The Stable Marriage Problem and its many variants have been widely studied in the literature [6, 22, 15], partly because of the inherent appeal of the problem, partly because of the elegance of the associated structures and algorithms, and partly because of important practical applications, such as the National Resident Matching Program [20] and similar large-scale matching schemes. Here, we present the first comprehensive study of variants of the problem in which the preference lists of the participants are not necessarily complete and not necessarily totally ordered. We show that, under surprisingly restrictive assumptions, a number of these variants are hard, and hard to approximate. The key observation is that, in contrast to the case where preference lists are complete or strictly ordered (or both), a given problem instance may admit stable matchings of different sizes. In this setting, examples of problems that are hard are: finding a stable matching of maximum or minimum size, determining whether a given pair is stable-even if the indifference takes the form of ties on one side only, the ties are at the tails of lists, there is at most one tie per list, and each tie is of length 2; and finding, or approximating, both an 'egalitarian' and a 'minimum regret' stable matching. However, we give a 2-approximation algorithm for the problems of finding a stable matching of maximum or minimum size. We also discuss the significant implications of our results for practical matching schemes.

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Since quantum information is continuous, its handling is sometimes surprisingly harder than the classical counterpart. A typical example is cloning; making a copy of digital information is straightforward but it is not possible exactly for quantum information. The question in this paper is whether or not quantum network coding is possible. Its classical counterpart is another good example to show that digital information flow can be done much more efficiently than conventional (say, liquid) flow. Our answer to the question is similar to the case of cloning, namely, it is shown that quantum network coding is possible if approximation is allowed, by using a simple network model called Butterfly. In this network, there are two flow paths, s_1 to t_1 and s_2 to t_2, which shares a single bottleneck channel of capacity one. In the classical case, we can send two bits simultaneously, one for each path, in spite of the bottleneck. Our results for quantum network coding include: (i) We can send any quantum state |psi_1> from s_1 to t_1 and |psi_2> from s_2 to t_2 simultaneously with a fidelity strictly greater than 1/2. (ii) If one of |psi_1> and |psi_2> is classical, then the fidelity can be improved to 2/3. (iii) Similar improvement is also possible if |psi_1> and |psi_2> are restricted to only a finite number of (previously known) states. (iv) Several impossibility results including the general upper bound of the fidelity are also given.Comment: 27pages, 11figures. The 12page version will appear in 24th International Symposium on Theoretical Aspects of Computer Science (STACS 2007

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, S 1 and S 2 , there is a sequence of transformations, each of them in the rule set, which changes S 1 to S 2 . 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 CNOTbased circuits.

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 manwoman 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.

The k-f(k) dense subgraph problem((k; f(k))-DSP) asks whether there is a k-vertex subgraph of a given graph G which has at least f(k) edges. When f(k) = k(k − 1)=2, (k; f(k))-DSP is equivalent to the well-known k-clique problem. The main purpose of this paper is to discuss the problem of ÿnding slightly dense subgraphs. Note that f(k) is about k 2 for the k-clique problem. It is shown that (k; f(k))-DSP remains NP-complete for f(k) = (k 1+) where may be any constant such that 0 ¡ ¡ 1. It is also NP-complete for "relatively" slightly-dense subgraphs, i.e., (k; f(k))-DSP is NP-complete for f(k) = ek 2 =v 2 (1+O(v −1)), where v is the number of G's vertices and e is the number of G's edges. This condition is quite tight because the answer to (k; f(k))-DSP is always yes for f(k) = ek 2 =v 2 (1 − (v − k)=(vk − k)) that is the average number of edges in a subgraph of k vertices. Also, we show that the hardness of (k; f(k))-DSP remains for regular graphs: (k; f(k))-DSP is NP-complete for (v 1)-regular graphs if f(k) = (k 1+ 2) for any 0 ¡ 1; 2 ¡ 1.

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