Seiichiro Tani

The Complexity of the Optimal Variable Ordering Problems of Shared Binary Decision Diagrams

A binary decision diagram (BDD) is a directed acyclic graph for representing a Boolean function. BDDs are widely used in various areas which require Boolean function manipulation, since BDDs can represent efficiently many of practical Boolean functions and have other desirable properties. However the complexity of constructing BDDs has hardly been researched theoretically. In this paper, we prove that the optimal variable ordering problem of shared BDDs is NP-complete, and touch on the hardness of this problem and related problems of BDDs.

Computing the Tutte Polynomial of a Graph of Moderate Size

The problem of computing the Tutte polynomial of a graph is #P-hard in general, and any known algorithm takes exponential time at least. This paper presents a new algorithm by exploiting a fact that many 2-isomorphic minors appear in the process of computation. The complexity of the algorithm is analyzed in terms of Bell numbers and Catalan numbers. This algorithm enables us to compute practically the Tutte polynomial of any graph with at most 14 vertices and 91 edges, and that of a planar graph such as 12×12 lattice graph with 144 vertices and 264 edges.

Exact Quantum Algorithms for the Leader Election Problem

This article gives a separation between quantum and classical models in pure (i.e., noncryptographic) computing abilities with no restriction on the amount of available computing resources, by considering the exact solvability of the leader election problem in anonymous networks, a celebrated unsolvable problem in classical distributed computing. The goal of the leader election problem is to elect a unique leader from among distributed parties. In an anonymous network, all parties with the same number of communication links are identical. It is well-known that no classical algorithm can exactly solve (i.e., in bounded time without error) the leader election problem in anonymous networks, even if the number of parties is given. This article devises a quantum algorithm that, if the number of parties is given, exactly solves the problem for any network topology in polynomial rounds with polynomial communication/time complexity with respect to the number of parties, when the parties are connected with quantum communication links and they have the ability of quantum computing. Our algorithm works even when only an upper bound of the number of parties is given. In such a case, no classical algorithm can solve the problem even under the zero-error setting, the setting in which error is not allowed but running time may be unbounded.

Claw finding algorithms using quantum walk

The claw finding problem has been studied in terms of query complexity as one of the problems closely connected to cryptography. Given two functions, f and g, with domain sizes N and M(N@?M), respectively, and the same range, the goal of the problem is to find x and y such that f(x)=g(y). This problem has been considered in both quantum and classical settings in terms of query complexity. This paper describes an optimal algorithm that uses quantum walk to solve this problem. Our algorithm can be slightly modified to solve the more general problem of finding a tuple consisting of elements in the two function domains that has a prespecified property. It can also be generalized to find a claw of k functions for any constant integer k>1, where the domain sizes of the functions may be different.

A Reordering Operation for an Ordered Binary Decision Diagram and an Extended Framework for Combinatorics of Graphs

Binary decision diagrams have been shown as a powerful paradigm in handling Boolean functions and have been applied to many fields such as VLSI CAD, AI, combinatorics, etc. This paper proposes a new operation on an ordered binary decision diagram (OBDD), called reordering, and demonstrates its usefulness with presenting an extended algorithmic framework of applying OBDDs to combinatorial graph enumeration problems.

Collapse of the Hierarchy of Constant-Depth Exact Quantum Circuits

We study the quantum complexity class QNC0f of quantum operations implement able exactly by constant-depth polynomial-size quantum circuits with unbounded fan-out gates. Our main result is that the quantum OR operation is in QNC0f, which is an affirmative answer to the question of Hoyer and Spalek. In sharp contrast to the strict hierarchy of the classical complexity classes: NC0 ⊊ AC0 ⊊ TC0, our result with Hoyer and Spaleks one implies the collapse of the hierarchy of the corresponding quantum ones: QNC0f = QAC0f = QTC0f. Then, we show that there exists a constant-depth sub quadratic-size quantum circuit for the quantum threshold operation. This allows us to obtain a better bound on the size difference between the QNC0f and QTC0f circuits for implementing the same quantum operation. Lastly, we show that, if the quantum Fourier transform modulo a prime is in QNC0f, there exists a polynomial-time exact classical algorithm for a discrete logarithm problem using a QNC0f oracle. This implies that, under a plausible assumption, there exists a classically hard problem that is solvable exactly by a QNC0f circuit with gates for the quantum Fourier transform.

Quantum Query Complexity of Boolean Functions with Small On-Sets

The main objective of this paper is to show that the quantum query complexity Q(f) of an N-bit Boolean function f is bounded by a function of a simple and natural parameter, i.e., M = |{x|f(x) = 1}| or the size of fs on-set. We prove that: (i) For

Output-size Sensitiveness of OBDD Construction Through Maximal Independent Set Problem

poly(N)\le M\le 2^{N^d}

Commuting Quantum Circuits with Few Outputs are Unlikely to be Classically Simulatable

for some constant 0 < d < 1, the upper bound of Q(f) is

Brief announcement: exactly electing a unique leader is not harder than computing symmetric functions on anonymous quantum networks

O(\sqrt{N\log M / \log N})