcet-Consistent and Approximately Strategyproof Tournament Rules

We consider the manipulability of tournament rules for round-robin tournaments of $n$ competitors. Specifically, $n$ competitors are competing for a prize, and a tournament rule $r$ maps the result of all $\binom{n}{2}$ pairwise matches (called a tournament, $T$) to a distribution over winners. Rule $r$ is Condorcet-consistent if whenever $i$ wins all $n-1$ of her matches, $r$ selects $i$ with probability $1$. We consider strategic manipulation of tournaments where player $j$ might throw their match to player $i$ in order to increase the likelihood that one of them wins the tournament. Regardless of the reason why $j$ chooses to do this, the potential for manipulation exists as long as $\Pr[r(T) = i]$ increases by more than $\Pr[r(T) = j]$ 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 $\Pr[r(T) = i]$ and decrease in $\Pr[r(T) = j]$ for any potential manipulating pair. We show that every Condorcet-consistent rule is in fact $1/3$-manipulable, and that selecting a winner according to a random single elimination bracket is not $α$-manipulable for any $α> 1/3$. We also show that many previously studied tournament formats are all $1/2$-manipulable, and the popular class of Copeland rules (any rule that selects a player with the most wins) are all in fact $1$-manipulable, the worst possible. Finally, we consider extensions to match-fixing among sets of more than two players.

cs.GT↗

arXiv2016

Competitive analysis of the top-K ranking problem

Motivated by applications in recommender systems, web search, social choice and crowdsourcing, we consider the problem of identifying the set of top $K$ items from noisy pairwise comparisons. In our setting, we are non-actively given $r$ pairwise comparisons between each pair of $n$ items, where each comparison has noise constrained by a very general noise model called the strong stochastic transitivity (SST) model. We analyze the competitive ratio of algorithms for the top-$K$ problem. In particular, we present a linear time algorithm for the top-$K$ problem which has a competitive ratio of $\tilde{O}(\sqrt{n})$; i.e. to solve any instance of top-$K$, our algorithm needs at most $\tilde{O}(\sqrt{n})$ times as many samples needed as the best possible algorithm for that instance (in contrast, all previous known algorithms for the top-$K$ problem have competitive ratios of $\tildeΩ(n)$ or worse). We further show that this is tight: any algorithm for the top-$K$ problem has competitive ratio at least $\tildeΩ(\sqrt{n})$.

cs.DS↗

arXiv2015

Tight space-noise tradeoffs in computing the ergodic measure

In this note we obtain tight bounds on the space-complexity of computing the ergodic measure of a low-dimensional discrete-time dynamical system affected by Gaussian noise. If the scale of the noise is $\varepsilon$, and the function describing the evolution of the system is not by itself a source of computational complexity, then the density function of the ergodic measure can be approximated within precision $δ$ in space polynomial in $\log 1/\varepsilon+\log\log 1/δ$. We also show that this bound is tight up to polynomial factors. In the course of showing the above, we prove a result of independent interest in space-bounded computation: that it is possible to exponentiate an $n$ by $n$ matrix to an exponentially large power in space polylogarithmic in $n$.

cs.CC↗

arXiv2015

Information complexity is computable

The information complexity of a function $f$ is the minimum amount of information Alice and Bob need to exchange to compute the function $f$. In this paper we provide an algorithm for approximating the information complexity of an arbitrary function $f$ to within any additive error $α> 0$, 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 $f$ when restricted to $b$-bit protocols to the (unrestricted) information complexity of $f$.

cs.IT↗

arXiv2014

Fast Dynamic Pointer Following via Link-Cut Trees

In this paper, we study the problem of fast dynamic pointer following: given a directed graph $G$ where each vertex has outdegree $1$, efficiently support the operations of i) changing the outgoing edge of any vertex, and ii) find the vertex $k$ vertices `after' a given vertex. We exhibit a solution to this problem based on link-cut trees that requires $O(\lg n)$ time per operation, and prove that this is optimal in the cell-probe complexity model.

cs.DS↗

arXiv2012

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 $n\times n$ toroidal chessboard such that no two polyominos overlap is eventually a polynomial in $n$, 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).

math.CO↗

arXiv2011

Enumeration and Quasipolynomiality of Chip-Firing Configurations

In this paper we explore enumeration problems related to the number of reachable configurations in a chip-firing game on a finite connected graph G. We define an auxiliary notion of debt-reachability and prove that the number of debt-reachable configurations from an initial configuration with c chips on one vertex is a quasipolynomial in c. For the cycle graph C_n, we apply these results to compute a near explicit formula for the number of debt-reachable configurations. We then derive polynomial asymptotic bounds for the number of debt-reachable and reachable configurations, and finally provide evidence for a quasipolynomiality conjecture regarding the number of reachable configurations.

math.CO↗