Given a set @math of @math points in the plane, we solve the problems of constructing a geometric planar graph spanning @math 1) of minimum degree 2, and 2) which is 2-edge connected, respectively, and has max edge length bounded by a factor of 2 times the optimal; we also show that the factor 2 is best possible given appropriate connectivity conditions on the set @math , respectively. First, we construct in @math time a geometric planar graph of minimum degree 2 and max edge length bounded by 2 times the optimal. This is then used to construct in @math time a 2-edge connected geometric planar graph spanning @math with max edge length bounded by @math times the optimal, assuming that the set @math forms a connected Unit Disk Graph. Second, we prove that 2 times the optimal is always sufficient if the set of points forms a 2 edge connected Unit Disk Graph and give an algorithm that runs in @math time. We also show that for @math , there exists a set @math of @math points in the plane such that even though the Unit Disk Graph spanning @math is @math -vertex connected, there is no 2-edge connected geometric planar graph spanning @math even if the length of its edges is allowed to be up to 17 16.
A related problem is studied in @cite_2 . The authors prove that it is NP-hard to decide whether @math contains a spanning planar graph of minimum degree 2 even if @math itself has minimum degree 2. They also posed and studied the problem of finding the minimum radius @math so that @math has a geometric planar spanning subgraph of minimum degree 3 provided that @math has a spanning subgraph of minimum degree 3.
@cite_2
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An efficient heuristic is presented for the problem of finding a minimum-size k-connected spanning subgraph of an (undirected or directed) simple graph G=(V,E). There are four versions of the problem, and the approximation guarantees are as follows: minimum-size k-node connected spanning subgraph of an undirected graph 1 + [1 k], minimum-size k-node connected spanning subgraph of a directed graph 1 + [1 k], minimum-size k-edge connected spanning subgraph of an undirected graph 1+[2 (k+1)], minimum-size k-edge connected spanning subgraph of a directed graph 1 + [4 k ]. The heuristic is based on a subroutine for the degree-constrained subgraph (b-matching) problem. It is simple and deterministic and runs in time O(k|E|2). The following result on simple undirected graphs is used in the analysis: The number of edges required for augmenting a graph of minimum degree k to be k-edge connected is at most k,|V| (k+1). For undirected graphs and k=2, a (deterministic) parallel NC version of the heuristic finds a 2-node connected (or 2-edge connected) spanning subgraph whose size is within a factor of ( @math ) of minimum, where @math is a constant. We present algorithmic and hardness results for network design problems with degree or order constraints. The first problem we consider is the Survivable Network Design problem with degree constraints on vertices. The objective is to find a minimum cost subgraph which satisfies connectivity requirements between vertices and also degree upper bounds @math on the vertices. This includes the well-studied Minimum Bounded Degree Spanning Tree problem as a special case. Our main result is a @math -approximation algorithm for the edge-connectivity Survivable Network Design problem with degree constraints, where the cost of the returned solution is at most twice the cost of an optimum solution (satisfying the degree bounds) and the degree of each vertex @math is at most @math . This implies the first constant factor (bicriteria) approximation algorithms for many degree constrained network design problems, including the Minimum Bounded Degree Steiner Forest problem. Our results also extend to directed graphs and provide the first constant factor (bicriteria) approximation algorithms for the Minimum Bounded Degree Arborescence problem and the Minimum Bounded Degree Strongly @math -Edge-Connected Subgraph problem. In contrast, we show that the vertex-connectivity Survivable Network Design problem with degree constraints is hard to approximate, even when the cost of every edge is zero. A striking aspect of our algorithmic result is its simplicity. It is based on the iterative relaxation method, which is an extension of Jain's iterative rounding method. This provides an elegant and unifying algorithmic framework for a broad range of network design problems. We also study the problem of finding a minimum cost @math -edge-connected subgraph with at least @math vertices, which we call the @math -subgraph problem. This generalizes some well-studied classical problems such as the @math -MST and the minimum cost @math -edge-connected subgraph problems. We give a polylogarithmic approximation for the @math -subgraph problem. However, by relating it to the Densest @math -Subgraph problem, we provide evidence that the @math -subgraph problem might be hard to approximate for arbitrary @math . A problem arising in the design of vacuum systems and having applications to some natural problems of interconnection design is described as follows. (1) Given a set X and subsets @math of @math , satisfying @math , find a graph G with vertex set X and the minimum number of edges such that for any i, the subgraph induced by @math has a connected component containing @math .Two other problems related to this one are the following ones. (2) Given a set X and subsets @math such that @math , find a graph G with vertex set X and the minimum number of edges such that for any i the subgraph @math induced by @math in G is connected. (3) Given a set X and subsets @math such that @math , find a graph G with vertex set X, find a graph G with vertex set X and the minimum number of edges such that for any subset I of @math , the subgraph induced by @math is co... 2-Edge connectivity is an important fault tolerance property of a network because it maintains network communication despite the deletion of a single arbitrary edge. Planar spanning subgraphs have been shown to play a significant role for achieving local decentralized routing in wireless networks. Existing algorithmic constructions of spanning planar subgraphs of unit disk graphs (UDGs) such as Minimum Spanning Tree, Gabriel Graph, Nearest Neighborhood Graph, etc. do not always ensure connectivity of the resulting graph under single edge deletion. Furthermore, adding edges to the network so as to improve its edge connectivity not only may create edge crossings (at points which are not vertices) but it may also require edges of unbounded length. Thus we are faced with the problem of constructing 2-edge connected geometric planar spanning graphs by adding edges of bounded length without creating edge crossings (at points which are not vertices). To overcome this difficulty, in this paper we address the problem of augmenting the edge set (i.e., adding new edges) of planar geometric graphs with straight line edges of bounded length so that the resulting graph is planar and 2-edge connected. We provide bounds on the number of newly added straight-line edges, prove that such edges can be of length at most 3 times the max length of an edge of the original graph, and also show that the factor 3 is optimal. It is shown to be NP-Complete to augment a geometric planar graph to a 2-edge connected geometric planar graph with the minimum number of new edges of a given bounded length. We also provide a constant time algorithm that works in location-aware settings to augment a planar graph into a 2-edge connected planar graph with straight-line edges of length bounded by 3 times the longest edge of the original graph. It turns out that knowledge of vertex coordinates is crucial to our construction and in fact we prove that this problem cannot be solved locally if the vertices do not know their coordinates. Moreover, we provide a family of k-connected UDGs which does not have 2-edge connected spanning planar subgraphs, for any . We consider the problem of finding a minimum edge cost subgraph of a graph satisfying both given node-connectivity requirements and degree upper bounds on nodes. We present an iterative rounding algorithm of the biset linear programming relaxation for this problem. For directed graphs and @math -out-connectivity requirements from a root, our algorithm computes a solution that is a 2-approximation on the cost, and the degree of each node @math in the solution is at most @math , where @math is the degree upper bound on @math . For undirected graphs and element-connectivity requirements with maximum connectivity requirement @math , our algorithm computes a solution that is a @math -approximation on the cost, and the degree of each node @math in the solution is at most @math . These ratios improve the previous @math -approximation on the cost and @math -approximation on the degrees. Our algorithms can be used to improve approximation ratios for other node-connectivity problems such as undirected $k... Given two graphs @math and @math , the Subgraph Isomorphism problem asks if @math is isomorphic to a subgraph of @math . While NP-hard in general, algorithms exist for various parameterized versions of the problem: for example, the problem can be solved (1) in time @math using the color-coding technique of Alon, Yuster, and Zwick; (2) in time @math using Courcelle's Theorem; (3) in time @math using a result on first-order model checking by Frick and Grohe; or (4) in time @math for connected @math using the algorithm of Matou s ek and Thomas. Already this small sample of results shows that the way an algorithm can depend on the parameters is highly nontrivial and subtle. We develop a framework involving 10 relevant parameters for each of @math and @math (such as treewidth, pathwidth, genus, maximum degree, number of vertices, number of components, etc.), and ask if an algorithm with running time [ f_1(p_1,p_2,..., p_ ) n^ f_2(p_ +1 ,..., p_k) ] exist, where each of @math is one of the 10 parameters depending only on @math or @math . We show that all the questions arising in this framework are answered by a set of 11 maximal positive results (algorithms) and a set of 17 maximal negative results (hardness proofs); some of these results already appear in the literature, while others are new in this paper. On the algorithmic side, our study reveals for example that an unexpected combination of bounded degree, genus, and feedback vertex set number of @math gives rise to a highly nontrivial algorithm for Subgraph Isomorphism. On the hardness side, we present W[1]-hardness proofs under extremely restricted conditions, such as when @math is a bounded-degree tree of constant pathwidth and @math is a planar graph of bounded pathwidth. We study the following problem: given a geometric graph G and an integer k, determine if G has a planar spanning subgraph (with the original embedding and straight-line edges) such that all nodes have degree at least k. If G is a unit disk graph, the problem is trivial to solve for k = 1. We show that even the slightest deviation from the trivial case (e.g., quasi unit disk graphs or k = 2) leads to NP-hard problems. Influence problem is one of the central problems in the study of online social networks, the goal of which is to influence all nodes with the minimum number of seeds. However, in the real world, it might be too expensive to influence all nodes. In many cases, it is satisfactory to influence nodes only up to some percent p. In this paper, we study the minimum partial positive influence dominating set (MPPIDS) problem. In fact, we presented an approximation algorithm for a more general problem called minimum partial set multicover problem. As a consequence, the MPPIDS problem admits an approximation with performance ratio @math źH(Δ), where @math H(·) is the Harmonic number, @math ź=1 (1-(1-p)ź),źźΔ2 ź, and @math Δ,ź are the maximum degree and the minimum degree of the graph, respectively. For power-law graphs, we show that our algorithm has a constant performance ratio. We present a probabilistic algorithm that, given a connected graph G (represented by adjacency lists) of average degree d, with edge weights in the set 1,...,w , and given a parameter @math , estimates in time @math the weight of the minimum spanning tree (MST) of G with a relative error of at most @math . Note that the running time does not depend on the number of vertices in G. We also prove a nearly matching lower bound of @math on the probe and time complexity of any approximation algorithm for MST weight. The essential component of our algorithm is a procedure for estimating in time @math the number of connected components of an unweighted graph to within an additive error of @math . (This becomes @math for @math .) The time bound is shown to be tight up to within the @math factor. Our connected-components algorithm picks @math vertices in the graph and then grows "local spanning trees" whose sizes are specified by a stochastic process. From the local information collected in this way, the algorithm is able to infer, with high confidence, an estimate of the number of connected components. We then show how estimates on the number of components in various subgraphs of G can be used to estimate the weight of its MST. The problem of finding a minimum spanning tree connecting n points in a k-dimensional space is discussed under three common distance metrics -- Euclidean, rectilinear, and @math . By employing a subroutine that solves the post office problem, we show that, for fixed k @math 3, such a minimum spanning tree can be found in time O( @math ), where a(k) = @math . The bound can be improved to O( @math ) for points in the 3-dimensional Euclidean space. We also obtain o( @math ) algorithms for finding a farthest pair in a set of n points and for other related problems. We consider the distributed construction of a minimum weight 2- edge-connected spanning subgraph (2-ECSS) of a given weighted or unweighted graph. A 2-ECSS of a graph is a subgraph that, for each pair of vertices, contains at least two edge-disjoint paths connecting these vertices. The problem of finding a minimum weight 2-ECSS is NP-hard and a natural extension of the distributed MST construction problem, one of the most fundamental problems in the area of distributed computation. We present a distributed 3 2-approximation algorithm for the unweighted 2-ECSS construction problem that requires O(n) communication rounds and O(m) messages. Moreover, we present a distributed 3-approximation algorithm for the weighted 2-ECSS construction problem that requires O(nlogn) communication rounds and O(nlog2 n+m) messages. An efficient heuristic is presented for the problem of finding a minimum-size k-connected spanning subgraph of a given (undirected or directed) graph G=(V,E). There are four versions of the problem, depending on whether G is undirected or directed, and whether the spanning subgraph is required to be k-node connected (k-NCSS) or k-edge connected (k-ECSS). The approximation guarantees are as follows: min-size k-NCSS of an undirected graph 1+[1 k], min-size k-NCSS of a directed graph 1+[1 k], min-size k-ECSS of an undirected graph 1+[7 k], & min-size k-ECSS of a directed graph 1+[4 spl radic k]. The heuristic is based on a subroutine for the degree-constrained subgraph (b-matching) problem. It is simple, deterministic, and runs in time O(k|E| sup 2 ). For undirected graphs and k=2, a (deterministic) parallel NC version of the heuristic finds a 2-node connected (or a-edge connected) spanning subgraph whose size is within a factor of (1.5+ spl epsiv ) of minimum, where spl epsiv >0 is a constant. In the survivable network design problem (SNDP), the goal is to find a minimum-cost spanning subgraph satisfying certain connectivity requirements. We study the vertex-connectivity variant of SNDP in which the input specifies, for each pair of vertices, a required number of vertex-disjoint paths connecting them. We give the first strong lower bound on the approximability of SNDP, showing that the problem admits no efficient @math ratio approximation for any fixed @math , unless @math . We show hardness of approximation results for some important special cases of SNDP, and we exhibit the first lower bound on the approximability of the related classical NP-hard problem of augmenting the connectivity of a graph using edges from a given set. We study network-design problems with multiple design objectives. In particular, we look at two cost measures to be minimized simultaneously: the total cost of the network and the maximum degree of any node in the network. Our main result can be roughly stated as follows: given an integer @math , we present approximation algorithms for a variety of network-design problems on an @math -node graph in which the degree of the output network is @math and the cost of this network is @math times that of the minimum-cost degree- @math -bounded network. These algorithms can handle costs on nodes as well as edges. Moreover, we can construct such networks so as to satisfy a variety of connectivity specifications including spanning trees, Steiner trees and generalized Steiner forests. The performance guarantee on the cost of the output network is nearly best-possible unless @math . We also address the special case in which the costs obey the triangle inequality. In this case, we obtain a polynomial-time approximation algorithm with a stronger performance guarantee. Given a bound @math on the degree, the algorithm finds a degree- @math -bounded network of cost at most a constant time optimum. There is no algorithm that does as well in the absence of triangle inequality unless @math . We also show that in the case of spanning networks, we can simultaneously approximate within a constant factor yet another objective: the maximum cost of any edge in the network, also called the bottleneck cost of the network. We extend our algorithms to find TSP tours and @math -connected spanning networks for any fixed @math that simultaneously approximate all these three cost measures. In this paper we study the problem of computing subgraphs of a certain configuration in a given topological graph G such that the number of crossings in the subgraph is minimum. The configurations that we consider are spanning trees, s-t paths, cycles, matchings, and @k-factors for @[email protected]? 1,2 . We show that it is NP-hard to approximate the minimum number of crossings for these configurations within a factor of k^1^-^@e for any @e>0, where k is the number of crossings in G. We then give a simple fixed-parameter algorithm that tests in O^@?(2^k) time whether G has a crossing-free configuration for any of the above, where the O^@?-notation neglects polynomial terms. For some configurations we have faster algorithms. The respective running times are O^@?(1.9999992^k) for spanning trees and O^@?((3)^k) for s-t paths and cycles. For spanning trees we also have an O^@?(1.968^k)-time Monte-Carlo algorithm. Each O^@?(@b^k)-time decision algorithm can be turned into an O^@?((@b+1)^k)-time optimization algorithm that computes a configuration with the minimum number of crossings. We present the best known algorithms for approximating the minimum-size undirected @math -edge connected spanning subgraph. For simple graphs our approximation ratio is @math . The more precise version of this bound requires @math , and for all such @math it improves the long-standing performance ratio of Cheriyan and Thurimella [SIAM J. Comput., 30 (2000), pp. 528-560], @math . The improvement comes in two steps. First we show that for simple @math -edge connected graphs, any laminar family of degree @math sets is smaller than the general bound ( @math versus @math ). This immediately implies that iterated rounding improves the performance ratio of Cheriyan and Thurimella. The second step carefully chooses good edges for rounding. For multigraphs our approximation ratio is @math . This improves the previous ratio @math [H. N. Gabow, M. X. Goemans, E. Tardos, and D. P. Williamson, Networks, 53 (2009), pp. 345-357]. It is of interest since it is known that for some constant @math , an approximation ratio @math implies @math . Our approximation ratio extends to the minimum-size Steiner network problem, where @math denotes the average vertex demand. The algorithm exploits rounding properties of the first two linear programs in iterated rounding. Abstract The problem of constructing a spanning tree for a graph G = ( V , E ) with n vertices whose maximal degree is the smallest among all spanning trees of G is considered. This problem is easily shown to be NP-hard. In the Steiner version of this problem, along with the input graph, a set of distinguished vertices D ⊆ V is given. A minimum-degree Steiner tree is a tree of minimum degree which spans at least the set D . Iterative polynomial time approximation algorithms for the problems are given. The algorithms compute trees whose maximal degree is at most Δ* + 1, where Δ* is the degree of some optimal tree for the respective problems. Unless P = NP, this is the best bound achievable in polynomial time. Completely independent spanning trees (T_1,T_2, ,T_k ) in a graph G are spanning trees in G such that for any pair of distinct vertices u and v, the k paths in the spanning trees between u and v mutually have no common edge and no common vertex except for u and v. The concept finds applications in fault-tolerant communication problems in a network. Recently, it was shown that Dirac’s condition for a graph to be hamiltonian is also a sufficient condition for a graph to have two completely independent spanning trees. In this paper, we generalize this result to three or more completely independent spanning trees. Namely, we show that for any graph G with (n 7 ) vertices, if the minimum degree of a vertex in G is at least (n-k ), where (3 k n 2 ), then there are ( n k ) completely independent spanning trees in G. Besides, we improve the lower bound of ( n 2 ) on the Dirac’s condition for completely independent spanning trees to ( n-1 2 ) except for some specific graph. Our results are theoretical ones, since these minimum degree conditions can be applied only to a very dense graph. We then present constructions of symmetric regular graphs which include optimal graphs with respect to the number of completely independent spanning trees. We present an @math -approximation algorithm for the problem of finding a @math -vertex connected spanning subgraph of minimum cost, where @math is the number of vertices in an input graph, and @math is a connectivity requirement. Our algorithm is the first that achieves a polylogarithmic approximation ratio for all values of @math and @math , and it works for both directed and undirected graphs. As in previous works, we use the Frank--Tardos algorithm for finding @math -outconnected subgraphs as a subroutine. However, with our structural lemmas, we are able to show that we need only partial solutions returned by the Frank--Tardos algorithm; thus, we can avoid paying the whole cost of an optimal solution every time the algorithm is applied. We give a ( 17 12 ) -approximation algorithm for the following NP- hard problem: Given a simple undirected graph, find a 2-edge connected span- ning subgraph that has the minimum number of edges. The best previous approximation guarantee was ( 3 2 ) . If the well known TSP ( 4 3 ) conjecture holds, then there is a ( 4 3 ) -approximation algorithm. Thus our main result gets half-way to this target.