The Complexity of the List Partition Problem for Graphs

The k-partition problem is as follows: Given a graph G and a positive integer k, partition the vertices of G into at most k parts A1, A2, . . . , Ak, where it may be specified that Ai induces a stable set, a clique, or an arbitrary subgraph, and pairs Ai, Aj (i≠j) be completely nonadjacent, completely adjacent, or arbitrarily adjacent. The list k-partition problem generalizes the k-partition problem by specifying for each vertex x, a list L(x) of parts in which it is allowed to be placed. Many well-known graph problems can be formulated as list k-partition problems: e.g., 3-colorability, clique cutset, stable cutset, homogeneous set, skew partition, and 2-clique cutset. We classify, with the exception of two polynomially equivalent problems, each list 4-partition problem as either solvable in polynomial time or NP-complete. In doing so, we provide polynomial-time algorithms for many problems whose polynomial-time solvability was open, including the list 2-clique cutset problem. This also allows us to classify each list generalized 2-clique cutset problem and list generalized skew partition problem as solvable in polynomial time or NP-complete

Stability in one-sided matching markets

The stable roommates problem may be unsolvable for sorne instances, therefore we study a relaxation, when it is allowed to form groups of any size (the stable partition problem). Two extensions of preferences over individuals to preferences over sets are suggested. For the first one, derived from the most prefered member of a set, it is shown that a stable partition always existis if the original preferences are strict and a simple algorithm for its computation is derived. This algorithm turns out to be strategy proof. The second extension, based on the least prefered member of a set, produces solutions very similar to those for the stable roornmates problem

Cost allocation in partition functionform games

A cooperative game in partition function form is proposed for a cost allocation problem. The game describes a real situation in which a payoff of any coalition does not only depend on the players in the coalition but also on the coalition structure of the other players. Solution concepts like the stable set and the core are analyzed. Relations of the concepts in the case of the game in partition function form and of an appropriately formulated game in characteristic function form are shown.partition function form game, core, stable set, cost allocation

Ergodic Theory Meets Polarization. I: An Ergodic Theory for Binary Operations

An open problem in polarization theory is to determine the binary operations that always lead to polarization (in the general multilevel sense) when they are used in Ar{\i}kan style constructions. This paper, which is presented in two parts, solves this problem by providing a necessary and sufficient condition for a binary operation to be polarizing. This (first) part of the paper introduces the mathematical framework that we will use in the second part to characterize the polarizing operations. We define uniformity preserving, irreducible, ergodic and strongly ergodic operations and we study their properties. The concepts of a stable partition and the residue of a stable partition are introduced. We show that an ergodic operation is strongly ergodic if and only if all its stable partitions are their own residues. We also study the products of binary operations and the structure of their stable partitions. We show that the product of a sequence of binary operations is strongly ergodic if and only if all the operations in the sequence are strongly ergodic. In the second part of the paper, we provide a foundation of polarization theory based on the ergodic theory of binary operations that we develop in this part.Comment: 34 pages, 1 figure. Accepted to IEEE Trans. Inform. Theory and presented in part at ISIT'1

Two Combinatorial Models with identical Statics yet different Dynamics

Motivated by the problem of sorting, we introduce two simple combinatorial models with distinct Hamiltonians yet identical spectra (and hence partition function) and show that the local dynamics of these models are very different. After a deep quench, one model slowly relaxes to the sorted state whereas the other model becomes blocked by the presence of stable local minima.Comment: 23 pages, 11 figure

Stable marriage with general preferences

We propose a generalization of the classical stable marriage problem. In our model, the preferences on one side of the partition are given in terms of arbitrary binary relations, which need not be transitive nor acyclic. This generalization is practically well-motivated, and as we show, encompasses the well studied hard variant of stable marriage where preferences are allowed to have ties and to be incomplete. As a result, we prove that deciding the existence of a stable matching in our model is NP-complete. Complementing this negative result we present a polynomial-time algorithm for the above decision problem in a significant class of instances where the preferences are asymmetric. We also present a linear programming formulation whose feasibility fully characterizes the existence of stable matchings in this special case. Finally, we use our model to study a long standing open problem regarding the existence of cyclic 3D stable matchings. In particular, we prove that the problem of deciding whether a fixed 2D perfect matching can be extended to a 3D stable matching is NP-complete, showing this way that a natural attempt to resolve the existence (or not) of 3D stable matchings is bound to fail.Comment: This is an extended version of a paper to appear at the The 7th International Symposium on Algorithmic Game Theory (SAGT 2014

"Almost stable" matchings in the Roommates problem

An instance of the classical Stable Roommates problem (SR) need not admit a stable matching. This motivates the problem of finding a matching that is “as stable as possible”, i.e. admits the fewest number of blocking pairs. In this paper we prove that, given an SR instance with n agents, in which all preference lists are complete, the problem of finding a matching with the fewest number of blocking pairs is NP-hard and not approximable within n^{\frac{1}{2}-\varepsilon}, for any \varepsilon>0, unless P=NP. If the preference lists contain ties, we improve this result to n^{1-\varepsilon}. Also, we show that, given an integer K and an SR instance I in which all preference lists are complete, the problem of deciding whether I admits a matching with exactly K blocking pairs is NP-complete. By contrast, if K is constant, we give a polynomial-time algorithm that finds a matching with at most (or exactly) K blocking pairs, or reports that no such matching exists. Finally, we give upper and lower bounds for the minimum number of blocking pairs over all matchings in terms of some properties of a stable partition, given an SR instance I