Node-weighted Steiner tree approximation in unit disk graphs

Given a graph G=(V,E) with node weight w:V→R ⁺ and a subset S⊆V, find a minimum total weight tree interconnecting all nodes in S. This is the node-weighted Steiner tree problem which will be studied in this paper. In general, this problem is NP-hard and cannot be approximated by a polynomial time algorithm with performance ratio aln n for any 0<a<1 unless NP⊆DTIME(n ^O(log n)), where n is the number of nodes in s. In this paper, we are the first to show that even though for unit disk graphs, the problem is still NP-hard and it has a polynomial time constant approximation. We present a 2.5ρ-approximation where ρ is the best known performance ratio for polynomial time approximation of classical Steiner minimum tree problem in graphs. As a corollary, we obtain that there is a polynomial time (9.875+ε)-approximation algorithm for minimum weight connected dominating set in unit disk graphs, and also there is a polynomial time (4.875+ε)-approximation algorithm for minimum weight connected vertex cover in unit disk graphs.     

Two Variations of the Minimum Steiner Problem

Given a set S of starting vertices and a set T of terminating vertices in a graph G = (V,E) with non-negative weights on edges, the minimum Steiner network problem is to find a subgraph of G with the minimum total edge weight. In such a subgraph, we require that for each vertex s S and t T, there is a path from s to a terminating vertex as well as a path from a starting vertex to t. This problem can easily be proven NP-hard. For solving the minimum Steiner network problem, we first present an algorithm that runs in time and space that both are polynomial in n with constant degrees, but exponential in |S|+|T|, where n is the number of vertices in G. Then we present an algorithm that uses space that is quadratic in n and runs in time that is polynomial in n with a degree O(max {max {|S|,|T|}–2,min {|S|,|T|}–1}). In spite of this degree, we prove that the number of Steiner vertices in our solution can be as large as |S|+|T|–2. Our algorithm can enumerate all possible optimal solutions. The input graph G can either be undirected or directed acyclic. We also give a linear time algorithm for the special case when min {|S|,|T|} = 1 and max {|S|,|T|} = 2.The minimum union paths problem is similar to the minimum Steiner network problem except that we are given a set H of hitting vertices in G in addition to the sets of starting and terminating vertices. We want to find a subgraph of G with the minimum total edge weight such that the conditions required by the minimum Steiner network problem are satisfied as well as the condition that every hitting vertex is on a path from a starting vertex to a terminating vertex. Furthermore, G must be directed acyclic. For solving the minimum union paths problem, we also present algorithms that have a time and space tradeoff similar to algorithms for the minimum Steiner network problem. We also give a linear time algorithm for the special case when |S| = 1, |T| = 1 and |H| = 2.An extended abstract of part of this paper appears in Hsu et al. (1996).Supported in part by the National Science Foundation under Grants CCR-9309743 and INT-9207212, and by the Office of Naval Research under Grant No. N00014-93-1-0272.Supported in part by the National Science Council, Taiwan, ROC, under Grant No. NSC-83-0408-E-001-021.     

Finding an anti-risk path between two nodes in undirected graphs

Given a weighted graph G=(V,E) with a source s and a destination t, a traveler has to go from s to t. However, some of the edges may be blocked at certain times, and the traveler only observes that upon reaching an adjacent site of the blocked edge. Let ℘={P _G (s,t)} be the set of all paths from s to t. The risk of a path is defined as the longest travel under the assumption that any edge of the path may be blocked. The paper will propose the Anti-risk Path Problem of finding a path P _G (s,t) in ℘ such that it has minimum risk. We will show that this problem can be solved in O(mn+n ²log n) time suppose that at most one edge may be blocked, where n and m denote the number of vertices and edges in G, respectively. This research is supported by NSF of China under Grants 70525004, 60736027, 70121001 and Postdoctoral Science Foundation of China under Grant 20060401003.     

Lower bounds and a tabu search algorithm for the minimum deficiency problem

An edge coloring of a graph G=(V,E) is a function c:E→ℕ that assigns a color c(e) to each edge e∈E such that c(e)≠c(e′) whenever e and e′ have a common endpoint. Denoting S _v (G,c) the set of colors assigned to the edges incident to a vertex v∈V, and D _v (G,c) the minimum number of integers which must be added to S _v (G,c) to form an interval, the deficiency D(G,c) of an edge coloring c is defined as the sum ∑ _v∈V D _v (G,c), and the span of c is the number of colors used in c. The problem of finding, for a given graph, an edge coloring with a minimum deficiency is NP-hard. We give new lower bounds on the minimum deficiency of an edge coloring and on the span of edge colorings with minimum deficiency. We also propose a tabu search algorithm to solve the minimum deficiency problem and report experiments on various graph instances, some of them having a known optimal deficiency.     

Restricted domination parameters in graphs

In a graph G, a vertex dominates itself and its neighbors. A subset S ⊂eqV(G) is an m-tuple dominating set if S dominates every vertex of G at least m times, and an m-dominating set if S dominates every vertex of G−S at least m times. The minimum cardinality of a dominating set is γ, of an m-dominating set is γ _m , and of an m-tuple dominating set is mtupledom. For a property π of subsets of V(G), with associated parameter f_π, the k-restricted π-number r _k (G,f_π) is the smallest integer r such that given any subset K of (at most) k vertices of G, there exists a π set containing K of (at most) cardinality r. We show that for 1< k < n where n is the order of G: (a) if G has minimum degree m, then r _k (G,γ _m ) < (mn+k)/(m+1); (b) if G has minimum degree 3, then r _k (G,γ) < (3n+5k)/8; and (c) if G is connected with minimum degree at least 2, then r _k (G,ddom) < 3n/4 + 2k/7. These bounds are sharp. Research supported in part by the South African National Research Foundation and the University of KwaZulu-Natal.     

Labelling algorithms for paired-domination problems in block and interval graphs

Let G=(V,E) be a graph without isolated vertices. A set S⊆V is a paired-dominating set if every vertex in V−S is adjacent to a vertex in S and the subgraph induced by S contains a perfect matching. The paired-domination problem is to determine the paired-domination number, which is the minimum cardinality of a paired-dominating set. Motivated by a mistaken algorithm given by Chen, Kang and Ng (Discrete Appl. Math. 155:2077–2086, 2007), we present two linear time algorithms to find a minimum cardinality paired-dominating set in block and interval graphs. In addition, we prove that paired-domination problem is NP-complete for bipartite graphs, chordal graphs, even for split graphs.     

Polynomial Time Approximation Scheme for the Rectilinear Steiner Arborescence Problem

Given a set N of n terminals in the first quadrant of the Euclidean plane E ², find a minimum length directed tree rooted at the origin o, connecting to all terminals in N, and consisting of only horizontal and vertical arcs oriented from left to right or from bottom to top. This problem is called rectilinear Steiner arborescence problem, which has been proved to be NP-complete recently (Shi and Su, 11th ACM-SIAM Symposium on Discrete Algorithms (SODA), January 2000, to appear). In this paper, we present a polynomial time approximation scheme for this problem.     

Hardness of <Emphasis Type="Italic">k</Emphasis>-Vertex-Connected Subgraph Augmentation Problem

Given a k-connected graph G=(V,E) and V ^′⊂V, k-Vertex-Connected Subgraph Augmentation Problem (k-VCSAP) is to find S⊂V∖V ^′ with minimum cardinality such that the subgraph induced by V ^′∪S is k-connected. In this paper, we study the hardness of k-VCSAP in undirect graphs. We first prove k-VCSAP is APX-hard. Then, we improve the lower bound in two ways by relying on different assumptions. That is, we prove no algorithm for k-VCSAP has a PR better than O(log (log n)) unless P=NP and O(log n) unless NP⊆DTIME(n ^{O(log log n)}), where n is the size of an input graph.     

A 6.55 factor primal-dual approximation algorithm for the connected facility location problem

In the connected facility location (ConFL) problem, we are given a graph G=(V,E) with nonnegative edge cost c _e on the edges, a set of facilities ??V, a set of demands (i.e., clients) $\mathcal{D}\subseteq VIn the connected facility location (ConFL) problem, we are given a graph G=(V,E) with nonnegative edge cost c _e on the edges, a set of facilities ℱ⊆V, a set of demands (i.e., clients) D í V\mathcal{D}\subseteq V , and a parameter M≥1. Each facility i has a nonnegative opening cost f _i and each client j has d _j units of demand. Our objective is to open some facilities, say F⊆ℱ, assign each demand j to some open facility i(j)∈F and connect all open facilities using a Steiner tree T such that the total cost, which is ?_{i ? F}f_i+?_{j ? D}d_jc_i(j)j+M?_{e ? T}c_e\sum_{i\in F}f_{i}+\sum_{j\in \mathcal{D}}d_{j}c_{i(j)j}+M\sum_{e\in T}c_{e} , is minimized. We present a primal-dual 6.55-approximation algorithm for the ConFL problem which improves the previous primal-dual 8.55-approximation algorithm given by Swamy and Kumar (Algorithmica 40:245–269, 2004).     

Approximation algorithms for multicast routing in ad hoc wireless networks

Energy efficient multicast problem is one of important issues in ad hoc networks. In this paper, we address the energy efficient multicast problem for discrete power levels in ad hoc wireless networks. The problem of our concern is: given n nodes deployed over 2-D plane and each node v has l(v) transmission power levels and a multicast request (s,D) (clearly, when D is V∖{s}, the multicast request is a broadcast request), how to find a multicast tree rooted at s and spanning all destinations in D such that the total energy cost of the multicast tree is minimized. We first prove that this problem is NP-hard and it is unlikely to have an approximation algorithm with performance ratio ρlnn(ρ<1). Then, we propose a general algorithm for the multicast/broadcast tree problem. And based on the general algorithm, we propose an approximation algorithm and a heuristics for multicast tree problem. Especially, we also propose an efficient heuristic for broadcast tree problem. Simulations ensure our algorithms are efficient.     

On minimum <Emphasis Type="Italic">m</Emphasis>-connected <Emphasis Type="Italic">k</Emphasis>-dominating set problem in unit disc graphs

Minimum m-connected k-dominating set problem is as follows: Given a graph G=(V,E) and two natural numbers m and k, find a subset S⊆V of minimal size such that every vertex in V∖S is adjacent to at least k vertices in S and the induced graph of S is m-connected. In this paper we study this problem with unit disc graphs and small m, which is motivated by the design of fault-tolerant virtual backbone for wireless sensor networks. We propose two approximation algorithms with constant performance ratios for m≤2. We also discuss how to design approximation algorithms for the problem with arbitrarily large m. This work was supported in part by the Research Grants Council of Hong Kong under Grant No. CityU 1165/04E, the National Natural Science Foundation of China under Grant No. 70221001, 10531070 and 10771209.     

Restrained domination in cubic graphs

Let G=(V,E) be a graph. A set S⊆V is a restrained dominating set if every vertex in V−S is adjacent to a vertex in S and to a vertex in V−S. The restrained domination number of G, denoted γ _r (G), is the smallest cardinality of a restrained dominating set of G. A graph G is said to be cubic if every vertex has degree three. In this paper, we study restrained domination in cubic graphs. We show that if G is a cubic graph of order n, then g_r(G) 3 \fracn4\gamma_{r}(G)\geq \frac{n}{4} , and characterize the extremal graphs achieving this lower bound. Furthermore, we show that if G is a cubic graph of order n, then g_r(G) ￡ \frac5n11.\gamma _{r}(G)\leq \frac{5n}{11}. Lastly, we show that if G is a claw-free cubic graph, then γ _r (G)=γ(G).     

On the distance paired domination of generalized Petersen graphs <Emphasis Type="Italic">P</Emphasis>(<Emphasis Type="Italic">n</Emphasis>,1) and <Emphasis Type="Italic">P</Emphasis>(<Emphasis Type="Italic">n</Emphasis>,2)

Let G=(V,E) be a graph without an isolated vertex. A set D⊆V(G) is a k -distance paired dominating set of G if D is a k-distance dominating set of G and the induced subgraph 〈D〉 has a perfect matching. The minimum cardinality of a k-distance paired dominating set for graph G is the k -distance paired domination number, denoted by γ _p ^k (G). In this paper, we determine the exact k-distance paired domination number of generalized Petersen graphs P(n,1) and P(n,2) for all k≥1.     

Approximation algorithms for connected facility location problems

We study Connected Facility Location problems. We are given a connected graph G=(V,E) with nonnegative edge cost c _e for each edge e∈E, a set of clients D⊆V such that each client j∈D has positive demand d _j and a set of facilities F⊆V each has nonnegative opening cost f _i and capacity to serve all client demands. The objective is to open a subset of facilities, say , to assign each client j∈D to exactly one open facility i(j) and to connect all open facilities by a Steiner tree T such that the cost is minimized for a given input parameter M≥1. We propose a LP-rounding based 8.29 approximation algorithm which improves the previous bound 8.55 (Swamy and Kumar in Algorithmica, 40:245–269, 2004). We also consider the problem when opening cost of all facilities are equal. In this case we give a 7.0 approximation algorithm.     

Minimum d-blockers and d-transversals in graphs

We consider a set V of elements and an optimization problem on V: the search for a maximum (or minimum) cardinality subset of V verifying a given property ℘. A d-transversal is a subset of V which intersects any optimum solution in at least d elements while a d-blocker is a subset of V whose removal deteriorates the value of an optimum solution by at least d. We present some general characteristics of these problems, we review some situations which have been studied (matchings, s–t paths and s–t cuts in graphs) and we study d-transversals and d-blockers of stable sets or vertex covers in bipartite and in split graphs.     

An Edge-Splitting Algorithm in Planar Graphs

For a multigraph G = (V, E), let s V be a designated vertex which has an even degree, and let _G (V – s) denote min{c _G(X) | Ø X V – s}, where c _G(X) denotes the size of cut X. Splitting two adjacent edges (s, u) and (s, v) means deleting these edges and adding a new edge (u, v). For an integer k, splitting two edges e ₁ and e ₂ incident to s is called (k, s)-feasible if _G(V – s) k holds in the resulting graph G. In this paper, we prove that, for a planar graph G and an even k or k = 3 with k _G (V – s), there exists a complete (k, s)-feasible splitting at s such that the resulting graph G is still planar, and present an O(n ³ log n) time algorithm for finding such a splitting, where n = |V|. However, for every odd k 5, there is a planar graph G with a vertex s which has no complete (k, s)-feasible and planarity-preserving splitting. As an application of this result, we show that for an outerplanar graph G and an even integer k the problem of optimally augmenting G to a k-edge-connected planar graph can be solved in O(n ³ log n) time.     

An Approximation for Minimum Multicast Route in Optical Networks with Nonsplitting Nodes

Consider the problem of computing the minimum-weight multicast route in an optical network with both nonsplitting and splitting nodes. This problem can be reduced to the minimum Hamiltonian path problem when all nodes are nonsplitting, and the Steiner minimum tree problem when all nodes are splitting. Therefore, the problem is NP-hard. Previously, the best known polynomial-time approximation has the performance ratio 3. In this paper, we present a new polynomial-time approximation with performance ratio of 1+ρ, where ρ is the best known approximation performance ratio for the Steiner minimum tree in graph and it has been known that ρ < 1.55. Support in part by National Science Foundation under grants CCF-0514796 and CNS-0524429     

Finding the anti-block vital edge of a shortest path between two nodes

Let P _G (s,t) denote a shortest path between two nodes s and t in an undirected graph G with nonnegative edge weights. A detour at a node u∈P _G (s,t)=(s,…,u,v,…,t) is defined as a shortest path P _G−e (u,t) from u to t which does not make use of (u,v). In this paper, we focus on the problem of finding an edge e=(u,v)∈P _G (s,t) whose removal produces a detour at node u such that the ratio of the length of P _G−e (u,t) to the length of P _G (u,t) is maximum. We define such an edge as an anti-block vital edge (AVE for short), and show that this problem can be solved in O(mn) time, where n and m denote the number of nodes and edges in the graph, respectively. Some applications of the AVE for two special traffic networks are shown. This research is supported by NSF of China under Grants 70471035, 70525004, 701210001 and 60736027, and PSF of China under Grant 20060401003.     

Galaxy cutsets in graphs

Given a network G=(V,E), we say that a subset of vertices S⊆V has radius r if it is spanned by a tree of depth at most r. We are interested in determining whether G has a cutset that can be written as the union of k sets of radius r. This generalizes the notion of k-vertex connectivity, since in the special case r=0, a set spanned by a tree of depth at most r is a single vertex.     

An upper bound on the total restrained domination number of a tree

Let G=(V,E) be a graph. A set of vertices S?V is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex of $V-\nobreak S$ is adjacent to a vertex in V?S. The total restrained domination number of G, denoted by γ _tr (G), is the smallest cardinality of a total restrained dominating set of G. A support vertex of a graph is a vertex of degree at least two which is adjacent to a leaf. We show that $\gamma_{\mathit{tr}}(T)\leq\lfloor\frac{n+2s+\ell-1}{2}\rfloor$ where T is a tree of order n≥3, and s and ? are, respectively, the number of support vertices and leaves of T. We also constructively characterize the trees attaining the aforementioned bound.