The Interlace Polynomial

In this paper, we survey results regarding the interlace polynomial of a graph, connections to such graph polynomials as the Martin and Tutte polynomials, and generalizations to the realms of isotropic systems and delta-matroids.Comment: 18 pages, 5 figures, to appear as a chapter in: Graph Polynomials, edited by M. Dehmer et al., CRC Press/Taylor & Francis Group, LL

Stable Matchings with Restricted Preferences: Structure and Complexity

In the stable marriage (SM) problem, there are two sets of agents–traditionally referred to as men and women–and each agent has a preference list that ranks (a subset of) agents of the opposite sex. The goal is to find a matching between men and women that is stable in the sense that no man-woman pair mutually prefer each other to their assigned partners. In a seminal work, Gale and Shapley showed that stable matchings always exist, and described an efficient algorithm for finding one. Irving and Leather defined the rotation poset of an SM instance and showed that it determines the structure of the set of stable matchings of the instance. They further showed that every finite poset can be realized as the rotation poset of some SM instance. Consequently, many problems–such as counting stable matchings and finding certain “fair” stable matchings–are computationally intractable (NP-hard) in general. In this paper, we consider SM instances in which certain restrictions are placed on the preference lists. We show that three natural preference models?k-bounded, k-attribute, and (k1, k2)-list–can realize arbitrary rotation posets for constant values of k. Hence even in these highly restricted preference models, many stable matching problems remain intractable. In contrast, we show that for any fixed constant k, the rotation posets of k-range instances are highly restricted. As a consequence, we show that exactly counting and uniformly sampling stable matchings, finding median, sex-equal, and balanced stable matchings are fixed-parameter tractable when parameterized by the range of the instance. Thus, these problems can be solved in polynomial time on instances of the k-range model for any fixed constant k

Stable Matchings with Restricted Preferences: Structure and Complexity

It is well known that every stable matching instance $I$ has a rotation poset $R(I)$ that can be computed efficiently and the downsets of $R(I)$ are in one-to-one correspondence with the stable matchings of $I$. Furthermore, for every poset $P$, an instance $I(P)$ can be constructed efficiently so that the rotation poset of $I(P)$ is isomorphic to $P$. In this case, we say that $I(P)$ realizes $P$. Many researchers exploit the rotation poset of an instance to develop fast algorithms or to establish the hardness of stable matching problems. In order to gain a parameterized understanding of the complexity of sampling stable matchings, Bhatnagar et al. [SODA 2008] introduced stable matching instances whose preference lists are restricted but nevertheless model situations that arise in practice. In this paper, we study four such parameterized restrictions; our goal is to characterize the rotation posets that arise from these models: $k$-bounded, $k$-attribute, $(k_1, k_2)$-list, $k$-range. We prove that there is a constant $k$ so that every rotation poset is realized by some instance in the first three models for some fixed constant $k$. We describe efficient algorithms for constructing such instances given the Hasse diagram of a poset. As a consequence, the fundamental problem of counting stable matchings remains $\#$BIS-complete even for these restricted instances. For $k$-range preferences, we show that a poset $P$ is realizable if and only if the Hasse diagram of $P$ has pathwidth bounded by functions of $k$. Using this characterization, we show that the following problems are fixed parameter tractable when parametrized by the range of the instance: exactly counting and uniformly sampling stable matchings, finding median, sex-equal, and balanced stable matchings.Comment: Various updates and improvements in response to reviewer comment

An extensive English language bibliography on graph theory and its applications, supplement 1

Graph theory and its applications - bibliography, supplement

Networks, (K)nots, Nucleotides, and Nanostructures

Designing self-assembling DNA nanostructures often requires the identification of a route for a scaffolding strand of DNA through the target structure. When the target structure is modeled as a graph, these scaffolding routes correspond to Eulerian circuits subject to turning restrictions imposed by physical constraints on the strands of DNA. Existence of such Eulerian circuits is an NP-hard problem, which can be approached by adapting solutions to a version of the Traveling Salesperson Problem. However, the author and collaborators have demonstrated that even Eulerian circuits obeying these turning restrictions are not necessarily feasible as scaffolding routes by giving examples of nontrivially knotted circuits which cannot be traced by the unknotted scaffolding strand. Often, targets of DNA nanostructure self-assembly are modeled as graphs embedded on surfaces in space. In this case, Eulerian circuits obeying the turning restrictions correspond to A-trails, circuits which turn immediately left or right at each vertex. In any graph embedded on the sphere, all A-trails are unknotted regardless of the embedding of the sphere in space. We show that this does not hold in general for graphs on the torus. However, we show this property does hold for checkerboard-colorable graphs on the torus, that is, those graphs whose faces can be properly 2-colored, and provide a partial converse to this result. As a consequence, we characterize (with one exceptional family) regular triangulations of the torus containing unknotted A-trails. By developing a theory of sums of A-trails, we lift constructions from the torus to arbitrary n-tori, and by generalizing our work on A-trails to smooth circuit decompositions, we construct all torus links and certain sums of torus links from circuit decompositions of rectangular torus grids. Graphs embedded on surfaces are equivalent to ribbon graphs, which are particularly well-suited to modeling DNA nanostructures, as their boundary components correspond to strands of DNA and their twisted ribbons correspond to double-helices. Every ribbon graph has a corresponding delta-matroid, a combinatorial object encoding the structure of the ribbon-graph\u27s spanning quasi-trees (substructures having exactly one boundary component). We show that interlacement with respect to quasi-trees can be generalized to delta-matroids, and use the resulting structure on delta-matroids to provide feasible-set expansions for a family of delta-matroid polynomials, both recovering well-known expansions of this type (such as the spanning-tree expansion of the Tutte polynnomial) as well as providing several previously unknown expansions. Among these are expansions for the transition polynomial, a version of which has been used to study DNA nanostructure self-assembly, and the interlace polynomial, which solves a problem in DNA recombination

Aspects of practical implementations of PRAM algorithms

The PRAM is a shared memory model of parallel computation which abstracts away from inessential engineering details. It provides a very simple architecture independent model and provides a good programming environment. Theoreticians of the computer science community have proved that it is possible to emulate the theoretical PRAM model using current technology. Solutions have been found for effectively interconnecting processing elements, for routing data on these networks and for distributing the data among memory modules without hotspots. This thesis reviews this emulation and the possibilities it provides for large scale general purpose parallel computation. The emulation employs a bridging model which acts as an interface between the actual hardware and the PRAM model. We review the evidence that such a scheme crn achieve scalable parallel performance and portable parallel software and that PRAM algorithms can be optimally implemented on such practical models. In the course of this review we presented the following new results: 1. Concerning parallel approximation algorithms, we describe an NC algorithm for finding an approximation to a minimum weight perfect matching in a complete weighted graph. The algorithm is conceptually very simple and it is also the first NC-approximation algorithm for the task with a sub-linear performance ratio. 2. Concerning graph embedding, we describe dense edge-disjoint embeddings of the complete binary tree with n leaves in the following n-node communication networks: the hypercube, the de Bruijn and shuffle-exchange networks and the 2-dimcnsional mesh. In the embeddings the maximum distance from a leaf to the root of the tree is asymptotically optimally short. The embeddings facilitate efficient implementation of many PRAM algorithms on networks employing these graphs as interconnection networks. 3. Concerning bulk synchronous algorithmics, we describe scalable transportable algorithms for the following three commonly required types of computation; balanced tree computations. Fast Fourier Transforms and matrix multiplications