To be or not to be Yutsis: Algorithms for the decision problem

https://doi.org/10.1016/j.cpc.2005.07.008Get rights and content

Abstract

Generalized recoupling coefficients or 3nj-coefficients can be expressed as multiple sums over products of Racah or 6j-coefficients [L.C. Biedenharn, J.D. Louck, Coupling of n angular momenta: recoupling theory, in: The Racah–Wigner Algebra in Quantum Theory, Encyclopedia of Mathematics and its Applications, vol. 9, Addison-Wesley, 1981, pp. 435–481]. The problem of finding an optimal summation formula (i.e. with a minimal number of Racah coefficients) for a given 3nj-coefficient is equivalent to finding an optimal reduction of a so-called Yutsis graph [A.P. Yutsis, I.B. Levinson, V.V. Vanagas, Mathematical Apparatus of the Theory of Angular Momentum, Israel Program for Scientific Translation, Jerusalem, 1962].

In terms of graph theory Yutsis graphs are connected simple cubic graphs which can be partitioned into two vertex induced trees. The two parts are necessarily of the same size. In this area Yutsis graphs are also studied under the name of cubic dual Hamiltonian graphs [F. Jaeger, On vertex-induced forests in cubic graphs, in: Proc. 5th Southeastern Conference, Congr. Numer. (1974) 501–512]. We present algorithms for determining whether a cubic graph is a Yutsis graph. This is interesting for generating large test cases for programs (as in [P.M. Lima, Comput. Phys. Comm. 66 (1991) 89; S. Fritzsche, T. Inghoff, T. Bastug, M. Tomaselli, Comput. Phys. Comm. 139 (2001) 314; D. Van Dyck, V. Fack, GYutsis: heuristic based calculation of general recoupling coefficients, Comput. Phys. Comm. 154 (2003) 219–232]) that determine a summation formula for a 3nj-coefficient.

Moreover, we give the numbers of Yutsis and non-Yutsis cubic graphs with up to 30 vertices and cubic polyhedra with up to 40 vertices. All these numbers have been computed by two independent programs in order to reduce the probability of error. Since the decision problem whether a given cubic graph is Yutsis or not is NP-complete, we could not hope for a polynomial time worst case performance of our programs. Nevertheless the programs described in this article are very fast on average.

Introduction

In various fields of theoretical physics the quantum mechanical description of many-particle processes often requires an explicit transformation of the angular momenta of the subsystems among different coupling schemes. Such transformations are described by general recoupling coefficients and arise mostly in atomic and nuclear structure and scattering calculations [1]. Several algorithms have been described to generate a summation formula expressing the recoupling coefficient as a multiple sum over products of Wigner 6j symbols multiplied by phase factors and square root factors [4], [5], [6], [7], [8], [9], [10]. It is desirable to find an optimal summation formula, i.e. with a minimum number of summation variables and Wigner 6j symbols.

The best algorithms at present are based on techniques developed by Yutsis, Levinson and Vanagas [2] and manipulate a graphical representation of the recoupling coefficient called a Yutsis graph. Reduction rules are defined for these graphs, which allow a stepwise transformation of the graph by reduction and removal of cycles. Each reduction step contributes part of the final summation formula. Section 2 summarizes some notions from the quantum theory of angular momenta and shows how a Yutsis graph is constructed for a given recoupling coefficient. For the general theory of Yutsis graphs we refer to [1], [2].

For our purposes a Yutsis graph can be defined as follows. A binary coupling tree on n+1 leaves is an unordered binary tree in which each leaf has a distinct label. By taking two binary coupling trees on n+1 leaves in which the unique leaf vertices with the same label are identified and then removed and where the root nodes are connected by an additional edge, we obtain a cubic multigraph with 2n nodes and 3n edges. In this multigraph the internal vertices of the coupling trees define two vertex induced trees and the former leaf and root edges form an edge-cut on n+2 edges. Fig. 1 shows an example. A multigraph that can be constructed this way is called a Yutsis graph or simply Yutsis. Two vertex induced trees coming from such a construction are called the defining trees and the edge-cut is called the defining cut. Note that a given Yutsis graph can in general be obtained from more than one pair of trees, so the defining trees and the defining edge-cut are in general not uniquely determined. Since the two endpoints of multiple edges in a Yutsis graph must obviously belong to different trees, they are trivial from the viewpoint of the decision problem and we will restrict ourselves to simple graphs when discussing the decision problem.

In mathematics, Yutsis graphs are also known as dual Hamiltonian cubic graphs [3].

Up to now no better method is known to determine whether a cubic graph is Yutsis than to search for a defining tree (or cut). For the quantum theory of angular momenta, we are interested in obtaining large test cases by generating large cubic graphs at random and filter out those graphs which are not Yutsis. In addition we would like to identify the non-Yutsis graphs and study their structure.

All graphs in this article are assumed to be connected.

Section snippets

Graphical representation of recoupling coefficients

In [1, Topic 12], recoupling theory is considered from the point of view of binary coupling schemes. A binary coupling scheme is the rooted binary tree representing the order of coupling of a state vector in the tensor product of n+1 angular momentum multiplets, labelled respectively by the angular momenta j1,j2,,jn+1. The leaves of the binary tree are labelled by these angular momenta j1,j2,,jn+1, and the remaining vertices of the tree can be labelled by the intermediate angular momenta. For

The decision problem is NP-complete

In this section we will prove that the problem of deciding whether a given graph is Yutsis or not is a very hard problem in the worst case. To be exact: The problem is NP-complete and it is even NP-complete when restricted to the subclass of cubic polyhedra, i.e. 3-connected planar cubic graphs.

Theorem 1

The decision problem whether a given cubic polyhedron is Yutsis or not is NP-complete.

Proof

Since it is easy to see that this problem is in NP (take, e.g., n random vertices and check whether they and their

Preliminaries

In this section we will give and prove some lemmas and remarks that we will use in the algorithm. G=(V,E) always denotes a simple cubic graph with 2n nodes and 3n edges.

Lemma 2

Let G be a cubic graph with 2n nodes, T an induced subgraph on n vertices that is a tree and S the complement of T.

Then S,T are defining trees for a Yutsis decomposition if and only if S is connected.

Proof

If T is an induced tree on n vertices in a cubic graph, then there are 3n2(n1)=n+2 edges between T and its complement S.

Fast heuristics—a greedy approach

In spite of the fact that this decision problem is NP-complete, in most cases a set of defining trees can be found very quickly by a heuristic we will now describe. Both of our filters work by first applying a heuristic a couple of times. Only in cases where the heuristics do not find a tree decomposition do we apply exhaustive search methods.

Given a connected cubic graph G, we start with a random vertex forming a one-vertex tree T1 and a list L1 of all its neighbours. We increase the tree

An exhaustive search method

Our testing method works in 4 phases: first the greedy heuristic is applied to the graphs, then the remaining graphs are tested for having bridges (such graphs are trivially non-Yutsis), then the heuristic is applied again to the graphs still remaining, and finally an exhaustive search is applied to the graphs.

The number of applications of the heuristic is determined by the input graphs. We found that (1/5)|V| trials for the first series and (1/10)|V| trials for the second series are

Results

The following numbers of Yutsis and non-Yutsis graphs were computed independently by the program described here1 and another somewhat slower approach that we have not described. The number of graphs with bridges and graphs with too many triangles were only computed by the first program. The programs used to generate the graphs were minibaum for all cubic graphs (see [15]), plantri for all cubic polyhedra (see [16]) and genrang

References (17)

There are more references available in the full text version of this article.

Cited by (0)

View full text