A Fixed-Parameter Tractable (FPT) ρ-approximation algorithm for a minimization (resp. maximization) parameterized problem P is an FPT-algorithm that, given an instance (x, k) ∈ P computes a solution of cost at most k · ρ(k) (resp. k/ρ(k)) if a solution of cost at most (resp. at least) k exists; otherwise the output can be arbitrary. For well-known intractable problems such as the W[1]-hard Cliq...

In an n by n complete bipartite graph with independent exponentially distributed edge costs, we ask for the minimum total cost of a set of edges of which each vertex is incident to at least one. This so-called minimum edge cover problem is a relaxation of perfect matching. We show that the large n limit cost of the minimum edge cover is W (1)2 + 2W (1) ≈ 1.456, where W is the Lambert W -functio...

We investigate the problem of packing and covering odd (u, v)-trails in a graph. A (u, v)-trail is a (u, v)-walk that is allowed to have repeated vertices but no repeated edges. We call a trail odd if the number of edges in the trail is odd. Let ν(u, v) denote the maximum number of edge-disjoint odd (u, v)-trails, and τ(u, v) denote the minimum size of an edge-set that intersects every odd (u, ...

Confluent drawing is a technique that allows some non-planar graphs to be visualized in a planar way. This approach merges edges together, drawing groups of them as single tracks, similar to train tracks. In the general case, producing confluent drawings automatically has proven quite difficult. We introduce the biclique edge cover graph that represents a graph G as an interconnected set of cli...

Markov Chain Monte Carlo (MCMC) method is a widely used algorithm design scheme with many applications. To make efficient use of this method, the key step is to prove that the Markov chain is rapid mixing. Canonical paths is one of the two main tools to prove rapid mixing. However, there are much fewer success examples comparing to coupling, the other main tool. The main reason is that there is...

It is a classical result that any finite tree with positively weighted edges, and without vertices of degree 2, is uniquely determined by the weighted path distance between each pair of leaves. Moreover, it is possible for a (small) strict subset L of leaf pairs to suffice for reconstructing the tree and its edge weights, given just the distances between the leaf pairs in L. It is known that an...

A 3-path vertex cover in a graph is a vertex subset C such that every path of three vertices contains at least one vertex from C. The parameterized 3-path vertex cover problem asks whether a graph has a 3-path vertex cover of size at most k. In this paper, we give a kernel of 5k vertices and an O(1.7485)-time polynomial-space algorithm for this problem, both new results improve previous known b...

The well known correspondence between even cycles of an undirected graph and polynomials in a binomial ideal associated to a graph is extended to odd cycles and polynomials in another binomial ideal. Other binomial ideals associated to an undirected graph are also introduced. The results about them with topics on monomial ideals are used in order to show decision procedures for bipartite graphs...