The acyclic directed Steiner tree problem (ADSP) requires a minimal outward tree within an acyclic digraph with edge costs G = (V; E; d) which connects a root r with a distinguished subset S V , #S = k. The best possible performance guarantee of any polynomial approximation algorithm for ADSP cannot be less than 1 4 log k unless ~ P NP. The presented series of heuristics A n has a performance g...

Greene and Zaslavsky proved that the number of acyclic orientations of a graph G with a unique sink at a given vertex is, up to sign, the linear coefficient of the chromatic polynomial. We give three proofs of this result using pure induction, noncommutative symmetric functions, and an algorithmic bijection. We also prove their result that if e=u0v0 is an edge of G then the number of acyclic or...

Expanding on a recent definition by Bialostocki and Voxman, we define the rainbow ramsey number RR(G1, G2) of two graphs G1 and G2 to be the minimum integer N such that any edge-coloring of the complete graph KN with any number of colors must contain either a copy of G1 with every edge the same color or a copy of G2 with every edge a different color. This number is well-defined if G1 is a star ...

We prove the NP-completeness of the integer multiflow problem in planar graphs, with the following restrictions: there are only two demand edges, both lying on the infinite face of the routing graph. This was one of the open challenges concerning disjoint paths, explicitly asked by Müller [5]. It also strengthens Schwärzler’s recent proof of one of the open problems of Schrijver’s book [9], abo...

A coloring of the vertices of a graph is called acyclic if the ends of each edge are colored in distinct colors, and there are no two-colored cycles. Suppose each face of rank k , k ≥ 4 , in a map on a surface S is replaced by the clique having the same number of vertices. It is proved in [1] that the resulting pseudograph admits an acyclic coloring with the number of colors depending linearly ...

The arboricity of a graph G is the minimum number of colours needed to colour the edges of G so that every cycle gets at least two colours. Given a positive integer p, we define the generalized p-arboricity Arbp(G) of a graph G as the minimum number of colours needed to colour the edges of a multigraph G in such a way that every cycle C gets at least min(|C|, p + 1) colours. In the particular c...

Scheduling by Edge Reversal (SER) is a fully distributed scheduling mechanism based on the manipulation of acyclic orientations of a graph. This work uses SER to perform constraint partitioning of Constraint Satisfaction Problems (CSP). In order to apply the SER mechanism, the graph representing the constraints must receive an acyclic orientation. Since obtaining an optimal acyclic orientation ...

We develop terminology and methods for working with maximally oriented partially directed acyclic graphs (maximal PDAGs). Maximal PDAGs arise from imposing restrictions on a Markov equivalence class of directed acyclic graphs, or equivalently on its graphical representation as a completed partially directed acyclic graph (CPDAG), for example when adding background knowledge about certain edge o...

Most data networks contain cycles, but so far most attention in the literature of network coding has been addressed to multicast in acyclic networks. The original paper on network coding [1] did consider cyclic networks but there it was suggested to transform the cyclic network into an acyclic network using the idea of unrolling the network into a layered network. This approach has many drawbac...

We consider the maximization version of the edge disjoint path problem (EDP). In undirected graphs and directed acyclic graphs, we obtain anO( √ n) upper bound on the approximation ratio where n is the number of nodes in the graph. We show this by establishing the upper bound on the integrality gap of the natural multicommodity flow based relaxation. Our upper bound matches to within a constant...