Thang Nguyen Bui

Graph bisection algorithms with good average case behavior

In the paper, we describe a polynomial time algorithm that, for every input graph, either outputs the minimum bisection of the graph or halts without output. More importantly, we show that the algorithm chooses the former course with high probability for many natural classes of graphs. In particular, for every fixedd≧3, all sufficiently largen and allb=o(n1−1/[(d+1)/2]), the algorithm finds the minimum bisection for almost alld-regular labelled simple graphs with 2n nodes and bisection widthb. For example, the algorithm succeeds for almost all 5-regular graphs with 2n nodes and bisection widtho(n2/3). The algorithm differs from other graph bisection heuristics (as well as from many heuristics for other NP-complete problems) in several respects. Most notably:(i)the algorithm provides exactly the minimum bisection for almost all input graphs with the specified form, instead of only an approximation of the minimum bisection,(ii)whenever the algorithm produces a bisection, it is guaranteed to be optimal (i.e., the algorithm also produces a proof that the bisection it outputs is an optimal bisection),(iii)the algorithm works well both theoretically and experimentally,(iv)the algorithm employs global methods such as network flow instead of local operations such as 2-changes, and(v)the algorithm works well for graphs with small bisections (as opposed to graphs with large bisections, for which arbitrary bisections are nearly optimal).

Finding good approximate vertex and edge partitions is NP-hard

Abstract In this paper we show that for n-vertex graphs with maximum degree 3, and for any fixed e> 0, it is NP-hard to find α-edge separators and α-vertex separators of size no more than OPT+n 1 2 − e, where OPT is the size of the optimal solutio n. For general graphs we show that it is NP-hard to find an α-edge separator of size no more than OPT+n2 - e. We also show that an O(ƒ(n))-approximation algorithm for finding α-vertex separators of maximum degree 3 graphs can be used to find an O(ƒ(n5))-approximation algorithm for finding α-edge separators of general graphs. Since it is NP-hard to find optimal α-edge separators for general graphs this means that it is NP-hard to find optimal vertex separators even when restricted to maximum degree 3 graphs.

Improving the Performance of the Kernighan-Lin and Simulated Annealing Graph Bisection Algorithms

In this paper, we compare the performance of two popular graph bisection algorithms. We also present an empirical study of a new heuristic, first proposed in [B87], that dramatically improves the performance of these bisection algorithms on graphs with small (≤ 4) average degree.

A new genetic approach for the traveling salesman problem

A new genetic algorithm (GA) for the traveling salesman problem (TSP) is given. Two novel features of this algorithm are: (i) a new locus-based encoding/crossover pair, and (ii) a static preprocessing step which changes the encoding order of the vertices. It is believed that this algorithm is also applicable to other ordering problems, not just TSP. Experimental results on the standard benchmarks for TSP are favorable.<<ETX>>

An ant-based algorithm for coloring graphs

This paper presents an ant-based algorithm for the graph coloring problem. An important difference that distinguishes this algorithm from previous ant algorithms is the manner in which ants are used in the algorithm. Unlike previous ant algorithms where each ant colors the entire graph, each ant in this algorithm colors just a portion of the graph using only local information. These individual coloring actions by the ants form a coloring of the graph. Even with the lack of pheromone laying capacity by the ants, the algorithm performed well on a set of 119 benchmark graphs. Furthermore, the algorithm produced very consistent results, having very small standard deviations over 50 runs of each graph tested.

An ant-based algorithm for finding degree-constrained minimum spanning tree

A spanning tree of a graph such that each vertex in the tree has degree at most d is called a degree-constrained spanning tree. The problem of finding the degree-constrained spanning tree of minimum cost in an edge weighted graphis well known to be NP-hard. In this paper we give an Ant-Based algorithm for finding low cost degree-constrained spanning trees. Ants are used to identify a set of candidate edges from which a degree-constrained spanning tree can be constructed. Extensive experimental results show that the algorithm performs very well against other algorithms on a set of 572 problem instances.

A Fast and Stable Hybrid Genetic Algorithm for the Ratio-Cut Partitioning Problem on Hypergraphs

A genetic algorithm (GA) for partitioning a hypergraph into two disjoint graphs of least ratio-cut is presented. Two notable features of this algorithm are: (i) a fast local optimizer, and (ii) a preprocessing step. Some supporting combinatorial arguments for the preprocessing heuristic are also provided. Experimental results on industrial benchmarks circuits are favorable when compared with recently published algorithms [25], [26], [19].

Partitioning planar graphs

A common problem in graph theory is that of dividing the vertices of a graph into two sets of prescribed size while cutting a minimum number of edges. In this paper this problem is considered as it is restricted to the class of planar graphs.Let G be a planar graph on n vertices and

GRCA: a hybrid genetic algorithm for circuit ratio-cut partitioning

s \in [0,n]

Graph Bisection Algorithins With Good Average Case Behavior

be given. An s-partition of G is a partition of the vertex set of G into sets of size s and