We consider two variants of the classical Stable Roommates problem with Incomplete (but strictly ordered) preference lists (sri) that are degree constrained, i.e., preference lists are of bounded length. The first variant, egald-sri, involves finding an egalitarian stable matching in solvable instances of sri with preference lists of length at most d. We show that this problem is NP-hard even if d = 3. On the positive side we give a $\frac {2d+3}{7}$ -approximation algorithm for d ∈{3,4,5} which improves on the known bound of 2 for the unbounded preference list case. In the second variant of sri, called d-srti, preference lists can include ties and are of length at most d. We show that the problem of deciding whether an instance of d-srti admits a stable matching is NP-complete even if d = 3. We also consider the “most stable” version of this problem and prove a strong inapproximability bound for the d = 3 case. However for d = 2 we show that the latter problem can be solved in polynomial time.

We study two-sided matching markets in which participants are initially endowed with partial preference orderings, lacking precise information about their true, strictly ordered list of preferences. We wish to reason about matchings that are stable with respect to agents' true preferences, and which are furthermore optimal for one given side of the market. We present three main results. First, one can decide in polynomial time whether there exists a matching that is stable and optimal under all strict preference orders that refine the given partial orders, and can construct this matching in polynomial time if it does exist. We show, however, that deciding whether a given pair of agents are matched in all or no such optimal stable matchings is co-NP-complete, even under quite severe restrictions on preferences. Finally, we describe a polynomial-time algorithm that decides, given a matching that is stable under the partial preference orderings, whether that matching is stable and optimal for one side of the market under some refinement of the partial orders.

We consider the problem of matching a set of applicants to a set of posts, where each applicant has a preference list, ranking a nonempty subset of posts in order of preference, possibly involving ties. We say that a matching M is popular if there is no matching $M'$ such that the number of applicants preferring $M'$ to M exceeds the number of applicants preferring M to $M'$. In this paper, we give the first polynomial-time algorithms to determine if an instance admits a popular matching and to find a largest such matching, if one exists. For the special case in which every preference list is strictly ordered (i.e., contains no ties), we give an $O(n + m)$ time algorithm, where n is the total number of applicants and posts and m is the total length of all of the preference lists. For the general case in which preference lists may contain ties, we give an $O(\sqrt{n}m)$ time algorithm.

Suppose that each member of a set A of applicants ranks a subset of a set P of posts in an order of preference, possibly involving ties. A matching is a set of (applicant, post) pairs such that each applicant and each post appears in at most one pair. A rank-maximal matching is one in which the maximum possible number of applicants are matched to their first choice post, and subject to that condition, the maximum possible number are matched to their second choice post, and so on. This is a relevant concept in any practical matching situation and it was first studied by Irving [2003].We give an algorithm to compute a rank-maximal matching with running time O (min( n + C , C √ n ) m ), where C is the maximal rank of an edge used in a rank-maximal matching, n is the number of applicants and posts and m is the total size of the preference lists.

The problems of sorting by reversals and sorting by transpositions have been studied because of their applications to genome comparison. Prior studies of both problems have assumed that the sequences to be compared (or sorted) contain no duplicates, but there is a natural generalization in which the sequences are allowed to contain repeated characters. In this paper we study primarily the versions of these problems in which the strings to be compared are drawn from a binary alphabet. We obtain upper and lower bounds for reversal and transposition distance and show that the problem of finding reversal distance between binary strings, and therefore between strings over an arbitrary fixed-size alphabet, is NP-hard.

Suppose there is a three-dimensional table of cross-tabulated nonnegative integer statistics, and suppose that all of the row, column, and “file” sums are revealed together with the values in some of the individual cells in the table. The question arises as to whether, as a consequence, the values contained in some of the other (suppressed) cells can be deduced from the information revealed. The corresponding problem in two dimensions has been comprehensively studied by Gusfield [SIAM J. Comput., 17 (1988), pp. 552–571], who derived elegant polynomial-time algorithms for the identification of any such “compromised” cells, and for calculating the tightest bounds on the values contained in all cells that follow from the information revealed. In this note it is shown, by contrast, that the three-dimensional version of the problem is NP-complete. It is also shown that if the suggested row, column, and file sums for an unknown three-dimensional table are given, with or without the values in some of the cells, the problem of determining whether there exists any table with the given sums is NP-complete. In the course of proving these results, the NP-completeness of some constrained Latin square construction problems, which are of some interest in their own right, is established.

In an instance of size n of the stable marriage problem, each of n men and n women ranks the members of the opposite sex in order of preference. A stable matching is a complete matching of men and women such that no man and woman who are not partners both prefer each other to their actual partners under the matching. It is well known [2] that at least one stable matching exists for every stable marriage instance. However, the classical Gale-Shapley algorithm produces a marriage that greatly favors the men at the expense of the women, or vice versa. The problem arises of finding a stable matching that is optimal under some more equitable or egalitarian criterion of optimality. This problem was posed by Knuth [6] and has remained unsolved for some time. Here, the objective of maximizing the average (or, equivalently, the total) “satisfaction” of all people is used. This objective is achieved when a person's satisfaction is measured by the position of his/her partner in his/her preference list. By exploiting the structure of the set of all stable matchings, and using graph-theoretic methods, an O ( n 4 ) algorithm for this problem is derived.

In an instance of size n of the stable marriage problem, each of n men and n women ranks the members of the opposite sex in order of preference. A stable matching is a complete matching of men and women such that no man and woman who are not partners both prefer each other to their actual partners under the matching. It is well known that at least one stable matching exists for every stable marriage instance, so that the decision version of the problem always has a “yes” answer. Furthermore, efficient algorithms are known for the determination of such a stable matching, so that the search version of the problem is polynomially solvable. However, by exploring the structure of the set of stable matchings for any particular instance of the problem, and exploiting its relationship with the set of antichains of an associated partially ordered set, we prove that the enumeration version of the problem—determining the number of stable matchings—is # P-complete, and therefore cannot be solved in polynomial time if ${\bf P} \ne {\bf NP}$.

An algorithm is presented for the lexicographic generation of permutations, which readily lends itself to the conduction of a backtrack search in the space of permutations of a set of items. The method is similar to that employed in an algorithm of Rohl, but achieves lexicographic order at no extra cost by storing unused items throughout in a linked linear list. Following the specification of the permutation generation algorithm itself, its application to backtracking is illustrated with reference to the n-queens problem, and some figures are given to compare its efficiency with that of related algorithms.