The pseudoflow algorithm for solving the maximum flow and minimum cut problems was devised in Hochbaum in [13]. The complexity of the algorithm was shown in [13] to be O(nm log n). In [5] Chandran and Hochbaum demonstrated that the pseudoflow algorithm is very efficient in practice, and that the highest label version of the algorithm tends to perform best. Here we improve the running time of th...

Given a partition of the vertices of a graph into two sets, a flip is a move of a vertex from its own set to the other, under the condition that it has more incident edges to vertices in its own set than in the other. Every sequence of flips eventually produces a bipartite subgraph capturing more than half of the edges in the graph. Each flip gains at least one edge. For an n-vertex loopless mu...

In this report we show, that an arbitrary MinSum problem (i.e. a MinSum problem with an arbitrary finite set of states) can be adequately transformed into a binary one (i.e. into a MinSum problem with only two states). Consequently all known results for binary MinSum problems can be easily extended to the general case. For instance it gives the possibility to solve exactly submodular MinSum pro...

Polynomial algorithms are given for the following two problems: • given a graph with n vertices and m edges, where m ≥ 3n3/2, find a complete balanced bipartite subgraph with parts about lnn ln(n/m) , • given a graph with n vertices, find a decomposition of its edges into complete balanced bipartite graphs having altogether O(n2/ lnn) vertices. Previous proofs of the existence of such objects, ...

We describe the links existing between a recently introduced semideenite relaxation for the max-cut problem and the well known semideenite relaxation for the stable set problem underlying the Lovv asz's theta function. It turns out that the connection between the convex bodies deening the semideenite relaxations mimics the connection existing between the corresponding poly-hedra. We also show h...

The max-cut problem is to partition the vertices of a weighted graph G = (V,E) into two subsets such that the weight sum of the edges crossing the two subsets is maximized. This paper presents a memetic max-cut algorithm (MACUT) that relies on a dedicated multi-parent crossover operator and a perturbation-based tabu search procedure. Experiments on 30 G-set benchmark instances show that MACUT c...

A biclique is a complete bipartite subgraph of a graph. This paper investigates the fractional biclique cover number, bc∗(G), and the fractional biclique partition number, bp∗(G), of a graph G. It is observed that bc∗(G) and bp∗(G) provide lower bounds on the biclique cover and partition numbers respectively, and conditions for equality are given. It is also shown that bc∗(G) is a better lower ...

The biclique problem asks, given a graph G and a parameter k, whether G has a complete bipartite subgraph of k vertices in each part (a biclique of order k). Fixed-parameter tractability of this problem is a longstanding open question in parameterized complexity that received a lot of attention from the community. In this paper we consider a restricted version of this problem by introducing an ...

2. Max-Cut Revisited As in last week’s lecture, we approximate solutions to Max-Cut using Goemans’s and Williamson’s αGW = 0.878-approximation. Specifically, we seek max ∑ (i,j)∈E 1 4 ∥∥vi − vj∥∥2 subject to the constraint that ∀i, ‖vi‖ = 1. We can visualize this by drawing the vectors restricted to a unit circle, as seen in the figure to the left. There is an appealing geometric intuition here...

A string graph is the intersection graph of a collection of continuous arcs in the plane. It is shown that any string graph with m edges can be separated into two parts of roughly equal size by the removal of O(m √ log m) vertices. This result is then used to deduce that every string graph with n vertices and no complete bipartite subgraph Kt,t has at most ctn edges, where ct is a constant depe...