On the Optimum Requirement Graph Problem
Bang Ye Wu
Dept. of Computer Science and Information Engineering, Shu-Te Univeristy,
YenChau, KaoShiung, Taiwan 824, R.O.C. Email: bangye@mail.stu.edu.tw
Kun–Mao Chao
Dept. of Life Science, National Yang-Ming University,
Taipei , Taiwan 112 , R.O.C. Email: kmchao@ym.edu.tw
Chuan Yi Tang
Dept. of Computer Science, National Tsing Hua University,
Hsinchu, Taiwan, R.O.C. Email: cytang@cs.nthu.edu.tw.
Abstract
Given a requirement aij for each pair of nodes i,j ∈ {1..n}, the optimum requirement graph prob-lem looks for a k-edge graph H such that the routing cost i<jaijdH(i, j) is minimum, where
dH(i, j) is the number of edges of the shortest path
between i and j on H. In this paper, we show that the problem is NP-hard and give a constant ratio approximation algorithm. Furthermore, we define a sufficient condition for the problem to be polynomial-time solvable.
1
Introduction
Finding spanning subgraphs of a given graph is a classical problem of network design. Typi-cally, we are given a nonnegative edge-weighted graph G. The weight on each edge represents the distance and reflects both the cost to install the link (building cost) and the cost to traverse it after the link is installed (routing cost). Let
G = ({1..n}, E, w) be an undirected graph, where w is a nonnegative edge weight function. The
building cost of G is w(G) = e∈Ew(e), and the routing cost ofG is c(G) = i<jaijdG(i, j), whereaij is the requirement between the two ver-tices, anddG(i, j) is the length of the shortest path between the two vertices. For the case where all the requirements are one, the routing cost of a graph is the sum of all distances in the graph, and was studied under various name in graph theory. For example, it was named as the transmission of a graph in [15].
When considering the building cost only, we are looking for the minimum weight spanning
sub-graph. The optimal solution is the minimum
weight spanning tree (MST) or forest of the graph. However, removing any edge from a graph will in-crease the routing cost unless such an edge can be replaced by a path with the same or smaller length. Therefore graphs with smaller routing cost might come up with larger building cost, and we often need to make a tradeoff between the two costs.
A large number of researchers have studied the problems of finding a spanning tree with minimum weight or minimum routing cost. Efficient polyno-mial time algorithms for the MST were developed (for example, see [4, 7]). Hu showed how to con-struct a spanning tree with minimum routing cost for two special cases in [10]. When the distances between every two vertices are the same, he gave a polynomial time algorithm to find the optimal solution. When all the requirements are the same, he gave a sufficient condition for the optimal tree to be a star. The optimal tree for a graph with identical requirements is referred as a minimum
routing cost spanning tree(MRT), or a shortest
to-tal path length spanning tree. In [8, 11], the MRT problem has been shown to be NP-hard , and 2-approximation algorithms were given in [5, 16]. Approximation algorithms with better error ratios were developed in [17], and a polynomial time
ap-proximation scheme(PTAS) for the MRT problem was presented in [18]. Some extensions of the MRT problem were also investigated. Given nonnega-tive weights of vertices, the product-requirement (or the sum-requirement) MRT problem assumes
Proc. of the 19th Workshop on Combinatorial Mathematics and Computation Theory
that the requirement between two vertices is the product (or the sum, respectively) of their weights. Constant ratio approximation algorithms for these two generalized problems were given in [19], and a PTAS for the product-requirement MRT was shown in [20].
In this paper, we consider the problems of find-ing spannfind-ing subgraphs with small buildfind-ing and routing costs. Instead of the general problem with arbitrary requirements and arbitrary distances, we shall focus on the special case that the distances
dG(i, j) are all equal to one, while the
require-mentsaij are arbitrary. We shall call the optimum graph in this case the optimum requirement graph (ORG). An example is illustrated in Figure 1.
Letn be the number of vertices and k the build-ing cost constraint. Ifk = n−1, the ORG problem is equivalent to the optimum requirement spanning
tree problem in [10], and can be optimally solved in polynomial time. Here we consider a more gen-eral case whenn − 1 ≤ k ≤ n(n − 1)/2 and obtain the following results.
• The NP-hardness of the ORG problem for
generalk.
• A sufficient condition such that the ORG can
be found in polynomial time. Interestingly, it includes the product requirement and sum requirement cases.
• An approximation algorithm with error ratio
1 + (n − 1)/k. It should be noted that this error ratio is no more than 2.
Several results for trees realizing tradeoffs be-tween weight and some distance requirements were studied before. In [12], Khuller et al showed that it is possible to construct a spanning tree balanc-ing the minimum spannbalanc-ing tree and the short-path tree of a specified node. The NP-hardness of find-ing the minimum diameter subgraph with budget constraint was established in [14], while a poly-nomial time algorithm for finding the minimum diameter spanning tree of a graph with arbitrary edge weights was given in [9]. Considerable work has been done on the spanner of a graph. In gen-eral, a t-spanner of G is a low-weight subgraph ofG such that, for any two vertices, the distance on the spanner is at mostt times the distance in
G. Some results of finding spanner of a weighted
graph can be found in [1]. Obviously, the spanners can be used to approximate the minimum rout-ing cost subgraph problem with arbitrary require-ments. But since the criteria for a spanner are often much stricter, we may often come up with
better results if we wish to minimize the routing cost only.
The remaining sections are organized as follows. Some basic assumptions are given in Section 2. The NP-hardness of the ORG problem is shown in Section 3, and the approximation algorithm and polynomial-time solvable cases are in Section 4.
2
Preliminaries
In this paper, a graph is a simple and undi-rected graph with unweighted edge. The require-ments are assumed to be nonnegative. Let G = (V, E) be a graph with nodes set V and edge set
E, and n be the number of vertices of G. The
distance dG(u, v) between vertices u and v in G is the number of edges of the shortest path in G between them. The weight of graphG is the total number of edges inG.
Definition 1. Let G = ({1..n}, E) be a graph
andaij be the requirement between verticesi and
j. The routing cost of a graph G is defined as c(G) =i<jaijdG(i, j).
Note that the summation is over all unordered
pairs of vertices, i.e. the routing cost between two vertices i and j is counted only once. In some other papers, the summation is over all pairs, and the routing cost is therefore exactly twice of that defined in this paper. Since we consider only undi-rected graph, the two definitions are equivalent. Also, the input requirements are assumed to be symmetry , i.e. aij = aji for all i and j, and only one of them appeared in the definition. If the input requirements aij is not symmetry, we can set bij =bji =aij +aji. Since the path is undi-rected, the routing costijaijdG(i, j) is the same as i<jbijdG(i, j).
Definition 2. Given a set of nonnegative re-quirements A = {aij|1 ≤ i < j ≤ n} and an integerk, the optimal requirement graph, denoted byORG, is a graph G = ({1..n}, E) such that the routing cost of G is minimum among all graphs withk edges, where n − 1 ≤ k ≤ n(n − 1)/2. If the vertex set can be partitioned into two sub-sets such that there is no requirement cross the two subsets, we may solve the problem individu-ally. Here we assume that there is at least one positive requirement for any cut.
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10
5
10
5
2
5
2
1
2
N
1N
2N
3N
4N
5N
6N
1N
2N
3N
4N
5N
6(a)The requirements among nodes
(b)The optimum requirement graph
Figure 1: An instance of the optimum requirement graph problem. (a)The input requirements among nodes. Requirements not shown are assumed to be zero. (b)The optimum graph fork = 6. The routing cost is (10 + 10 + 5 + 5 + 5 + 2)× 1 + (2 + 2 + 1) × 2 = 47. The distances between N1 andN3, N3and
N5,N5and N6are twos; while distances between node pairs with nonzero requirements are all ones.
3
The NP-hardness
In this section, we shall show that the ORG problem is NP-hard by reducing the 2-spanner problem to it. The definition of the 2-spanner problem is as follows:
Definition 3. Let G = (V, E) be a connected
graph. A subgraph H = (V, F ), F ⊂ E, of G
is a 2-spanner of G if dH(i, j) ≤ 2dG(i, j) for all verticesi and j.
Definition 4 . Given a graph G = (V, E), the minimum spanner problem is to find the 2-spanner with minimum number of edges. Given an integerk ≥ n − 1, the decision version of the min-imum spanner problem asks if there exists a 2-spanner with no more thank edges. The decision problem is referred to as the 2-spanner problem in this paper.
Theorem 1. The ORG problem is NP-hard even
for the case that all requirements are either 1 or 0.
Proof: We transform the 2-spanner problem to the ORG problem. Given an instance of the 2-spanner problem by the graph G = (V, E) and an integerk, we construct an instance of the ORG problem as follows. For any edge (i, j) ∈ E, we set the requirementaij= 1 andaij = 0 otherwise. The edge constraint of the ORG problem is set to
k. We claim that the ORG problem has a solution
with routing cost at most 2a∗− k if and only if there is a 2-spanner ofG with k edges, where a∗=
i<jaij is the total requirement.
Suppose that H = (V, F ) is a 2-spanner of
G and |F | = k. Consider H as a solution of
the ORG problem. Every edge in H corresponds to one requirement. There are k requirements are directly routed by one edge. Since H is a 2-spanner, the other requirements are routed by two edges. Consequently the routing cost ofH is
k + 2(a∗− k) = 2a∗− k. Conversely, suppose that
the ORG problem has a solution H with routing cost 2a∗− k. For any k-edge graph, there are at most k requirements can be routed by only one edge and the others needs at least two edges. The lower bound of the routing cost of a k-edge sub-graph is 2a∗−k, and such a routing cost is achieved only whenk requirements are routed by one edge and all others by two edges. Therefore every edge in H is also an edge in G and dH(i, j) = 2 for any edge (i, j) in G but not in H. It also implies that H is a 2-spanner of G. Since the 2-spanner problem is NP-complete [3, 13], by the above re-duction, the ORG problem is NP-hard.
4
Polynomial-time algorithms
In the following, we shall define a sufficient con-dition that the ORG problem can be solved in polynomial time.
Definition 5. For a set of requirements {aij},
a vertex m is a heavy vertex if aim ≥ aij for all verticesi and j.
Theorem 2. The optimal requirement graph problem can be solved optimally in O(n2) time if the input contains a heavy vertex.
Proc. of the 19th Workshop on Combinatorial Mathematics and Computation Theory
Proof: Without loss of the generality, we assume that vertex n is the heavy vertex. Let
Y = {ain|1 ≤ i < n} and X be the subset of
the largest k − (n − 1) requirements of the set
{aij|1 ≤ i < j ≤ n − 1}. Let a∗ be the total
requirement, and x and y be the total require-ment in X and Y respectively. For a graph with
k edges, there are only k requirements may be
routed directly and others will be routed by at least two edges. Since there is at least one edge incident to each vertex and n is the heavy ver-tex, for any graph G with k edges, the routing cost c(G) ≥ (x + y) + 2(a∗− (x + y)), and the equality holds when the requirement in X ∪ Y are routed directly and others are routed by ex-actly two edges. However, the graph with edge set
{(i, j)|aij ∈ X ∪ Y } achieves the lower bound of
the routing cost. Therefore such a graph is opti-mal and can be constructed in O(n2) time by a linear time algorithm [2] finding the k-th largest element in a set ofO(n2) numbers.
The next two corollaries can be proved simi-larly.
Corollary 3. Any graph with diameter 2 is an
optimal solution of the ORG problem with identi-cal requirements.
Corollary 4. Let A be a set of requirements over a vertex set {1..n} and A1 be the subset of the largest k requirements in A. Assume H = ({1..n}, E), where E = {(i, j)|aij ∈ A1}. If the
diameter ofH is 2, then H is the ORG.
Letqi be a nonnegative weight of vertex i. As defined in [19], requirements are called product
re-quirements(or sum requirements) ifaij =qiqj (or
aij =qi+qj respectively) for all verticesi and j.
Corollary 5. The ORG problem with product
requirements or sum-requirements can be solved inO(n log n + k) time.
Proof: Letqibe the given nonnegative weight of vertex i. Sort the vertices by their weights in nondecreasing order. We have qi ≤ qi+1 for all
i < n. The requirement matrix is a sorted matrix, i.e. all rows and columns are sorted. Obviously the vertexn is a heavy vertex. By theorem 2, all we need to do is to find the k − (n − 1) largest numbers in the set{qij|1 ≤ i < j < n}. Since it is a sorted matrix, the selection can be done inO(n) time by the algorithm in [6]. Then it takesO(k) time to report the selected requirements , and the time complexity isO(n log n + k).
We now turn to an approximation algorithm for the ORG problem with arbitrary requirements. The idea is to take a vertexm as if it was a heavy vertex. We connect all other vertices to it, i.e.
m is the center of the constructed graph, and the
remainingk−(n−1) edges are added to the vertex pairs with larger requirements. Since the diameter of the constructed graph is two, the cost is within twice of that of the optimum. The next theorem gives us a slightly better result for largek.
Theorem 6. The ORG problem with arbitrary
requirements can be approximated with error ratio 1 + (n − 1)/k in O(n2) time.
Proof: LetA1be the summation of the largest
k requirements and the remaining total
require-ment be A2. Obviously the routing cost of the optimal graph is no less than A1+ 2A2. LetA3 be the summation of the largest k − (n − 1) re-quirements. For the constructed graph, since its diameter is two and the largest k − (n − 1) re-quirements are routed directly, its routing cost is no more than A3 + 2(A1 − A3+A2). Since
A3 ≥ ((k − n + 1)/k)A1, the relative error ratio
is 1 + (n − 1)/k. The time complexity is domi-nated by the time to select the largestk − (n − 1) requirements. By a linear time algorithm for the selection problem, the time complexity is O(n2) since there areO(n2) requirements.
In fact, there exists a vertexm such that there are at least 2k/n of the largest k requirements incident to m. By choosing m as the center, we may get a little better ratio but it is not asymp-totically better.
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