Given a graph G = (V; E), the metric polytope S(G) is deened by the inequalities x(F) ? x(C n F) jFj ? 1 for F C; jFj odd ; C cycle of G, and 0 x e 1 for e 2 E. Optimization over S(G) provides an approximation for the max-cut problem. The graph G is called 1 d-integral if all the vertices of S(G) have their coordinates in f i d j 0 i dg. We prove that the class of 1 d-integral graphs is closed ...

Given positive integers m,n, s, t, let z (m,n, s, t) be the maximum number of ones in a (0, 1) matrix of size m× n that does not contain an all ones submatrix of size s× t. We show that if s ≥ 2 and t ≥ 2, then for every k = 0, . . . , s− 2, z (m,n, s, t) ≤ (s− k − 1) nm + kn+ (t− 1)m. This generic bound implies the known bounds of Kövari, Sós and Turán, and of Füredi. As a consequence, we also...

Absfmcf-We consider the real-weight maximum cut of a planar graph. Given an undirected planar graph with real-valued weights associated with its edges, find a partition of the vertices into two nonemply sets such that the sum of the weights of the edges connecting the two sets is maximum. The conventional maximum cut and minimum cut problems assume nonnegative edge weights, and thus are special...

The bipartite vertex (resp. edge) frustration of a graph G, denoted by ψ(G) (resp. φ(G)), is the smallest number of vertices (resp. edges) that have to be deleted from G to obtain a bipartite subgraph of G. A sharp lower bound of the bipartite vertex frustration of the line graph L(G) of every graph G is given. In addition, the exact value of ψ(L(G)) is calculated when G is a forest.

Let MaxCut(G) be the value of the maximum cut of a graph G. Let f(x, n) be the expectation of MaxCut(G)/xn for random graphs with n vertices and xn edges and let r(x, n) be the expectation of MaxCut(G)/xn for random 2x-regular graphs with n vertices. We prove, for sufficiently large x: 1. limn→∞ f(x, n) ≤ 12 + √ ln 2 2x , 2. limn→∞ r(x, n) ≤ 12 + 1 √ x + 1 2 ln x x .

Laurent and Poljak introduced a very general class of valid linear inequalities, called gap inequalities, for the max-cut problem. We show that an analogous class of inequalities can be defined for general nonconvex mixed-integer quadratic programs. These inequalities dominate some inequalities arising from a natural semidefinite relaxation.

We present the results of a computational investigation of the pseudoflow and push-relabel algorithms for the maximum flow and minimum s-t cut problems. The two algorithms were tested on several problem instances from the literature. Our results show that our implementation of the pseudoflow algorithm is faster than the best-known implementation of push-relabel on most of the problem instances ...

We will create the partitioning by adding the vertices of V to A or B one by one. Let us assume V ′ is the set of vertices we have already added and at least half of the edges in G[V ′] are in between A and B. Now if u is the next vertex to insert we will greedily add it to A if |Γ(u) ∩ B| > |Γ(u) ∩ A| otherwise we add it to B. Thus after inserting u the invariant that at least half of the edge...

The following algorithm partitions road networks surprisingly well: (i) sort the vertices by longitude (or latitude, or some linear combination) and (ii) compute the maximum flow from the first k nodes (forming the source) to the last k nodes (forming the sink). Return the corresponding minimum cut as an edge separator (or recurse until the resulting subgraphs are sufficiently small).

In this paper, we report record nanosecond output energies of gain-switched Cr:ZnSe lasers pumped by Q-switched Cr:Tm:Ho:YAG (100 ns @ 2.096 μm) and Raman shifted Nd:YAG lasers (7 ns @ 1.906 μm). In these experiments we used Brewster cut Cr:ZnSe gain elements with a chromium concentration of 8x10 cm. Under Cr:Tm:Ho:YAG pumping, the first Cr:ZnSe laser demonstrated 3.1 mJ of output energy, 52% s...