Lucky labelings of graphs

Suppose the vertices of a graph G were labeled arbitrarily by positive integers, and let S(v) denote the sum of labels over all neighbors of vertex v. A labeling is lucky if the function S is a proper coloring of G, that is, if we have S(u)≠S(v) whenever u and v are adjacent. The least integer k for which a graph G has a lucky labeling from the set {1,2,…,k} is the lucky number of G, denoted by η(G).Using algebraic methods we prove that η(G)?k+1 for every bipartite graph G whose edges can be oriented so that the maximum out-degree of a vertex is at most k. In particular, we get that η(T)?2 for every tree T, and η(G)?3 for every bipartite planar graph G. By another technique we get a bound for the lucky number in terms of the acyclic chromatic number. This gives in particular that for every planar graph G. Nevertheless we offer a provocative conjecture that η(G)?χ(G) for every graph G.     

Approximation Algorithms for Bounded Degree Phylogenetic Roots

The Degree- Δ Closest Phylogenetic k th Root Problem (ΔCPR _k ) is the problem of finding a (phylogenetic) tree T from a given graph G=(V,E) such that (1) the degree of each internal node in T is at least 3 and at most Δ, (2) the external nodes (i.e. leaves) of T are exactly the elements of V, and (3) the number of disagreements, i.e., |E ⊕{{u,v} : u,v are leaves of T and d _T (u,v)≤k}|, is minimized, where d _T (u,v) denotes the distance between u and v in tree T. This problem arises from theoretical studies in evolutionary biology and generalizes several important combinatorial optimization problems such as the maximum matching problem. Unfortunately, it is known to be NP-hard for all fixed constants Δ,k such that either both Δ≥3 and k≥3, or Δ>3 and k=2. This paper presents a polynomial-time 8-approximation algorithm for Δ CPR ₂ for any fixed Δ>3, a quadratic-time 12-approximation algorithm for 3CPR ₃, and a polynomial-time approximation scheme for the maximization version of Δ CPR _k for any fixed Δ and k.     

Informative labeling schemes for graphs

This paper introduces the notion of informative labeling schemes for arbitrary graphs. Let f(W) be a function on subsets of vertices W. An f labeling scheme labels the vertices of a weighted graph G in such a way that f(W) can be inferred (or at least approximated) efficiently for any vertex subset W of G by merely inspecting the labels of the vertices of W, without having to use any additional information sources.A number of results illustrating this notion are presented in the paper. We begin by developing f labeling schemes for three functions f over the class of n-vertex trees. The first function, SepLevel, gives the separation level of any two vertices in the tree, namely, the depth of their least common ancestor. The second, LCA, provides the least common ancestor of any two vertices. The third, Center, yields the center of any three given vertices v₁,v₂,v₃ in the tree, namely, the unique vertex z connected to them by three edge-disjoint paths. All of these three labeling schemes use O-bit labels, which is shown to be asymptotically optimal.Our main results concern the function Steiner(W), defined for weighted graphs. For any vertex subset W in the weighted graph G, Steiner(W) represents the weight of the Steiner tree spanning the vertices of W in G. Considering the class of n-vertex trees with M-bit edge weights, it is shown that for this class there exists a Steiner labeling scheme using O((M+logn)logn) bit labels, which is asymptotically optimal. It is then shown that for the class of arbitrary n-vertex graphs with M-bit edge weights, there exists an approximate-Steiner labeling scheme, providing an estimate (up to a factor of O(logn)) for the Steiner weight Steiner(W) of a given set of vertices W, using O bit labels.     

The Swap Edges of a Multiple-Sources Routing Tree

Let T be a spanning tree of a graph G and S⊂V(G) be a set of sources. The routing cost of T is the total distance from all sources to all vertices. For an edge e of T, the swap edge of e is the edge f minimizing the routing cost of the tree formed by replacing e with f. Given an undirected graph G and a spanning tree T of G, we investigate the problem of finding the swap edge for every tree edge. In this paper, we propose an O(mlog n+n ²)-time algorithm for the case of two sources and an O(mn)-time algorithm for the case of more than two sources, where m and n are the numbers of edges and vertices of G, respectively.     

Time slot scheduling of compatible jobs

A version of weighted coloring of a graph is introduced which is motivated by some types of scheduling problems: each node v of a graph G corresponds to some operation to be processed (with a processing time w(v)), edges represent nonsimultaneity requirements (incompatibilities). We have to assign each operation to one time slot in such a way that in each time slot, all operations assigned to this slot are compatible; the length of a time slot will be the maximum of the processing times of its operations. The number k of time slots to be used has to be determined as well. So, we have to find a k-coloring = of G such that w(S ₁) + ⋅s +w(S _k ) is minimized where w(S _i ) = max {w(v) :v∊V}. Properties of optimal solutions are discussed, and complexity and approximability results are presented. Heuristic methods are given for establishing some of these results. The associated decision problems are shown to be NP-complete for bipartite graphs, for line-graphs of bipartite graphs, and for split graphs.     

A Simple Optimal Parallel Algorithm for a Core of a Tree

A core of a tree T = (V, E) is a path in T which minimizes ∑_vVd(v, P), where d(v, P), the distance from a vertex v to path P, is defined as min_uPd(v, u). We present an optimal parallel algorithm to find a core of T in O(log n) time using O(n/log n) processors on an EREW PRAM machine, where n is the number of vertices of tree T.     

Minimum Cost Source Location Problems with Flow Requirements

In this paper, we consider source location problems and their generalizations with three connectivity requirements (arc-connectivity requirements λ and two kinds of vertex-connectivity requirements κ and ), where the source location problems are to find a minimum-cost set S⊆V in a given graph G=(V,A) with a capacity function u:A→ℝ₊ such that for each vertex v∈V, the connectivity from S to v (resp., from v to S) is at least a given demand d ⁻(v) (resp., d ⁺(v)). We show that the source location problem with edge-connectivity requirements in undirected networks is strongly NP-hard, which solves an open problem posed by Arata et al. (J. Algorithms 42: 54–68, 2002). Moreover, we show that the source location problems with three connectivity requirements are inapproximable within a ratio of cln D for some constant c, unless every problem in NP has an O(N ^{log log N} )-time deterministic algorithm. Here D denotes the sum of given demands. We also devise (1+ln D)-approximation algorithms for all the extended source location problems if we have the integral capacity and demand functions. By the inapproximable results above, this implies that all the source location problems are Θ(ln ∑ _v∈V (d ⁺(v)+d ⁻(v)))-approximable. An extended abstract of this paper appeared in Sakashita et al. (Proceedings of LATIN 2006, Chile, LNCS, vol. 3887, pp. 769–780, March 2006).     

On efficient implementation of an approximation algorithm for the Steiner tree problem

Summary This paper studies the design and implementation of an approximation algorithm for the Steiner tree problem. Given any undirected distance graph G and a set of Steiner points S, the algorithm produces a Steiner tree with total weight on its edges no more than 2(1–1/L) times the total weight on the optimal Steiner tree, where L is the number of leaves in the optimal Steiner tree. Our implementation of the algorithm, in the worst case, makes it run in 0(¦E _g¦+¦V _g–S¦log¦V _g–S¦+¦S¦log ¦S¦) time for general graph G and in 0(¦S¦ log¦S¦+M log (M,¦V _g–S¦)) time for sparse graph G, where E _g is the set of edges in G, V_g is the set of vertices in G, M = min {¦E _g, (¦V _g–S¦–1)²/2} and (x,y) = min {i¦log⁽ⁱ⁾ y x/y}.The implementation is not likely to be improved significantly without the improvement of the shortest paths algorithm and the minimum spanning tree algorithm as the algorithm essentially composes of the computation of the multiple sources shortest paths of a graph with ¦V _g¦ vertices and ¦E _g¦ edges and the minimum spanning tree of a graph with ¦V _g–S¦ vertices and M edges.     

On Clark and Suen bound-type results for k-domination and Roman domination of Cartesian product graphs

A set S of vertices of a graph G is a dominating set for G if every vertex of G is adjacent to at least one vertex of S. The domination number γ(G), of G, is the minimum cardinality of a dominating set in G. Moreover, if the maximum degree of G is Δ, then for every positive integer k≤Δ, the set S is a k-dominating set in G if every vertex outside of S is adjacent to at least k vertices of S. The k-domination number of G, denoted by γ _k (G), is the minimum cardinality of a k-dominating set in G. A map f: V→<texlscub>0, 1, 2</texlscub>is a Roman dominating function for G if for every vertex v with f(v)=0, there exists a vertex u∈N(v) such that f(u)=2. The weight of a Roman dominating function is f(V)=∑ _u∈V f(u). The Roman domination number γ_R(G), of G, is the minimum weight of a Roman dominating function on G. In this paper, we obtain that for any two graphs G and H, the k-domination number of the Cartesian product of G and H is bounded below by γ(G)γ _k (H)/2. Also, we obtain that the domination number of Cartesian product of G and H is bounded below by γ(G)γ_R(H)/3.     

Collective Tree Spanners in Graphs with Bounded Parameters

In this paper we study collective additive tree spanners for special families of graphs including planar graphs, graphs with bounded genus, graphs with bounded tree-width, graphs with bounded clique-width, and graphs with bounded chordality. We say that a graph G=(V,E) admits a system of μ collective additive tree r -spanners if there is a system $\mathcal{T}(G)In this paper we study collective additive tree spanners for special families of graphs including planar graphs, graphs with bounded genus, graphs with bounded tree-width, graphs with bounded clique-width, and graphs with bounded chordality. We say that a graph G=(V,E) admits a system of μ collective additive tree r -spanners if there is a system T(G)\mathcal{T}(G) of at most μ spanning trees of G such that for any two vertices x,y of G a spanning tree T ? T(G)T\in\mathcal{T}(G) exists such that d _T (x,y)≤d _G (x,y)+r. We describe a general method for constructing a “small” system of collective additive tree r-spanners with small values of r for “well” decomposable graphs, and as a byproduct show (among other results) that any weighted planar graph admits a system of O(?n)O(\sqrt{n}) collective additive tree 0-spanners, any weighted graph with tree-width at most k−1 admits a system of klog ₂ n collective additive tree 0-spanners, any weighted graph with clique-width at most k admits a system of klog _3/2 n collective additive tree (2w)(2\mathsf{w}) -spanners, and any weighted graph with size of largest induced cycle at most c admits a system of log ₂ n collective additive tree (2?c/2?w)(2\lfloor c/2\rfloor\mathsf{w}) -spanners and a system of 4log ₂ n collective additive tree (2(?c/3?+1)w)(2(\lfloor c/3\rfloor +1)\mathsf {w}) -spanners (here, w\mathsf{w} is the maximum edge weight in G). The latter result is refined for weighted weakly chordal graphs: any such graph admits a system of 4log ₂ n collective additive tree (2w)(2\mathsf{w}) -spanners. Furthermore, based on this collection of trees, we derive a compact and efficient routing scheme for those families of graphs.     

On the partial terminal Steiner tree problem

We investigate a practical variant of the well-known graph Steiner tree problem. For a complete graph G = ( V,E ) with length function l:E → R ⁺ and two vertex subsets R ⊂ V and R ′ ⊆ R, a partial terminal Steiner tree is a Steiner tree which contains all vertices in R such that all vertices in R ∖ R ′ belong to the leaves of this Steiner tree. The partial terminal Steiner tree problem is to find a partial terminal Steiner tree T whose total lengths ∑ _{(u,v) ∈T} l ( u,v ) is minimum. In this paper, we show that the problem is both NP-complete and MAX SNP-hard when the lengths of edges are restricted to either 1 or 2. We also provide an approximation algorithm for the problem.

Local colourings of Cartesian product graphs

A local colouring of a graph G is a function c: V(G)→? such that for each S ? V(G), 2≤|S|≤3, there exist u, v∈S with |c(u)?c(v)| at least the number of edges in the subgraph induced by S. The maximum colour assigned by c is the value χ_?(c) of c, and the local chromatic number of G is χ_?(G)=min {χ_?(c): c is a local colouring of G}. In this note the local chromatic number is determined for Cartesian products G □ H, where G and GH are 3-colourable graphs. This result in part corrects an error from Omoomi and Pourmiri [On the local colourings of graphs, Ars Combin. 86 (2008), pp. 147–159]. It is also proved that if G and H are graphs such that χ(G)≤? χ_?(H)/2 ?, then χ_?(G □ H)≤χ_?(H)+1.     

Constructing Labeling Schemes through Universal Matrices

Let f be a function on pairs of vertices. An f -labeling scheme for a family of graphs ℱ labels the vertices of all graphs in ℱ such that for every graph G∈ℱ and every two vertices u,v∈G, f(u,v) can be inferred by merely inspecting the labels of u and v. The size of a labeling scheme is the maximum number of bits used in a label of any vertex in any graph in ℱ. This paper illustrates that the notion of universal matrices can be used to efficiently construct f-labeling schemes.     

Electric S-brane solutions corresponding to rank-2 Lie algebras: Acceleration and small variation of <Emphasis Type="Italic">G</Emphasis>

Electric S-brane solutions with two non-composite electric branes and a set of l scalar fields are considered. The intersection rules for branes correspond to Lie algebras A ₂, C ₂ and G ₂. The solutions contain five factor spaces. One of them, M ₀, is interpreted as our 3-dimensional space. It is shown that there exists a time interval where accelerated expansion of our 3-dimensional space is compatible with a small enough variation of the effective gravitational constant G(τ). This interval contains τ ₀, a point of minimum of the function G(τ). A special solution with two phantom scalar fields is analyzed, and it is shown that, in the vicinity of the point τ ₀, the time variation of G(τ) (calculated in the linear approximation) decreases in the sequence of Lie algebras A ₂, C ₂ and G ₂.     

Approximating Minimum-Power Degree and Connectivity Problems

Power optimization is a central issue in wireless network design. Given a graph with costs on the edges, the power of a node is the maximum cost of an edge incident to it, and the power of a graph is the sum of the powers of its nodes. Motivated by applications in wireless networks, we consider several fundamental undirected network design problems under the power minimization criteria. Given a graph G=(V,E)\mathcal{G}=(V,\mathcal{E}) with edge costs {c(e):e∈ℰ} and degree requirements {r(v):v∈V}, the Minimum-Power Edge-Multi-Cover\textsf{Minimum-Power Edge-Multi-Cover} (MPEMC\textsf{MPEMC} ) problem is to find a minimum-power subgraph G of G\mathcal{G} so that the degree of every node v in G is at least r(v). We give an O(log n)-approximation algorithms for MPEMC\textsf{MPEMC} , improving the previous ratio O(log ⁴ n). This is used to derive an O(log n+α)-approximation algorithm for the undirected $\textsf{Minimum-Power $\textsf{Minimum-Power ($\textsf{MP$\textsf{MP ) problem, where α is the best known ratio for the min-cost variant of the problem. Currently, _boxclosen-k)\alpha=O(\log k\cdot \log\frac{n}{n-k}) which is O(log k) unless k=n−o(n), and is O(log ² k)=O(log ² n) for k=n−o(n). Our result shows that the min-power and the min-cost versions of the $\textsf{$\textsf{ problem are equivalent with respect to approximation, unless the min-cost variant admits an o(log n)-approximation, which seems to be out of reach at the moment.     

On the edge-tenacity of the middle graph of a graph

We consider the problem of efficiently breaking a graph into small components by removing edges. One measure of how easily this can be done is the edge-tenacity. Given a set of edges of G, the score of S is defined as sc(S)=[| S|+τ (G?S)]/[w(G?S)]. Formally, the edge-tenacity of a graph G is defined as T′(G)=min sc(S), where the minimum is taken over all edge-sets S of G. A subset S of E(G) is said to be a T′-set of G if T′(G)=sc(S). Note that if G is disconnected, the set S may be empty. For any graph G, τ(G?S) is the number of vertices in the largest component of G?S and w(G?S) is the number of components of G?S. The middle graph M(G) of a graph G is the graph obtained from G by inserting a new vertex into every edge of G and by joining by edges those pairs of these new vertices which lie on adjacent edges of G. In this paper, we give the edge-tenacity of the middle graph of specific families of graphs and its relationships with other parameters.     

Distributed Approximation of Capacitated Dominating Sets

We study local, distributed algorithms for the capacitated minimum dominating set (CapMDS) problem, which arises in various distributed network applications. Given a network graph G=(V,E), and a capacity cap(v)∈ℕ for each node v∈V, the CapMDS problem asks for a subset S⊆V of minimal cardinality, such that every network node not in S is covered by at least one neighbor in S, and every node v∈S covers at most cap(v) of its neighbors. We prove that in general graphs and even with uniform capacities, the problem is inherently non-local, i.e., every distributed algorithm achieving a non-trivial approximation ratio must have a time complexity that essentially grows linearly with the network diameter. On the other hand, if for some parameter ε>0, capacities can be violated by a factor of 1+ε, CapMDS becomes much more local. Particularly, based on a novel distributed randomized rounding technique, we present a distributed bi-criteria algorithm that achieves an O(log Δ)-approximation in time O(log ³ n+log (n)/ε), where n and Δ denote the number of nodes and the maximal degree in G, respectively. Finally, we prove that in geometric network graphs typically arising in wireless settings, the uniform problem can be approximated within a constant factor in logarithmic time, whereas the non-uniform problem remains entirely non-local.     

Multicolorings of Series-Parallel Graphs

Let G be a graph, and let each vertex v of G have a positive integer weight ω(v). A multicoloring of G is to assign each vertex v a set of ω(v) colors so that any pair of adjacent vertices receive disjoint sets of colors. This paper presents an algorithm to find a multicoloring of a given series-parallel graph G with the minimum number of colors in time O(n W), where n is the number of vertices and W is the maximum weight of vertices in G.     

On the locatic number of graphs

A vertex v of a connected graph G distinguishes a pair u, w of vertices of G if d(v, u)≠d(v, w), where d(·,·) denotes the length of a shortest path between two vertices in G. A k-partition Π={S ₁, S ₂, …, S _k } of the vertex set of G is said to be a locatic partition if for every pair of distinct vertices v and w of G, there exists a vertex s∈S _i for all 1≤i≤k that distinguishes v and w. The cardinality of a largest locatic partition is called the locatic number of G. In this paper, we study the locatic number of paths, cycles and characterize all the connected graphs of order n having locatic number n, n?1 and n?2. Some realizable results are also given in this paper.     

Injective Colorings of Graphs with Low Average Degree

Let mad (G) denote the maximum average degree (over all subgraphs) of G and let χ _i (G) denote the injective chromatic number of G. We prove that if Δ≥4 and mad(G) < \frac145\mathrm{mad}(G)<\frac{14}{5}, then χ _i (G)≤Δ+2. When Δ=3, we show that mad(G) < \frac3613\mathrm{mad}(G)<\frac{36}{13} implies χ _i (G)≤5. In contrast, we give a graph G with Δ=3, mad(G)=\frac3613\mathrm{mad}(G)=\frac{36}{13}, and χ _i (G)=6.