arXiv subjects

Pinyan Lu

Publications and source records attributed to Pinyan Lu.

At least 55 records · Page 3Linked to original sources

arXiv2021

An Algorithmic Framework for Approximating Maximin Share Allocation of Chores

In this paper, we consider the problem of how to fairly dividing $m$ indivisible chores among $n$ agents. The fairness measure we considered here is the maximin share. The previous best known result is that there always exists a $\frac{4}{3}$ approximation maximin share allocation. With a novel algorithm, we can always find a $\frac{11}{9}$ approximation maximin share allocation for any instances. We also discuss how to improve the efficiency of the algorithm and its connection to the job scheduling problem.

cs.GT↗

arXiv2021

Correlation Decay up to Uniqueness in Spin Systems

We give a complete characterization of the two-state anti-ferromagnetic spin systems which are of strong spatial mixing on general graphs. We show that a two-state anti-ferromagnetic spin system is of strong spatial mixing on all graphs of maximum degree at most $Δ$ if and only if the system has a unique Gibbs measure on infinite regular trees of degree up to $Δ$, where $Δ$ can be either bounded or unbounded. As a consequence, there exists an FPTAS for the partition function of a two-state anti-ferromagnetic spin system on graphs of maximum degree at most $Δ$ when the uniqueness condition is satisfied on infinite regular trees of degree up to $Δ$. In particular, an FPTAS exists for arbitrary graphs if the uniqueness is satisfied on all infinite regular trees. This covers as special cases all previous algorithmic results for two-state anti-ferromagnetic systems on general-structure graphs. Combining with the FPRAS for two-state ferromagnetic spin systems of Jerrum-Sinclair and Goldberg-Jerrum-Paterson, and the hardness results of Sly-Sun and independently of Galanis-Stefankovic-Vigoda, this gives a complete classification, except at the phase transition boundary, of the approximability of all two-state spin systems, on either degree-bounded families of graphs or family of all graphs.

cs.DS↗

arXiv2020

Generalized Sorting with Predictions

Generalized sorting problem, also known as sorting with forbidden comparisons, was first introduced by Huang et al. together with a randomized algorithm which requires $\tilde O(n^{3/2})$ probes. We study this problem with additional predictions for all pairs of allowed comparisons as input. We propose a randomized algorithm which uses $O(n \log n+w)$ probes with high probability and a deterministic algorithm which uses $O(nw)$ probes, where $w$ is the number of mistakes made by prediction.

cs.DS↗

arXiv2020

Bayesian Auctions with Efficient Queries

Generating good revenue is one of the most important problems in Bayesian auction design, and many (approximately) optimal dominant-strategy incentive compatible (DSIC) Bayesian mechanisms have been constructed for various auction settings. However, most existing studies do not consider the complexity for the seller to carry out the mechanism. It is assumed that the seller knows "each single bit" of the distributions and is able to optimize perfectly based on the entire distributions. Unfortunately, this is a strong assumption and may not hold in reality: for example, when the value distributions have exponentially large supports or do not have succinct representations. In this work we consider, for the first time, the query complexity of Bayesian mechanisms. We only allow the seller to have limited oracle accesses to the players' value distributions, via quantile queries and value queries. For a large class of auction settings, we prove logarithmic lower-bounds for the query complexity for any DSIC Bayesian mechanism to be of any constant approximation to the optimal revenue. For single-item auctions and multi-item auctions with unit-demand or additive valuation functions, we prove tight upper-bounds via efficient query schemes, without requiring the distributions to be regular or have monotone hazard rate. Thus, in those auction settings the seller needs to access much less than the full distributions in order to achieve approximately optimal revenue.

cs.GT↗

arXiv2020

Optimal Budget-Feasible Mechanisms for Additive Valuations

In this paper, we show a tight approximation guarantee for budget-feasible mechanisms with an additive buyer. We propose a new simple randomized mechanism with approximation ratio of $2$, improving the previous best known result of $3$. Our bound is tight with respect to either the optimal offline benchmark, or its fractional relaxation. We also present a simple deterministic mechanism with the tight approximation guarantee of $3$ against the fractional optimum, improving the best known result of $(2+ \sqrt{2})$ for the weaker integral benchmark.

cs.GT↗

arXiv2020

Tight Revenue Gaps among Simple Mechanisms

We consider a fundamental problem in microeconomics: selling a single item to a number of potential buyers, whose values are drawn from known independent and regular (not necessarily identical) distributions. There are four widely-used and widely-studied mechanisms in the literature: {\sf Myerson Auction}~({\sf OPT}), {\sf Sequential Posted-Pricing}~({\sf SPM}), {\sf Second-Price Auction with Anonymous Reserve}~({\sf AR}), and {\sf Anonymous Pricing}~({\sf AP}). {\sf OPT} is revenue-optimal but complicated, which also experiences several issues in practice such as fairness; {\sf AP} is the simplest mechanism, but also generates the lowest revenue among these four mechanisms; {\sf SPM} and {\sf AR} are of intermediate complexity and revenue. We explore revenue gaps among these mechanisms, each of which is defined as the largest ratio between revenues from a pair of mechanisms. We establish two tight bounds and one improved bound: 1. {\sf SPM} vs.\ {\sf AP}: this ratio studies the power of discrimination in pricing schemes. We obtain the tight ratio of $\mathcal{C^*} \approx 2.62$, closing the gap between $\big[\frac{e}{e - 1}, e\big]$ left before. 2. {\sf AR} vs.\ {\sf AP}: this ratio measures the relative power of auction scheme vs.\ pricing scheme, when no discrimination is allowed. We attain the tight ratio of $\frac{π^2}{6} \approx 1.64$, closing the previously known bounds $\big[\frac{e}{e - 1}, e\big]$. 3. {\sf OPT} vs.\ {\sf AR}: this ratio quantifies the power of discrimination in auction schemes, and is previously known to be somewhere between $\big[2, e\big]$. The lower-bound of $2$ was conjectured to be tight by Hartline and Roughgarden (2009) and Alaei et al.\ (2015). We acquire a better lower-bound of $2.15$, and thus disprove this conjecture.

cs.GT↗

arXiv2019

Approximating Permanent of Random Matrices with Vanishing Mean: Made Better and Simpler

The algorithm and complexity of approximating the permanent of a matrix is an extensively studied topic. Recently, its connection with quantum supremacy and more specifically BosonSampling draws special attention to the average-case approximation problem of the permanent of random matrices with zero or small mean value for each entry. Eldar and Mehraban (FOCS 2018) gave a quasi-polynomial time algorithm for random matrices with mean at least $1/\mathbf{\mathrm{polyloglog}} (n)$. In this paper, we improve the result by designing a deterministic quasi-polynomial time algorithm and a PTAS for random matrices with mean at least $1/\mathbf{\mathrm{polylog}}(n)$. We note that if it can be further improved to $1/\mathbf{\mathrm{poly}}(n)$, it will disprove a central conjecture for quantum supremacy. Our algorithm is also much simpler and has a better and flexible trade-off for running time. The running time can be quasi-polynomial in both $n$ and $1/ε$, or PTAS (polynomial in $n$ but exponential in $1/ε$), where $ε$ is the approximation parameter.

cs.DS↗

arXiv2019

Strategyproof Mechanism for Two Heterogeneous Facilities with Constant Approximation Ratio

In this paper, we study the two-facility location game on a line with optional preference where the acceptable set of facilities for each agent could be different and an agent's cost is his distance to the closest facility within his acceptable set. The objective is to minimize the total cost of all agents while achieving strategyproofness. We design a deterministic strategyproof mechanism for the problem with approximation ratio of 2.75, improving upon the earlier best ratio of n/2+1.

cs.DS↗

arXiv2019

Zeros of ferromagnetic 2-spin systems

We study zeros of the partition functions of ferromagnetic 2-state spin systems in terms of the external field, and obtain new zero-free regions of these systems via a refinement of Asano's and Ruelle's contraction method. The strength of our results is that they do not depend on the maximum degree of the underlying graph. Via Barvinok's method, we also obtain new efficient and deterministic approximate counting algorithms. In certain regimes, our algorithm outperforms all other methods such as Markov chain Monte Carlo and correlation decay.

cs.DS↗

arXiv2019

Counting hypergraph colorings in the local lemma regime

We give a fully polynomial-time approximation scheme (FPTAS) to count the number of $q$-colorings for $k$-uniform hypergraphs with maximum degree $Δ$ if $k\ge 28$ and $q >357Δ^{\frac{14}{k-14}}$ . We also obtain a polynomial-time almost uniform sampler if $q>931Δ^{\frac{16}{k-16/3}}$. These are the first approximate counting and sampling algorithms in the regime $q\llΔ$ (for large $Δ$ and $k$) without any additional assumptions. Our method is based on the recent work of Moitra (STOC, 2017). One important contribution of ours is to remove the dependency of $k$ and $Δ$ in Moitra's approach.

cs.DS↗

arXiv2019

Counting independent sets and colorings on random regular bipartite graphs

We give a fully polynomial-time approximation scheme (FPTAS) to count the number of independent sets on almost every $Δ$-regular bipartite graph if $Δ\ge 53$. In the weighted case, for all sufficiently large integers $Δ$ and weight parameters $λ=\tildeΩ\left(\frac{1}Δ\right)$, we also obtain an FPTAS on almost every $Δ$-regular bipartite graph. Our technique is based on the recent work of Jenssen, Keevash and Perkins (SODA, 2019) and we also apply it to confirm an open question raised there: For all $q\ge 3$ and sufficiently large integers $Δ=Δ(q)$, there is an FPTAS to count the number of $q$-colorings on almost every $Δ$-regular bipartite graph.

cs.DS↗

arXiv2019

Revenue Maximization with Imprecise Distribution

We study the revenue maximization problem with an imprecisely estimated distribution of a single buyer or several independent and identically distributed buyers given that this estimation is not far away from the true distribution. We use the earth mover's distance to capture the estimation error between those two distributions in terms of both values and their probabilities, i.e., the error in value space given a quantile, and the error in quantile space given a value. We give explicit characterization of the optimal mechanisms for the single buyer setting. For the multi-buyer case, we provide an algorithm that finds an approximately optimal mechanism (FPTAS) among the family of second price auctions with a fixed reserve.

cs.GT↗

arXiv2018

Approximability of the Eight-vertex Model

We initiate a study of the classification of approximation complexity of the eight-vertex model defined over 4-regular graphs. The eight-vertex model, together with its special case the six-vertex model, is one of the most extensively studied models in statistical physics, and can be stated as a problem of counting weighted orientations in graph theory. Our result concerns the approximability of the partition function on all 4-regular graphs, classified according to the parameters of the model. Our complexity results conform to the phase transition phenomenon from physics. We introduce a quantum decomposition of the eight-vertex model and prove a set of closure properties in various regions of the parameter space. Furthermore, we show that there are extra closure properties on 4-regular planar graphs. These regions of the parameter space are concordant with the phase transition threshold. Using these closure properties, we derive polynomial time approximation algorithms via Markov chain Monte Carlo. We also show that the eight-vertex model is NP-hard to approximate on the other side of the phase transition threshold.

cs.CC↗

arXiv2018

Tight Approximation Ratio of Anonymous Pricing

We consider two canonical Bayesian mechanism design settings. In the single-item setting, we prove tight approximation ratio for anonymous pricing: compared with Myerson Auction, it extracts at least $\frac{1}{2.62}$-fraction of revenue; there is a matching lower-bound example. In the unit-demand single-buyer setting, we prove tight approximation ratio between the simplest and optimal deterministic mechanisms: in terms of revenue, uniform pricing admits a $2.62$-approximation of item pricing; we further validate the tightness of this ratio. These results settle two open problems asked in~\cite{H13,CD15,AHNPY15,L17,JLTX18}. As an implication, in the single-item setting: we improve the approximation ratio of the second-price auction with anonymous reserve to $2.62$, which breaks the state-of-the-art upper bound of $e \approx 2.72$.

cs.GT↗

arXiv2018

Zeros of Holant problems: locations and algorithms

We present fully polynomial-time (deterministic or randomised) approximation schemes for Holant problems, defined by a non-negative constraint function satisfying a generalised second order recurrence modulo a couple of exceptional cases. As a consequence, any non-negative Holant problem on cubic graphs has an efficient approximation algorithm unless the problem is equivalent to approximately counting perfect matchings, a central open problem in the area. This is in sharp contrast to the computational phase transition shown by 2-state spin systems on cubic graphs. Our main technique is the recently established connection between zeros of graph polynomials and approximate counting. We also use the "winding" technique to deduce the second result on cubic graphs.

cs.DS↗

arXiv2018

Combinatorial Multi-Armed Bandit with General Reward Functions

In this paper, we study the stochastic combinatorial multi-armed bandit (CMAB) framework that allows a general nonlinear reward function, whose expected value may not depend only on the means of the input random variables but possibly on the entire distributions of these variables. Our framework enables a much larger class of reward functions such as the $\max()$ function and nonlinear utility functions. Existing techniques relying on accurate estimations of the means of random variables, such as the upper confidence bound (UCB) technique, do not work directly on these functions. We propose a new algorithm called stochastically dominant confidence bound (SDCB), which estimates the distributions of underlying random variables and their stochastically dominant confidence bounds. We prove that SDCB can achieve $O(\log{T})$ distribution-dependent regret and $\tilde{O}(\sqrt{T})$ distribution-independent regret, where $T$ is the time horizon. We apply our results to the $K$-MAX problem and expected utility maximization problems. In particular, for $K$-MAX, we provide the first polynomial-time approximation scheme (PTAS) for its offline problem, and give the first $\tilde{O}(\sqrt T)$ bound on the $(1-ε)$-approximation regret of its online problem, for any $ε>0$.

cs.LG↗

arXiv2018

Uniqueness, Spatial Mixing, and Approximation for Ferromagnetic 2-Spin Systems

We give fully polynomial-time approximation schemes (FPTAS) for the partition function of ferromagnetic 2-spin systems in certain parameter regimes. The threshold we obtain is almost tight up to an integrality gap. Our technique is based on the correlation decay framework. The main technical contribution is a new potential function, with which we establish a new kind of spatial mixing.

cs.DS↗

arXiv2017

Approximability of the Six-vertex Model

In this paper we take the first step toward a classification of the approximation complexity of the six-vertex model, an object of extensive research in statistical physics. Our complexity results conform to the phase transition phenomenon from physics. We show that the approximation complexity of the six-vertex model behaves dramatically differently on the two sides separated by the phase transition threshold. Furthermore, we present structural properties of the six-vertex model on planar graphs for parameter settings that have known relations to the Tutte polynomial $T(G, x, y)$.

cs.CC↗