An approximation algorithm dependent on edge-coloring number for minimum maximal matching problem

We consider the minimum maximal matching problem, which is NP-hard (Yannakakis and Gavril (1980) [18]). Given an unweighted simple graph G=(V,E), the problem seeks to find a maximal matching of minimum cardinality. It was unknown whether there exists a non-trivial approximation algorithm whose approximation ratio is less than 2 for any simple graph. Recently, Z. Gotthilf et al. (2008) [5] presented a -approximation algorithm, where c is an arbitrary constant.In this paper, we present a -approximation algorithm based on an LP relaxation, where χ^′(G) is the edge-coloring number of G. Our algorithm is the first non-trivial approximation algorithm whose approximation ratio is independent of |V|. Moreover, it is known that the minimum maximal matching problem is equivalent to the edge dominating set problem. Therefore, the edge dominating set problem is also -approximable. From edge-coloring theory, the approximation ratio of our algorithm is , where Δ(G) represents the maximum degree of G. In our algorithm, an LP formulation for the edge dominating set problem is used. Fujito and Nagamochi (2002) [4] showed the integrality gap of the LP formulation for bipartite graphs is at least . Moreover, χ^′(G) is Δ(G) for bipartite graphs. Thus, as far as an approximation algorithm for the minimum maximal matching problem uses the LP formulation, we believe our result is the best possible.     

The Pair Completion algorithm for the Homogeneous Set Sandwich Problem

A homogeneous set is a non-trivial module of a graph, i.e., a non-empty, non-unitary, proper vertex subset such that all its elements present the same outer neighborhood. Given two graphs G₁(V,E₁) and G₂(V,E₂), the Homogeneous Set Sandwich Problem (HSSP) asks whether there exists a graph GS(V,ES), E₁⊆ES⊆E₂, which has a homogeneous set. This paper presents an algorithm that uses the concept of bias graph [S. Tang, F. Yeh, Y. Wang, An efficient algorithm for solving the homogeneous set sandwich problem, Inform. Process. Lett. 77 (2001) 17-22] to solve the problem in time, thus outperforming the other known HSSP deterministic algorithms for inputs where .     

Speedup for a periodic subgraph miner

Given a series G={G₀,G₁,…,GT} of graphs encompassing V vertices and E edges, a periodic graph is a spatially as well as temporally maximal subgraph of a subsequence of G in the form , where n is not smaller than some predetermined threshold value σ. An algorithm for finding all such subgraphs is proposed taking time O((E+V)T²ln(T/σ)), which is faster by a factor of T than the method previously available.     

Enhanced algorithms for Local Search

Let G=(V,E) be a finite graph, and be any function. The Local Search problem consists in finding a local minimum of the function f on G, that is a vertex v such that f(v) is not larger than the value of f on the neighbors of v in G. In this note, we first prove a separation theorem slightly stronger than the one of Gilbert, Hutchinson and Tarjan for graphs of constant genus. This result allows us to enhance a previously known deterministic algorithm for Local Search with query complexity , so that we obtain a deterministic query complexity of , where n is the size of G, d is its maximum degree, and g is its genus. We also give a quantum version of our algorithm, whose query complexity is of . Our deterministic and quantum algorithms have query complexities respectively smaller than the algorithm Randomized Steepest Descent of Aldous and Quantum Steepest Descent of Aaronson for large classes of graphs, including graphs of bounded genus and planar graphs.     

Weighted Coloring: further complexity and approximability results

Given a vertex-weighted graph G=(V,E;w), w(v)?0 for any v∈V, we consider a weighted version of the coloring problem which consists in finding a partition S=(S₁,…,Sk) of the vertex set of G into stable sets and minimizing where the weight of S is defined as . In this paper, we continue the investigation of the complexity and the approximability of this problem by answering some of the questions raised by Guan and Zhu [D.J. Guan, X. Zhu, A coloring problem for weighted graphs, Inform. Process. Lett. 61 (2) (1997) 77-81].     

Conditional fault hamiltonian connectivity of the complete graph

A path in G is a hamiltonian path if it contains all vertices of G. A graph G is hamiltonian connected if there exists a hamiltonian path between any two distinct vertices of G. The degree of a vertex u in G is the number of vertices of G adjacent to u. We denote by δ(G) the minimum degree of vertices of G. A graph G is conditional k edge-fault tolerant hamiltonian connected if G−F is hamiltonian connected for every F⊂E(G) with |F|?k and δ(G−F)?3. The conditional edge-fault tolerant hamiltonian connectivity is defined as the maximum integer k such that G is k edge-fault tolerant conditional hamiltonian connected if G is hamiltonian connected and is undefined otherwise. Let n?4. We use Kn to denote the complete graph with n vertices. In this paper, we show that for n∉{4,5,8,10}, , , , and .     

About acyclic edge colourings of planar graphs

Let G=(V,E) be any finite simple graph. A mapping is called an acyclic edge k-colouring of G, if any two adjacent edges have different colours and there are no bichromatic cycles in G. In other words, for every pair of distinct colours i and j, the subgraph induced by all the edges which have either colour i or j is acyclic. The smallest number k of colours, such that G has an acyclic edge k-colouring is called the acyclic chromatic index of G and is denoted by .In 1991, Alon et al. [N. Alon, C.J.H. McDiarmid, B.A. Reed, Acyclic coloring of graphs, Random Structures and Algorithms 2 (1991) 277-288] proved that for any graph G of maximum degree Δ(G). This bound was later improved to 16Δ(G) by Molloy and Reed in [M. Molloy, B. Reed, Further algorithmic aspects of the local lemma, in: Proceedings of the 30th Annual ACM Symposium on Theory of Computing, 1998, pp. 524-529].In this paper we prove that for a planar graph G without cycles of length three and that the same holds if G has an edge-partition into two forests. We also show that if G is planar.     

Acyclic edge colourings of graphs with the number of edges linearly bounded by the number of vertices

Let G be any finite graph. A mapping c:E(G)→{1,…,k} is called an acyclic edge k-colouring of G, if any two adjacent edges have different colours and there are no bichromatic cycles in G. In other words, for every pair of distinct colours i and j, the subgraph induced in G by all the edges that have colour i or j is acyclic. The smallest number k of colours such that G has an acyclic edge k-colouring is called the acyclic chromatic index of G and is denoted by .Determining the acyclic chromatic index of a graph is a hard problem, both from theoretical and algorithmical point of view. In 1991, Alon et al. proved that for any graph G of maximum degree Δ(G). This bound was later improved to 16Δ(G) by Molloy and Reed. In general, the problem of computing the acyclic chromatic index of a graph is NP-complete. Only a few algorithms for finding acyclic edge colourings have been known by now. Moreover, these algorithms work only for graphs from particular classes.In our paper, we prove that for every graph G which satisfies the condition that |E(G^′)|?t|V(G^′)|−1 for each subgraph G^′⊆G, where t?2 is a given integer, the constant p=2t³−3t+2. Based on that result, we obtain a polynomial algorithm which computes such a colouring. The class of graphs covered by our theorem is quite rich, for example, it contains all t-degenerate graphs.     

Ranking numbers of graphs

Given a graph G, a vertex ranking (or simply, ranking) of G is a mapping f from V(G) to the set of all positive integers, such that for any path between two distinct vertices u and v with f(u)=f(v), there is a vertex w in the path with f(w)>f(u). If f is a ranking of G, the ranking number of G under f, denoted γf(G), is defined by , and the ranking number of G, denoted γ(G), is defined by . The vertex ranking problem is to determine the ranking number γ(G) of a given graph G. This problem is a natural model for the manufacturing scheduling problem. We study the ranking numbers of graphs in this paper. We consider the relation between the ranking numbers and the minimal cut sets, and the relation between the ranking numbers and the independent sets. From this, we obtain the ranking numbers of the powers of paths and the powers of cycles, the Cartesian product of P₂ with Pn or Cn, and the caterpilars. And we also find the vertex ranking numbers of the composition of two graphs in this paper.     

Spectral radius minimization for optimal average consensus and output feedback stabilization

In this paper, we consider two problems which can be posed as spectral radius minimization problems. Firstly, we consider the fastest average agreement problem on multi-agent networks adopting a linear information exchange protocol. Mathematically, this problem can be cast as finding an optimal such that x(k+1)=Wx(k), , and W∈S(E). Here, is the value possessed by the agents at the kth time step, is an all-one vector and S(E) is the set of real matrices in with zeros at the same positions specified by a network graph G(V,E), where V is the set of agents and E is the set of communication links between agents. The optimal W is such that the spectral radius is minimized. To this end, we consider two numerical solution schemes: one using the qth-order spectral norm (2-norm) minimization (q-SNM) and the other gradient sampling (GS), inspired by the methods proposed in [Burke, J., Lewis, A., & Overton, M. (2002). Two numerical methods for optimizing matrix stability. Linear Algebra and its Applications, 351-352, 117-145; Xiao, L., & Boyd, S. (2004). Fast linear iterations for distributed averaging. Systems & Control Letters, 53(1), 65-78]. In this context, we theoretically show that when E is symmetric, i.e. no information flow from the ith to the jth agent implies no information flow from the jth to the ith agent, the solution from the 1-SNM method can be chosen to be symmetric and is a local minimum of the function . Numerically, we show that the q-SNM method performs much better than the GS method when E is not symmetric. Secondly, we consider the famous static output feedback stabilization problem, which is considered to be a hard problem (some think NP-hard): for a given linear system (A,B,C), find a stabilizing control gain K such that all the real parts of the eigenvalues of A+BKC are strictly negative. In spite of its computational complexity, we show numerically that q-SNM successfully yields stabilizing controllers for several benchmark problems with little effort.     

Adjacent vertex-distinguishing edge and total chromatic numbers of hypercubes

An adjacent vertex-distinguishing edge coloring of a simple graph G is a proper edge coloring of G such that incident edge sets of any two adjacent vertices are assigned different sets of colors. A total coloring of a graph G is a coloring of both the edges and the vertices. A total coloring is proper if no two adjacent or incident elements receive the same color. An adjacent vertex-distinguishing total coloring h of a simple graph G=(V,E) is a proper total coloring of G such that H(u)≠H(v) for any two adjacent vertices u and v, where H(u)={h(wu)|wu∈E(G)}∪{h(u)} and H(v)={h(xv)|xv∈E(G)}∪{h(v)}. The minimum number of colors required for an adjacent vertex-distinguishing edge coloring (resp. an adjacent vertex-distinguishing total coloring) of G is called the adjacent vertex-distinguishing edge chromatic number (resp. adjacent vertex-distinguishing total chromatic number) of G and denoted by (resp. χat(G)). In this paper, we consider the adjacent vertex-distinguishing edge chromatic number and adjacent vertex-distinguishing total chromatic number of the hypercube Qn, prove that for n?3 and χat(Qn)=Δ(Qn)+2 for n?2.     

Acyclic chromatic index of planar graphs with triangles

A proper edge coloring of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic chromatic index of G, denoted by , is the least number of colors in an acyclic edge coloring of G. Let G be a planar graph with maximum degree Δ(G). In this paper, we show that , if G contains no 4-cycle; , if G contains no intersecting triangles; and if G contains no adjacent triangles.     

On domination number of Cartesian product of directed cycles

Let γ(G) denote the domination number of a digraph G and let Cm□Cn denote the Cartesian product of Cm and Cn, the directed cycles of length m,n?2. In this paper, we determine the exact values: γ(C₂□Cn)=n; γ(C₃□Cn)=n if , otherwise, γ(C₃□Cn)=n+1; if , otherwise, .     

The complexity of changing colourings with bounded maximum degree

Let G be a graph, x,y∈V(G), and ?:V(G)→[k] a k-colouring of G such that ?(x)=?(y). If then the following question is NP-complete: Does there exist a k-colouring ?^′ of G such that ?^′(x)≠?^′(y)? Conversely, if then the problem is polynomial time.