Isomorphic Problems Research Articles

Lattice Theoretic Ordering Properties for NP-Complete Optimization Problems

The ordering among classes of isomorphic NP-complete optimization problems is studied from a lattice theoretic point of view. It is shown that the set of such classes has the structure of an upper semilattice; while a top element can found be it is proved that no bottom element can exist.

Structure preserving reductions among convex optimization problems

In this paper we introduce the concept of convex optimization problem. Convex optimization problems are studied by giving a formalization of the concept of combinatorial structure, in terms of spectra of approximate solutions, and of reduction which preserves such structure. On this basis a classification of convex NP-optimization problems is introduced and is applied to study the combinatorial structure of several optimization problems associated to well-known NP-complete sets. Conditions on the approximability of such optimization problems are also given and it is shown that structurally isomorphic problems have similar approximability properties.

A state-space description of transfer effects in isomorphic problem situations

Previous research has shown significant transfer for subjects solving two problems of isomorphic structure. The state-space representation of these problems is used to characterize their structure, and in particular, the state-space decomposition modulo isomorphic sub-problems is used to define “stages” within the problem's solution. Changes within these “stages” provide a framework for a more complete description and understanding of the transfer phenomenon.

Transfer effects in isomorphic problem situations

The results of a recent experiment to assess the relationship between human problem solving behaviour and the structural properties of certain problems are presented. The results suggest that human performance in a problem-solving task is significantly influenced by immediately prior experience on a different but structurally identical problem. Specifically, transfer effects are demonstrated across two problems of isomorphic structure.

The Twenty Questions Game as a Form of Problem Solving

SIEGLER, ROBERT S. The Twenty Questions Game as a Form of Problem Solving. CHILD DEVELOPMENT, 1977, 48, 395-403. 3 experiments were performed in an attempt to integrate ideas and methods developed in the study of problem solving with those developed specifically to study children's inquiry strategies. The focus was on how experience with isomorphic problems affected subsequent inquiries. In Experiment 1, it was found that the order of presentation of 2 isomorphic problems greatly influenced the types of questions asked by 13and 14-year-olds. In Experiment 2, evidence was produced that this effect was due to the adolescents' conducting only limited searches of their existing knowledge in generating questions. The results of Experiment 3 ruled out 2 possible alternative explanations of the Experiment 2 findings-that they might be due to motivational factors or to superior knowledge about task demands-and also provided an independent replication of the results of Experiment 1. Together, the 3 experiments demonstrated the applicability of such problem-solving concepts as isomorphism, analogy, and depth of search to the study of children's inquiries.

The number of locally restricted directed graphs

1. Preliminaries. We shall be concerned with finite graphs of t directed lines on n points, or nodes. The lines are joinsfrom one point to another. To each point, Pi, is associated a non-negative number pair (ri, si), where ri is the number of lines from Pi, and si is the number of lines to Pi. These number pairs are considered to be fixed local restrictions. Obviously, ri= Ej sj, =t. The vectors r and s, with elements ri and si, respectively, are two n-part non-negative integer partitions of t, with vanishing parts permitted. In general, it is possible to construct many n-node graphs satisfying a specified set of local restrictions. The main result of this paper gives the number of distinct graphs of this kind. The enumeration is helpful in studying communication networks in which the ith element has ri receivers and si senders; here the pertinent question is whether such graphs exist. In the isomorphic problem in group organization theory, where the ith individual in the group gives information or orders or choices to ri others and receives from si others, the exact number of graphs is needed in order to construct the probability distributions for random variables defined over the graphs. All of these activities, like the graphs, are irreflexive and take values on a twoelement Boolean algebra. Previous work on numbers of graphs has been devoted exclusively to counts of graphs topologically distinct under permutations of the points which preserve joins. Thus, G. Polya [9] gave explicitly the numbers of trees and rooted trees on n points, R. Otter [8] gave simpler purely combinatorial methods for counting trees and rooted trees, and F. Harary and G. E. Uhlenbeck [6] gave the numbers of free pure and mixed Husimi trees. Also, F. Harary indicates in an abstract [5] that Polya's method of generating functions may be extended to give the numbers of ordinary and of directed graphs on n points and t lines. R. L. Davis [3] defines and gives a method for counting the numbers of various subclasses of directed graphs on n points. The problem we face is essentially different from all of these and requires quite different methods of enumeration. We shall make use of certain bipartitional functions due to Cayley