Laura A. Sanchis

Multiple-way network partitioning with different cost functions

An adaptation to multiple blocks of a two-block network partitioning algorithm by Krishnamurthy was previously presented and analyzed by the author (see ibid., vol.38, p.62-81, 1989). The algorithm assumed one of several possible generalizations of two-way partitioning to multiple-way partitioning. The problem of adapting this algorithm to work with different generalizations more suitable for other types of applications of network partitioning is considered. It is shown that certain portions of the algorithm must be revised in order to maintain a relatively low time complexity for the modified algorithms. Experimental results are given. >

Bounds related to domination in graphs with minimum degree two

A dominating set for a graph G = (V,E) is a subset of vertices V′ ⊆ V such that for all v E V − V′ there exists some u E V′ for which {v, u} E E. The domination number of G is the size of its smallest dominating set(s). We show that for almost all connected graphs with minimum degree at least 2 and q edges, the domination number is bounded by (q + 1)/3. From this we derive exact lower bounds for the number of edges of a connected graph with minimum degree at least 2 and a given domination number. We also generalize the bound to k-restricted domination numbers; these measure how many vertices are necessary to dominate a graph if an arbitrary set of k vertices must be incluced in the dominating set.

Experimental Analysis of Heuristic Algorithms for the Dominating Set Problem

Abstract We say a vertex v in a graph Gcovers a vertex w if v=w or if v and w are adjacent. A subset of vertices of G is a dominating set if it collectively covers all vertices in the graph. The dominating set problem, which is NP-hard, consists of finding a smallest possible dominating set for a graph. The straightforward greedy strategy for finding a small dominating set in a graph consists of successively choosing vertices which cover the largest possible number of previously uncovered vertices. Several variations on this greedy heuristic are described and the results of extensive testing of these variations is presented. A more sophisticated procedure for choosing vertices, which takes into account the number of ways in which an uncovered vertex may be covered, appears to be the most successful of the algorithms which are analyzed. For our experimental testing, we used both random graphs and graphs constructed by test case generators which produce graphs with a given density and a specified size for the smallest dominating set. We found that these generators were able to produce challenging graphs for the algorithms, thus helping to discriminate among them, and allowing a greater variety of graphs to be used in the experiments.

Some Experimental and Theoretical Results on Test Case Generators for the Maximum Clique Problem

We describe and analyze test case generators for the maximum clique problem (or equivalently for the maximum independent set or vertex cover problems). The generators produce graphs with specified number of vertices and edges, and known maximum clique size. The experimental hardness of the test cases is evaluated in relation to several heuristics for the maximum clique problem, based on neural networks, and derived from the work of A. Jagota. Our results show that the hardness of the graphs produced by this method depends in a crucial way on the construction parameters; for a given edge density, challenging graphs can only be constructed using this method for a certain range of maximum clique values; the location of this range depends on the expected maximum clique size for random graphs of that density; the size of the range depends on the density of the graph. We also show that one of the algorithms, based on reinforcement learning techniques, has more success than the others at solving the test cases produced by the generators. In addition, NP-completeness reductions are presented showing that (in spite of what might be suggested by the results just mentioned) the maximum clique problem remains NP-hard even if the domain is restricted to graphs having a constant edge density and a constant ratio of the maximum clique size to the number of vertices, for almost all valid combinations of such values. Moreover, the set of all graphs produced by the generators, and having a constant ratio and edge density, is also NP-hard for almost all valid parameter combinations.

Relating the size of a connected graph to its total and restricted domination numbers

Abstract A dominating set for a graph G =( V , E ) is a subset of vertices D ⊆ V such that for all v ∈ V − D there exists some u ∈ D adjacent to v . The domination number of G is the size of its smallest dominating set. A dominating set D is a total dominating set if every vertex in D has a neighbor in D . We give a tight upper bound on the number of edges that a connected graph with a given total domination number can have, and characterize the extremal graphs attaining the bound. We do the same for the k -restricted domination number, which is the smallest number d , such that for any subset U ⊆ V where | U |= k there exists a dominating set for G of size at most d , and containing all vertices in U .

Adaptive, Restart, Randomized Greedy Heuristics for Maximum Clique

This paper presents some adaptive restart randomized greedy heuristics for MAXIMUM CLIQUE. The algorithms are based on improvements and variations of previously-studied algorithms by the authors. Three kinds of adaptation are studied: adaptation of the initial state (AI) given to the greedy heuristic, adaptation of vertex weights (AW) on the graph, and no adaptation (NA). Two kinds of initialization of the vertex-weights are investigated: unweighted initialization (wi := 1) and degree-based initialization (wi := di where di is the degree of vertex i in the graph). Experiments are conducted on several kinds of graphs (random, structured) with six combinations: {NA, AI, and AW} × {unweighted initialization, degree-based initialization. A seventh state of the art semi-greedy algorithm, DMclique, is evaluated as a benchmark algorithm. We concentrate on the problem of finding large cliques in large, dense graphs in a relatively short amount of time. We find that the different strategies produce different effects, and that different algorithms work best on different kinds of graphs.

Generating hard and diverse test sets for NP-hard graph problems

Abstract In evaluating the performance of approximation algorithms for NP-hard problems, it is often necessary to resort to empirical testing. In order to do such testing it is useful to have test instances of the problem for which the correct answer is known. We present algorithms for efficiently generating test instances for some NP-hard graph problems in such a way that the sets of instances generated can be shown to be both diverse and computationally hard. The techniques used involve combining extremal graph theory results with NP-hardness reductions.

On the complexity of test case generation for NP-hard problems

Approximation algorithms for NP-hard problems are designed to run efficiently, i.e., in polynomial time for all or at least most instances. They may however fail to provide an answer for some instances or they may provide an approximation rather than the correct answer. We deal with the testing of approximation algorithms that run in polynomial time and that provide a (possibly approximate) answer for each problem instance.

On the efficient generation of language instances

Polynomial-time Turing machines that output instances of a given language are considered, where the instances are required to have a certain length specified by the input. Two types of generating machines are investigated. The first, called a constructor, is deterministic and outputs one string in the language having the specified input length, if such a string exists. A generator is nondeterministic and may output different strings in the language using different computations on the same input; it is required, however, that for any string in the language satisfying the input constraint, there be some computation of the generator on this input that produces the string. Although most P and NP languages examined appear to have such polynomial-time constructors and generators, it is shown that the question of whether all NP languages have such machines is related to other open questions in complexity theory and that even under the assumption that P is not equal to NP, the question cannot be resolved using te...

Parallel Algorithms for Counting and Randomly Generating Integer Partitions

This paper presents parallel algorithms for determining the number of partitions of a given integerN, where the partitions may be subject to restrictions, such as being composed of distinct parts, of a given number of parts, and/or of parts belonging to a specified set. We present a series of adaptive algorithms suitable for varying numbers of processors. The fastest of these algorithms computes the number of partitions ofnwith largest part equal tok, for 1 ?k?n?N, in timeO(log2(N)) usingO(N2/logN) processors. Parallel logarithmic time algorithms that generate partitions uniformly at random, using these quantities, are also presented.