Kazuo Iwama

Transformation rules for designing CNOT-based quantum circuits

This paper gives a simple but nontrivial set of local transformation rules for Control-NOT(CNOT)-based combinatorial circuits. It is shown that this rule set is complete, namely, for any two equivalent circuits, S1 and S2, there is a sequence of transformations, each of them in the rule set, which changes S1 to S2. Our motivation is to use this rule set for developing a design theory for quantum circuits whose Boolean logic parts should be implemented by CNOT based circuits. As a preliminary example, we give a design procedure based on our transformation rules which reduces the cost of CNOT-based circuits.

Stable Marriage with Incomplete Lists and Ties

The original stable marriage problem requires all men and women to submit a complete and strictly ordered preference list. This is obviously often unrealistic in practice, and several relaxations have been proposed, including the following two common ones: one is to allow an incomplete list, i.e., a man is permitted to accept only a subset of the women and vice versa. The other is to allow a preference list including ties. Fortunately, it is known that both relaxed problems can still be solved in polynomial time. In this paper, we show that the situation changes substantially if we allow both relaxations (incomplete lists and ties) at the same time: the problem not only becomes NP-hard, but also the optimal cost version has no approximation algorithm achieving the approximation ratio of N1-Ɛ, where N is the instance size, unless P=NP.

A Survey of the Stable Marriage Problem and Its Variants

The stable marriage problem is to find a matching between men and women, considering preference lists in which each person expresses his/her preference over the members of the opposite gender. The output matching must be stable, which intuitively means that there is no man- woman pair both of which have incentive to elope. This problem was introduced in 1962 in the seminal paper of Gale and Shapley, and has attracted researchers in several areas, including mathematics, economics, game theory, computer science, etc. This paper introduces old and recent results on the stable marriage problem and some other related problems.

CNF satisfiability test by counting and polynomial average time

The average-case performance of an algorithm for CNF SAT, recently introduced by the author, is discussed. It is shown that the algorithm takes polynomial average time for a class of CNF equations satisfying the condition that for, a constant c,

Removable Online Knapsack Problems

p^2 v \geqq \ln t - c

Approximability results for stable marriage problems with ties

, where v is the number of variables, t is the number of clauses, and p is the probability that a given literal appears in a clause. It was known that backtracking plus the pure literal rule, a common way of solving CNF SAT, takes polynomial average time if

An improved exact algorithm for cubic graph TSP

p \geqq \varepsilon

Tight bounds on the number of states of DFAs that are equivalent to n -state NFAs

(any small constant) or

Undecidability on quantum finite automata

p \leqq c(\ln {v / v})^{{3 / 2}}

Improved approximation results for the stable marriage problem

, but no algorithms were known to take polynomial average time (for all t) in the range