We prove new results for approximating Graphic TSP. Specifically, we provide a polynomialtime 9 7 -approximation algorithm for cubic bipartite graphs and a ( 9 7 + 1 21(k−2) )-approximation algorithm for k-regular bipartite graphs, both of which are improved approximation factors compared to previous results. Our approach involves finding a cycle cover with relatively few cycles, which we are a...

Given a directed graph D = (V; A) and a set S V , a bibranching is a set of arcs B A that contains a v{(V n S) path for every v 2 S and an S{v path for every v 2 V n S. In this paper, we describe a primal-dual algorithm that determines a minimum weight bibranching in a weighted digraph. It has running time O(n 0 (m + n log n)), where m = jAj, n = jV j and n 0 = minfjSj; jV n Sjg. Thus, our algo...

We prove that a generalization of the edge dominating set problem, in which each edge e needs to be covered be times for all e ∈ E, admits a linear time 2-approximation for general unweighted graphs and that it can be solved optimally for weighted trees. We show how to solve it optimally in linear time for unweighted trees and for weighted trees in which be ∈ {0, 1} for all e ∈ E. Moreover, we ...

The k-chain subgraph cover problem asks if the edge set of a given bipartite graph G is the union of the edge sets of k chain graphs, where each chain graph is a subgraph of G. Although the X--chain subgraph cover problem is known to be NP-complete for the class of bipartite graphs, it is still unknown whether this problem is NP-complete or polynomial-time solvable for subclasses of bipartite g...

Let T = (V,A) be a directed tree. Given a collection P of dipaths on T , we can look at the arc-intersection graph I(P, T ) whose vertex set is P and where two vertices are connected by an edge if the corresponding dipaths share a common arc. Monma and Wei, who started their study in a seminal paper on intersection graphs of paths on a tree, called them DE graphs (for directed edge path graphs)...

Given a graph G = (V, E), a set of vertices S ⊆ V covers v ∈ V if the edge connectivity between S and v is at least a given number k. Vertices in S are called sources. The source location problem is a problem of finding a minimum-size source set covering all vertices of a given graph. This paper presents a new variation of the problem, called maximum-cover source-location problem, which finds a...

A circuit cover of an edge-weighted graph (G, p) is a multiset of circuits in G such that every edge e is contained in exactly p(e) circuits in the multiset. A nonnegative integer valued weight vector p is admissible if the total weight of any edge-cut is even, and no edge has more than half the total weight of any edge-cut containing it. A graph G has the circuit cover property if (G, p) has a...

1 Approximation Algorithms Any known algorithm that finds the solution to an NP-hard optimization problem has exponential running time. However, sometimes polynomial time algorithms exist which find a " good " solution instead of an optimum solution. Given a minimization problem and an approximation algorithm, we can evaluate the algorithm as follows. First, we find a lower bound on the optimum...

It is shown that the Hilbert series of the face ring of a clique complex (equivalently, flag complex) of a graph G is, up to a factor, just a specialization of S G (x, y), the subgraph polynomial of the complement of G. We also find a simple relationship between the size of a minimum vertex cover of a graph G and its subgraph polynomial. This yields a formula for the h-vector of the flag comple...