Complete Undirected Graph Research Articles

Approximation theory in combinatorial optimization. Application to the generalized minimum spanning tree problem

We present an overview of the approximation theory in
 combinatorial optimization. As an application we consider the 
 Generalized Minimum Spanning Tree (GMST) problem which is defined on an undirected complete graph with the nodes partitioned into
 clusters and non-negative costs are associated to the edges. This
 problem is NP-hard and it is known that a
 polynomial approximation algorithm cannot exist. We present an
 in-approximability result for the GMST problem and under special
 assumptions: cost function satisfying the triangle inequality and
 with cluster sizes bounded by \(\rho\), we give an approximation
 algorithm with ratio \(2 \rho\).

Video summarization and scene detection by graph modeling

We propose a unified approach for video summarization based on the analysis of video structures and video highlights. Two major components in our approach are scene modeling and highlight detection. Scene modeling is achieved by normalized cut algorithm and temporal graph analysis, while highlight detection is accomplished by motion attention modeling. In our proposed approach, a video is represented as a complete undirected graph and the normalized cut algorithm is carried out to globally and optimally partition the graph into video clusters. The resulting clusters form a directed temporal graph and a shortest path algorithm is proposed to efficiently detect video scenes. The attention values are then computed and attached to the scenes, clusters, shots, and subshots in a temporal graph. As a result, the temporal graph can inherently describe the evolution and perceptual importance of a video. In our application, video summaries that emphasize both content balance and perceptual quality can be generated directly from a temporal graph that embeds both the structure and attention information.

Approximating the Degree-Bounded Minimum Diameter Spanning Tree Problem

We consider the problem of finding a minimum diameter spanning tree with maximum node degree B in a complete undirected edge-weighted graph. We prove that the problem is NP-complete, and provide an \(O(\sqrt{\log_Bn})\)-approximation algorithm for the problem. Our algorithm is purely combinatorial, and relies on a combination of filtering and divide and conquer.

On the convex hull of 3-cycles of the complete graph

Let Kn be the complete undirected graph with n vertices. A 3-cycle is a cycle consisting of 3 edges. The 3-cycle polytope is defined as the convex hull of the incidence vectors of all 3-cycles in Kn. In this paper, we present a polyhedral analysis of the 3-cycle polytope. In particular, we give several classes of facet defining inequalities of this polytope and we prove that the separation problem associated to one of these classes of inequalities is NP-complete. Finally, it is proved that the 3-cycle polytope is a 2-neighborly polytope.

On the linear description of the 3-cycle polytope

Let K n be the complete undirected graph with n vertices. A 3-cycle is a simple cycle consisting of exactly 3-edges in K n . The 3-cycle polytope, PC 3 n is defined as the convex hull of the incidence vectors of all 3-cycles in K n . This polytope is proved to be a facet of the dominant of all cycles in K n [Math. Oper. Res. 22 (1997) 110]. In this paper, we design some sequential lifting procedures for facet-defining inequalities of PC 3 n . Using these lifting procedures, we show that the number of distinct coefficients in facet-defining inequalities of PC 3 n increases strictly when n grows and the maximum difference between the greatest coefficient and the smallest coefficient in some facet-defining inequalities is exponential in n. We also discuss relationships between PC 3 n and other “cycle” polytopes. Finally in Appendix A we give a complete description of PC 3 8 and show that all of its facets except one can be lifted from the facets of PC 3 5, of PC 3 6 and PC 3 7.

The Diameters of Some Transition Graphs Constructed from Hamilton Cycles

A complete undirected graph of order n has \(\) Hamilton cycles. We consider the diameter of a transition graph whose vertices correspond to those Hamilton cycles and any of two vertices are adjacent if and only if the corresponding Hamilton cycles can be transformed each other by exchanging two edges. Moreover, we consider several transition graphs related to it.

On the Linear Description of the k‐cycle Polytope

We study a linear description of PCkn the convex hull of incidence vectors of all the cycles consisting of exactly k edges (k≥4) in Kn, the complete undirected graph with n vertices. First, we describe some properties of PCkn. Second, we discuss relations between PCkn and PCk′n′ with k′>k and n′>n. Then we give three lifting algorithms that transform a facet of PCkn into facets of PCkn′ with n′>n. Finally, we provide an integer formulation and a partial linear description of PCkn.

Steiner pentagon covering designs

Let K n denote the complete undirected graph on n vertices. A Steiner pentagon covering design (SPCD) of order n is a pair (K n, B) , where B is a collection of c( n)=⌈ n/5⌈ n−1/2⌉⌉ pentagons from K n such that any two vertices are joined by a path of length 1 in at least one pentagon of B , and also by a path of length 2 in at least one pentagon of B . The existence of SPCDs is investigated. The main approach is to use certain types of holey Steiner pentagon systems. For n even, the existence of SPCDs is established with a few possible exceptions. For n odd, new SPCDs are found which improve an earlier known result. In addition, new results are also found for Steiner pentagon packing designs.

Approximation Algorithms for Minimum K -Cut

Let G=(V,E) be a complete undirected graph, with node set V={v 1 , . . ., v n } and edge set E . The edges (v i ,v j ) ∈ E have nonnegative weights that satisfy the triangle inequality. Given a set of integers K = { k i } i=1 p $(\sum_{i=1}^p k_i \leq |V|$) , the minimum K-cut problem is to compute disjoint subsets with sizes { k i } i=1 p , minimizing the total weight of edges whose two ends are in different subsets. We demonstrate that for any fixed p it is possible to obtain in polynomial time an approximation of at most three times the optimal value. We also prove bounds on the ratio between the weights of maximum and minimum cuts.

The edge-weighted clique problem: Valid inequalities, facets and polyhedral computations

Let K n =( V, E) be the complete undirected graph with weights c e associated to the edges in E. We consider the problem of finding the subclique C=( U, F) of K n such that the sum of the weights of the edges in F is maximized and | U|⩽ b, for some b∈[1,…, n]. This problem is called the Maximum Edge-Weighted Clique Problem (MEWCP) and is NP-hard. In this paper we investigate the facial structure of the polytope associated to the MEWCP introducing new classes of facet defining inequalities. Computational experiments with a branch-and-cut algorithm are reported confirming the strength of these inequalities. All instances with up to 48 nodes could be solved without entering into the branching phase. Moreover, we show that some of these new inequalities also define facets of the Boolean Quadric Polytope and generalize many of the previously known inequalities for this well-studied polytope.

Restricted 2-factor polytopes

The optimal k-restricted 2-factor problem consists of finding, in a complete undirected graph K n , a minimum cost 2-factor (subgraph having degree 2 at every node) with all components having more than k nodes. The problem is a relaxation of the well-known symmetric travelling salesman problem, and is equivalent to it when ≤k≤n−1. We study the k-restricted 2-factor polytope. We present a large class of valid inequalities, called bipartition inequalities, and describe some of their properties; some of these results are new even for the travelling salesman polytope. For the case k=3, the triangle-free 2-factor polytope, we derive a necessary and sufficient condition for such inequalities to be facet inducing.

Approximation Algorithms with Bounded Performance Guarantees for the Clustered Traveling Salesman Problem

Let G=(V,E) be a complete undirected graph with vertex set V , edge set E , and edge weights l(e) satisfying triangle inequality. The vertex set V is partitioned into clusters V 1 , . . ., V k . The clustered traveling salesman problem is to compute a shortest Hamiltonian cycle (tour) that visits all the vertices, and in which the vertices of each cluster are visited consecutively. Since this problem is a generalization of the traveling salesman problem, it is NP-hard. In this paper we consider several variants of this basic problem and provide polynomial time approximation algorithms for them.

Small Min-Cut Polyhedra

We study small min-cut polyhedra for directed and undirected complete graphs. We give the ideal descriptions of the polyhedra for undirected graphs with up to seven nodes and for directed graphs with up to five nodes, and generalize the inequalities for complete graphs with arbitrary number of nodes. The facial structure of the polyhedra turns out to be unexpectedly rich.

The monotonic diameter of traveling salesman polytopes

The monotonic diameter of the polytopes arising in the asymmetric and the symmetric traveling salesman problem (TSP) are obtained. For the asymmetric TSP polytope associated with the complete directed graph on n nodes, the monotonic diameter is shown to be exactly ⌊ n/3⌋, for all n⩾3. For the symmetric TSP polytope associated with the complete undirected graph on n nodes, the monotonic diameter is shown to be exactly ⌊ n/2⌋−1, for all n⩾3, except for n=5, in which case it is ⌊ n/2⌋.

A spectral method for concordant polyhedral faces

For natural families of polytopes determined by substructures (e.g., tours or matchings) of a complete structure (e.g., undirected graph, directed graph, or uniform hypergraph), we show how D.G. Higman's theory of coherent configurations can be used to establish the spectra (and hence the dimension) of classes of faces induced by other substructures (e.g., cliques) of the structure. Furthermore, we show how the algebra can be used to derive a maximal irredundant set of equations satisfied by every point of such a face. We work out complete details of the proof machinery when the structure is the complete undirected graph and the faces are induced by cliques. As an example we consider the subtour elimination faces of the Hamiltonian tour polytope. Our approach extends the methodology introduced by J. Lee for determining the dimension of some combinatorially described polytopes.

Partial factoring: an efficient algorithm for approximating two-terminal reliability on complete graphs

An efficient network reliability approximation algorithm for two-terminal undirected complete graphs is presented. The method recursively applies a factoring theorem of network reliability in conjunction with series-parallel reliability-preserving reductions. After each pivot, one of the resulting subproblems is further factored, while the other is approximated. The final approximate solution is given in terms of an upper and a lower bound. In the worse case, the algorithm requires O( mod V mod /sup 2/ log mod V mod ) time and O( mod V/sup 2/ mod ) space. For 128-node complete graphs, the method requires less than 16 s on an IBM AT personal computer. The algorithm's mean error bound is both theoretically and empirically analyzed. The mean difference between the upper and lower bounds is guaranteed to be less than 10/sup -14/ for component reliabilities uniformly distributed between 0.99 and 1.00. Empirically, this difference is always less than 10/sup -17/. >

On multiplicative graphs and the product conjecture

We study the following problem: which graphsG have the property that the class of all graphs not admitting a homomorphism intoG is closed under taking the product (conjunction)? Whether all undirected complete graphs have the property is a longstanding open problem due to S. Hedetniemi. We prove that all odd undirected cycles and all prime-power directed cycles have the property. The former result provides the first non-trivial infinite family of undirected graphs known to have the property, and the latter result verifies a conjecture of Nesetřil and Pultr These results allow us (in conjunction with earlier results of Nesetřil and Pultr [17], cf also [7]) to completely characterize all (finite and infinite, directed and undirected) paths and cycles having the property. We also derive the property for a wide class of 3-chromatic graphs studied by Gerards, [5].

On Shortest Paths in Graphs with Random Weights

We consider the shortest paths between all pairs of nodes in a directed or undirected complete graph with edge lengths which are uniformly and independently distributed in [0, 1]. We show that die longest of these paths is bounded by c log n/n almost surely, where c is a constant and n is the number of nodes. Our bound is the best possible up to a constant. We apply this result to some well-known problems and obtain several algorithmic improvements over existing results. Our results hold with obvious modifications to random (as opposed to complete) graphs and to any distribution of weights whose density is positive and bounded from below at a neighborhood of zero. As a corollary of our proof we get a new result concerning the diameter of random graphs.

Steiner pentagon systems

A Steiner pentagon system is a pair (Kn, P) where Kn isthe complete undirected graph on n vertices. P is a collection of edge-disjoint pentagons which partition Kn, and such that every part of distinct vertices of Kn is joined by a path of length two in exactly one pentagon of the collection P. The number n is called the order of the system. This paper gives a somplete solution of the existence problem of Steiner pentagon systems. In particular it is shown that the spectrum for Steiner pentagon systems (=the set of all orders for which a Steiner pentagon system exists) is precisely the set of all n ≡ 1 or 5 (mod 10), except 15, for which no such system exists.

Graphical t-wise balanced designs

A pair ( X, B ) will be a t-wise balanced design ( tBD) of type t−( v, K, λ) if B = (B i: i ϵ I) is a family of subsets of X, called blocks, such that: (i) |X| = v ϵ N , where N is the set of positive integers; (ii) 1⩽t⩽|B i|ϵK⊆ N , for every i ϵ I; and (iii) if T ⊆ X, | T| = t, then there are λ ϵ N indices i ϵ I where T ⊆ B i . Throughout this paper we make three restrictions on our tBD's: (1) there are no repeated blocks, i.e. B will be a set of subsets of X; (2) t ∉ K or there are no blocks of size t; and (3) P k(X)⊈ B or B does not contain all k-subsets of X for any t< k⩽ v. Note then that X ∉ B . Also, if we give the parameters of a specific tBD, then we will choose a minimal K. We focus on the t−(( p 2 ), K, λ) designs with the symmetric group S p as automorphism group, i.e. X will be the set of v = ( p 2 ) labelled edges of the undirected complete graph K p and if B ϵ B then all subgraphs of K p isomorphic to B are also in B . Call such tBD's ‘graphical tBD's’. We determine all graphical tBD's with λ = 1 or 2 which will include one with parameters 4−(15,{5,7},1).