Reductions in Computational Complexity Using Clifford Algebras

A number of combinatorial problems are treated using properties of abelian null-square-generated and idempotent-generated subalgebras of Clifford algebras. For example, the problem of deciding whether or not a graph contains a Hamiltonian cycle is known to be NP-complete. By considering entries of Λ^k, where Λ is an appropriate nilpotent adjacency matrix, the k-cycles in any finite graph are recovered. Within the algebra context (i.e., considering the number of multiplications performed within the algebra), these problems are reduced to matrix multiplication, which is in complexity class P. The Hamiltonian cycle problem is one of many problems moved from classes NP-complete and #P-complete to class P in this context. Other problems considered include the set covering problem, counting the edge-disjoint cycle decompositions of a finite graph, computing the permanent of an arbitrary matrix, computing the girth and circumference of a graph, and finding the longest path in a graph.     

On orientation-preserving automorphisms of Hamiltonian cycles in the <Emphasis Type="Italic">N</Emphasis>-dimensional Boolean cube

We obtain an upper bound for the order of the group of orientation-preserving automorphisms of a Hamiltonian cycle in the Boolean n-cube. We prove that the existence of a Hamiltonian cycle with the order of the group of orientation-preserving automorphisms attaining this upper bound is equivalent to the existence of a Hamiltonian cycle with an additional condition on the graph of orbits of a fixed automorphism of the n-cube.     

An exact algorithm for subgraph homeomorphism

The subgraph homeomorphism problem is to decide if there is an injective mapping of the vertices of a pattern graph into vertices of a host graph so that the edges of the pattern graph can be mapped into (internally) vertex-disjoint paths in the host graph. The restriction of subgraph homeomorphism where an injective mapping of the vertices of the pattern graph into vertices of the host graph is already given in the input instance is termed fixed-vertex subgraph homeomorphism.We show that fixed-vertex subgraph homeomorphism for a pattern graph on p vertices and a host graph on n vertices can be solved in time 2^n−pn^O(1) or in time 3^n−pn^O(1) and polynomial space. In effect, we obtain new non-trivial upper bounds on the time complexity of the problem of finding k vertex-disjoint paths and general subgraph homeomorphism.     

Computing the maximum clique in the visibility graph of a simple polygon

In this paper, we present an algorithm for computing the maximum clique in the visibility graph G of a simple polygon P in O(n²e) time, where n and e are number of vertices and edges of G respectively. We also present an O(ne) time algorithm for computing the maximum hidden vertex set in the visibility graph G of a convex fan P. We assume in both algorithms that the Hamiltonian cycle in G that corresponds to the boundary of P is given as an input along with G.     

Finding minimum dominating cycles in permutation graphs

A dominating cycle for a graph G = (V, E) is a subset C of V which has the following properties: (i) the subgraph of G induced by C has a Hamiltonian cycle, and (ii) every vertex of V is adjacent to some vertex of C. In this paper, we develop an O(n²) algorithm for finding a minimum cardinality dominating cycle in a permutation graph. We also show that a minimum cardinality dominating cycle in a permutation graph always has an even number of vertices unless it is isomorphic to C₃.     

Finding hidden hamiltonian cycles

Consider a random graph G composed of a Hamiltonian cycle on n labeled vertices and dn random edges that “high” the cycle. Is it possible to unravel the structures, that is, to efficiently find a Himiltonian cycle in G? We describe an O(n³ log n)-step algorithm A for this purpose, and prove that it succeeds almost surely. Part one of A properly covers the “trouble spots” of G by a collection of disjoint paths. (This is the hard part to analyze). Part two of A extends this cover to a full cycle by the rotation-extension technique which is already classical for such problems. © 1994 John Wiley & Sons, Inc.     

Cycle Lengths in Hamiltonian Graphs with a Pair of Vertices Having Large Degree Sum

A graph of order n is said to be pancyclic if it contains cycles of all lengths from three to n. Let G be a Hamiltonian graph and let x and y be vertices of G that are consecutive on some Hamiltonian cycle in G. Hakimi and Schmeichel showed (J Combin Theory Ser B 45:99–107, 1988) that if d(x) + d(y) ≥ n then either G is pancyclic, G has cycles of all lengths except n − 1 or G is isomorphic to a complete bipartite graph. In this paper, we study the existence of cycles of various lengths in a Hamiltonian graph G given the existence of a pair of vertices that have a high degree sum but are not adjacent on any Hamiltonian cycle in G.     

A Degree Constraint for Uniquely Hamiltonian Graphs

A graph, G, is called uniquely Hamiltonian if it contains exactly one Hamilton cycle. We show that if G=(V, E) is uniquely Hamiltonian then Where #(G)=1 if G has even number of vertices and 2 if G has odd number of vertices. It follows that every n-vertex uniquely Hamiltonian graph contains a vertex whose degree is at most c log₂n+2 where c=(log₂3−1)⁻¹≈1.71 thereby improving a bound given by Bondy and Jackson [3].     

Finding a minimum independent dominating set in a permutation graph

We give a linear time reduction of the problem of finding a minimum independent dominating set in a permutation graph, into that of finding a shortest maximal increasing subsequence. We then give an O(n log²n)-time algorithm for solving the second (and hence the first) problem. This improves on the O(n³)-time algorithm given in [4] for solving the problem of finding a minimum independent dominating set in a permutation graph.     

Induced Circuits in Planar Graphs

In McDiarmid, B. Reed, A. Schrijver, and B. Shepherd (Univ. of Waterloo Tech. Rep., 1990) a polynomial-time algorithm is given for the problem of finding a minimum cost circuit without chords (induced circuit) traversing two given vertices of a planar graph. The algorithm is based on the ellipsoid method. Here we give an O(n²) combinatorial algorithm to determine whether two nodes in a planar graph lie on an induced circuit. We also give a min-max relation for the problem of finding a maximum number of paths connecting two given vertices in a planar graph so that each pair of these paths forms an induced circuit.     

On the “log rank”-conjecture in communication complexity

We show the existence of a non-constant gap between the communication complexity of a function and the logarithm of the rank of its input matrix. We consider the following problem: each of two players gets a perfect matching between twon-element sets of vertices. Their goal is to decide whether or not the union of the two matcliings forms a Hamiltonian cycle. We prove:

1. The rank of the input matrix over the reals for this problem is 2 ^O(n) .
2. The non-deterministic communication complexity of the problem is Ω(nloglogn).

Our result also supplies a superpolynomial gap between the chromatic number of a graph and the rank of its adjacency matrix. Another conclusion from the second result is an Ω(nloglogn) lower bound for the graph connectivity problem in the non-deterministic case. We make use of the theory of group representations for the first result. The second result is proved by an information theoretic argument.     

Graph powers and k-ordered Hamiltonicity

It is known that if G is a connected simple graph, then G³ is Hamiltonian (in fact, Hamilton-connected). A simple graph is k-ordered Hamiltonian if for any sequence v₁, v₂,…,v_k of k vertices there is a Hamiltonian cycle containing these vertices in the given order. In this paper, we prove that if k?4, then G^⌊3k/2⌋-2 is k-ordered Hamiltonian for every connected graph G on at least k vertices. By considering the case of the path graph P_n, we show that this result is sharp. We also give a lower bound on the power of the cycle C_n that guarantees k-ordered Hamiltonicity.     

Solving the path cover problem on circular-arc graphs by using an approximation algorithm

A path cover of a graph G=(V,E) is a family of vertex-disjoint paths that covers all vertices in V. Given a graph G, the path cover problem is to find a path cover of minimum cardinality. This paper presents a simple O(n)-time approximation algorithm for the path cover problem on circular-arc graphs given a set of n arcs with endpoints sorted. The cardinality of the path cover found by the approximation algorithm is at most one more than the optimal one. By using the result, we reduce the path cover problem on circular-arc graphs to the Hamiltonian cycle and Hamiltonian path problems on the same class of graphs in O(n) time. Hence the complexity of the path cover problem on circular-arc graphs is the same as those of the Hamiltonian cycle and Hamiltonian path problems on circular-arc graphs.     

Extremal maximal uniquely hamiltonian graphs

Let G be a graph of order n with exactly one Hamiltonian cycle and suppose that G is maximal with respect to this property. We determine the minimum number of edges G can have.     

Graphs with exactly one hamiltonian circuit

Let h(n) be the largest integer such that there exists a graph with n vertices having exactly one Hamiltonian circuit and exactly h(n) edges. We prove that h(n) = [n²/4]+1 (n ≧ 4) and discuss some related problems.     

Generalized fibonacci cubes are mostly hamiltonian

The Hamiltonian problem is to determine whether a graph contains a spanning (Hamiltonian) path or cycle. Here we study the Hamiltonian problem for the generalized Fibonacci cubes, which are a new family of graphs that have applications in interconnection topologies [J. Liuand W.-J. Hsu, ?Distributed Algorithms for Shortest-Path, Deadlock-Free Routing and Broadcasting in a Class of Interconnection Topologies,”? International Parallel Processing Symposium (1992)]. We show that each member of this family contains a Hamiltonian path. Furthermore, we also characterize the members of this family that contain a Hamiltonian cycle.     

Explicit Geometric Integration of Polynomial Vector Fields

We present a unified framework in which to study splitting methods for polynomial vector fields in R ⁿ . The vector field is to be represented as a sum of shears, each of which can be integrated exactly, and each of which is a function of k<n variables. Each shear must also inherit the structure of the original vector field: we consider Hamiltonian, Poisson, and volume-preserving cases. Each case then leads to the problem of finding an optimal distribution of points on an appropriate homogeneous space, generally the Grassmannians of k-planes or (in the Hamiltonian case) isotropic k-planes in R ⁿ . These optimization problems have the same structure as those of constructing optimal experimental designs in statistics.     

Analysis of parallel algorithms for finding a maximal independent set in a random hypergraph

It is well known [9] that finding a maximal independent set in a graph is in class NC and [10] that finding a maximal independent set in a hypergraph with fixed dimension is in RNC. It is not known whether this latter problem remains in NC when the dimension is part of the input. We will study the problem when the problem instances are randomly chosen. It was shown in [6] that the expected running time of a simple parallel algorithm for finding the lexicographically first maximal independent set (Ifmis) in a random simple graph is logarithmic in the input size. In this paper, we will prove a generalization of this result. We show that if a random k-uniform hypergraph has vertex set {1, 2, …, n} and its edges are chosen independently with probability p from the set of (ⁿ_k) possible edges, then our algorithm finds the Ifmis in O( ) expected time. The hidden constant is independent of k, p. © 1996 John Wiley & Sons, Inc. Random Struct. Alg., 9 , 359–377 (1996)     

Complete partitions of graphs

A complete partition of a graph G is a partition of its vertex set in which any two distinct classes are connected by an edge. Let cp(G) denote the maximum number of classes in a complete partition of G. This measure was defined in 1969 by Gupta [19], and is known to be NP-hard to compute for several classes of graphs. We obtain essentially tight lower and upper bounds on the approximability of this problem. We show that there is a randomized polynomial-time algorithm that given a graph G with n vertices, produces a complete partition of size Ω(cp(G)/√lgn). This algorithm can be derandomized. We show that the upper bound is essentially tight: there is a constant C > 1, such that if there is a randomized polynomial-time algorithm that for all large n, when given a graph G with n vertices produces a complete partition into at least C·cp(G)/√lgn classes, then NP ⊆ RTime(n ^{O(lg lg n)}). The problem of finding a complete partition of a graph is thus the first natural problem whose approximation threshold has been determined to be of the form Θ((lgn) ^c ) for some constant c strictly between 0 and 1. The work reported here is a merger of the results reported in [30] and [21].     

On κ—ordered Graphs Involved Degree Sum

A graph G is κ-ordered Hamiltonian 2≤κ≤n,if for every ordered sequence S of κ distinct vertices of G,there exists a Hamiltonian cycle that encounters S in the given order,In this article,we prove that if G is a graph on n vertices with degree sum of nonadjacent vertices at least n 3κ-9/2,then G is κ-ordered Hamiltonian for κ=3,4,…,[n/19].We also show that the degree sum bound can be reduced to n 2[κ/2]-2 if κ(G）≥3κ-1/2 or δ（G）≥5κ-4.Several known results are generalized.