arXiv subjects

Yang Cai

Publications and source records attributed to Yang Cai.

94 records · Page 6Linked to original sources

arXiv2012

Can Nondeterminism Help Complementation?

Complementation and determinization are two fundamental notions in automata theory. The close relationship between the two has been well observed in the literature. In the case of nondeterministic finite automata on finite words (NFA), complementation and determinization have the same state complexity, namely Theta(2^n) where n is the state size. The same similarity between determinization and complementation was found for Buchi automata, where both operations were shown to have 2^Θ(n lg n) state complexity. An intriguing question is whether there exists a type of omega-automata whose determinization is considerably harder than its complementation. In this paper, we show that for all common types of omega-automata, the determinization problem has the same state complexity as the corresponding complementation problem at the granularity of 2^Θ(.).

cs.LO↗

arXiv2012

Optimal Multi-Dimensional Mechanism Design: Reducing Revenue to Welfare Maximization

We provide a reduction from revenue maximization to welfare maximization in multi-dimensional Bayesian auctions with arbitrary (possibly combinatorial) feasibility constraints and independent bidders with arbitrary (possibly combinatorial) demand constraints, appropriately extending Myerson's result to this setting. We also show that every feasible Bayesian auction can be implemented as a distribution over virtual VCG allocation rules. A virtual VCG allocation rule has the following simple form: Every bidder's type t_i is transformed into a virtual type f_i(t_i), via a bidder-specific function. Then, the allocation maximizing virtual welfare is chosen. Using this characterization, we show how to find and run the revenue-optimal auction given only black box access to an implementation of the VCG allocation rule. We generalize this result to arbitrarily correlated bidders, introducing the notion of a second-order VCG allocation rule. We obtain our reduction from revenue to welfare optimization via two algorithmic results on reduced forms in settings with arbitrary feasibility and demand constraints. First, we provide a separation oracle for determining feasibility of a reduced form. Second, we provide a geometric algorithm to decompose any feasible reduced form into a distribution over virtual VCG allocation rules. In addition, we show how to execute both algorithms given only black box access to an implementation of the VCG allocation rule. Our results are computationally efficient for all multi-dimensional settings where the bidders are additive. In this case, our mechanisms run in time polynomial in the total number of bidder types, but not type profiles. For generic correlated distributions, this is the natural description complexity of the problem. The runtime can be further improved to poly(#items, #bidders) in item-symmetric settings by making use of recent techniques.

cs.GT↗

arXiv2011

A Tight Lower Bound for Streett Complementation

Finite automata on infinite words ($ω$-automata) proved to be a powerful weapon for modeling and reasoning infinite behaviors of reactive systems. Complementation of $ω$-automata is crucial in many of these applications. But the problem is non-trivial; even after extensive study during the past four decades, we still have an important type of $ω$-automata, namely Streett automata, for which the gap between the current best lower bound $2^{Ω(n \lg nk)}$ and upper bound $2^{Ω(nk \lg nk)}$ is substantial, for the Streett index size $k$ can be exponential in the number of states $n$. In arXiv:1102.2960 we showed a construction for complementing Streett automata with the upper bound $2^{O(n \lg n+nk \lg k)}$ for $k = O(n)$ and $2^{O(n^{2} \lg n)}$ for $k=ω(n)$. In this paper we establish a matching lower bound $2^{Ω(n \lg n+nk \lg k)}$ for $k = O(n)$ and $2^{Ω(n^{2} \lg n)}$ for $k = ω(n)$, and therefore showing that the construction is asymptotically optimal with respect to the $2^{Θ(\cdot)}$ notation.

cs.LO↗

arXiv2011

Tight Upper Bounds for Streett and Parity Complementation

Complementation of finite automata on infinite words is not only a fundamental problem in automata theory, but also serves as a cornerstone for solving numerous decision problems in mathematical logic, model-checking, program analysis and verification. For Streett complementation, a significant gap exists between the current lower bound $2^{Ω(n\lg nk)}$ and upper bound $2^{O(nk\lg nk)}$, where $n$ is the state size, $k$ is the number of Streett pairs, and $k$ can be as large as $2^{n}$. Determining the complexity of Streett complementation has been an open question since the late '80s. In this paper show a complementation construction with upper bound $2^{O(n \lg n+nk \lg k)}$ for $k = O(n)$ and $2^{O(n^{2} \lg n)}$ for $k = ω(n)$, which matches well the lower bound obtained in \cite{CZ11a}. We also obtain a tight upper bound $2^{O(n \lg n)}$ for parity complementation.

cs.LO↗