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Approximation hardness of min–max tree covers
Operations Research Letters 38 (2010) 169–173
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Operations Research Letters journal homepage: www.elsevier.com/locate/orl
Approximation hardness of min–max tree covers Zhou Xu a,∗ , Qi Wen b a
Department of Logistics and Maritime Studies, The Hong Kong Polytechnic University, Hong Kong
b
Department of Management Sciences, The City University of Hong Kong, Hong Kong
article
info
Article history: Received 4 August 2009 Accepted 29 January 2010 Available online 10 February 2010 Keywords: Inapproximability bound Min–max Vehicle routing Tree covers
abstract We prove the first inapproximability bounds to study approximation hardness for a min–max k-tree cover problem and its variants. The problem is to find a set of k trees to cover vertices of a given graph with metric edge weights, so as to minimize the maximum total edge weight of any of the k trees. Our technique can also be applied to improve inapproximability bounds for min–max problems that use other covering objectives, such as stars, paths, and tours. © 2010 Elsevier B.V. All rights reserved.
1. Introduction This paper follows a growing body of research on the min–max k-tree cover problem [5] and its variants [1,8,2,9]. The problem is to find a set of k trees, for k ≥ 1, to cover all vertices in a given graph, whose edge weights form a metric, so as to minimize the maximum total edge weight of any of the k trees. It has many applications, especially in routing multiple vehicles to service customers when the latest service time is critical [4,5]. Depending on different applications, each of the k subtrees may have to contain a given root, or to contain a distinct root in a given set R of k roots. This leads to two rooted variants of the problem that have been studied in the literature, which are called the min–max rooted tree cover problem [8], and the min–max R-rooted tree cover problem [5]. In this paper, we present the first results on approximation hardness of the min–max k-tree cover problem and its rooted variants. Related work. The problems studied in this paper have been shown to be NP-hard [3,5]. Therefore, it is of great practical value to develop ρ -approximation algorithms for them, where ρ are some constant approximation ratios. [8] proposed a 3-approximation algorithm for the rooted tree cover problem. [5] developed 4approximation algorithms for the k-tree cover problem, and for the R-rooted tree cover problem. For some special cases, where the given graph is a tree, approximation ratios can be further improved [2,9].
∗
Corresponding author. E-mail address: [email protected] (Z. Xu).
0167-6377/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2010.02.004
Besides trees, other covering objects, such as stars, paths, and tours, have also been studied in the literature [1,5]. The earliest such work was [6] in the 1970s. It presented a 5/2-approximation algorithm for the k-traveling salesman problem (k-TSP), which is under the min–max objective to find a set of k tours to cover vertices of a given graph with each tour containing a given root. Moreover, it is very common in the literature [1,5] to approximate a path or a tour cover from a tree cover, because by doubling each edge of the k trees that cover all vertices, one can obtain k Eulerian subgraphs, which embed k paths or k tours that cover all vertices. In contrast to approximation algorithms, very few results on approximation hardness are known in the literature for problems mentioned above. It is only known that the k-TSP is at least as hard as one of its special cases, the classic traveling salesman problem (TSP), which cannot be approximated with a ratio less than 117/116 unless NP = P [10]. Notice that when k = 1, the min–max k-tree cover problem and its rooted variants are all reduced to the problem of finding a minimum spanning tree of a given graph, which is well known for its tractability. However, when k is an arbitrary positive integer, the existing best approximation ratios for these problems are either 3 or 4 [8,5]. Thus, questions about their inapproximability bounds, i.e., the lower bounds on their best possible approximation ratios, becomes interesting to answer. Our results and techniques. We have obtained the first inapproximability bounds for the min–max k-tree cover problem and its rooted variants. We prove that unless NP = P, the k-rooted tree cover problem and the min–max rooted tree cover problem cannot be approximated with a ratio less than 3/2, and the min–max rooted tree cover problem cannot be approximated with a ratio less than 10/9.
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Our technique to derive inapproximability bounds for min–max problems of tree covers is based on reductions from the problem of three dimensional matching (3DM), a well known NP-complete problem [7]. The technique can also be applied to derive inapproximability bounds for problems that use other covering objects. For example, we have derived the first inapproximability bounds of 3/2 for two problems studied in [1] which use paths and stars to cover vertices of a given graph. Furthermore, we have shown that unless NP = P, the k-TSP cannot be approximated with a ratio less than 20/17, which enhances its best known inapproximability bound of 117/116 by more than an order of magnitude. Organization. We formulate the min–max problems of k-tree cover, R-rooted tree cover, and rooted tree cover in Section 2. By reductions from the 3DM, we prove inapproximability bounds for these three problems, in Sections 3–5, respectively. Techniques used in these proofs are applied, in Section 6, to derive inapproximability bounds for other related problems, such as the k-TSP. 2. Problem formulations Let G = (V , E ) be a complete undirected graph on a vertex set V . Each edge e ∈ E is weighted by d(e), where d forms a metric, and so satisfies the triangle inequality. A subtree of G is an acyclic connected subgraph of G. For any edge subset Q ⊆ E, let d(Q ) = P e∈Q d(e) denote the total weight of edges in Q . For any subgraph H of G, let V (H ) and E (H ) denote its vertex and its edge sets. Thus, d(E (Q )) indicates the total edge weight of Q . For any set H of subgraphs of G, let cost (H ) denote the cost of H , which is defined in (1) as the maximum total edge weight of any subgraph in H : cost (H ) = max d(E (H )).
Fig. 1. A component for each Mi ∈ M , where Mi = (w, x, y), used to transform the 3DM to the k-TCP.
Finally, we introduce as follows the 3DM, a well known NPcomplete problem, which is extensively used in the remainder of this paper. Three Dimensional Matching (3DM) [7] Instance: (W , X , Y , n, M , m), where W , X and Y are disjoint sets with |W | = |X | = |Y | = n, and M = {Mi : 1 ≤ i ≤ m} is a subset of W × X × Y with |M | = m. Objective: to decide whether M contains an exact matching M 0 ⊆ M such that |M 0 | = n and no two elements of M0 agree in any coordinate. 3. Hardness of k-TCP
(1)
The following theorem states an inapproximability bound of 3/2 for the k-TCP.
The followings formulate the min–max k-tree cover problem and its rooted variants, and present their approximation ratios known in the literature, and our inapproximability bounds.
Theorem 1. Unless NP = P, there is no polynomial time (3/2 − )approximation algorithm for the k-TCP, for any > 0.
H ∈H
Min–max k-Tree Cover Problem (k-TCP) Instance: (G, k, d), where G = (V , E ) is a complete undirected graph, k is a positive integer, d : E → R+ is a metric function representing edge weights. Feasible solution: A set of k subtrees T = {T1 , T2 , . . . , Tk } such Sk that V = i=1 V (Ti ). Objective: minT cost (T ). Approximation ratio: 4. [5,1] Our inapproximability bound: 3/2. Min–max R-Rooted Tree Cover Problem (R-RTCP) Instance: (G, k, R, d), where G = (V , E ) is a complete undirected graph, R ⊆ V is a given set of k roots (with |R| = k), d : E → R+ is a metric function representing edge weights. Feasible solution: A set of k subtrees T = {T1 , T2 , . . . , Tk } such that each subtree Ti for 1 ≤ i ≤ k contains a distinct root in R, Sk and that V = i=1 V (Ti ). Objective: minT cost (T ). Approximation ratio: 4. [5] Our inapproximability bound: 3/2. Min–max Rooted Tree Cover Problem (RTCP) Instance: (G, k, r , d), where G = (V , E ) is a complete undirected graph, k is a positive integer, r ∈ V is a given root, d : E → R+ is a metric function representing edge weights. Feasible solution: A set of k subtrees T = {T1 , T2 , . . . , Tk } such Sk that Ti contains root r for all 1 ≤ i ≤ k, and that V = i=1 V (Ti ). Objective: minT cost (T ). Approximation ratio: 3. [8] Our inapproximability bound: 10/9.
Proof. Suppose there exists a (3/2 − )-approximation algorithm for the k-TCP with > 0. We are going to show that this algorithm can be used to solve the 3DM in polynomial time, which contradicts to NP 6= P. Given any 3DM instance (W , X , Y , n, M , m), consider a k-TCP instance I = (G, k, d), defined as follows. Let G = (V , E ) be a complete graph on V , where V = W ∪ X ∪ Y ∪m i=1 {qi,j : 1 ≤ j ≤ 9}. Sm Let e E = i =1 e Ei , where e Ei , for each element Mi ∈ M where Mi = (w, x, y), denotes the set of the 11 edges shown in Fig. 1. Accordingly, we define d(e) = 1 for each e ∈ e E, and d(e) = 2 for each e ∈ E \e E. It can be verified that d forms a metric. Finally, let k = 3m + n. Define F (2, 3) as the set of subtrees T of G such that |V (T )| = 3 and d(E (T )) ≤ 2. For each Mi ∈ M , where Mi = (w, x, y), consider the following nine subtrees, denoted by Ti,j for 1 ≤ j ≤ 9: E (Ti,1 ) = {(qi,1 , qi,2 ), (qi,2 , w)}, E (Ti,2 ) = {(x, qi,8 ), (qi,8 , qi,7 )}, E (Ti,3 ) = {(qi,4 , qi,5 ), (qi,5 , y)}, E (Ti,4 ) = {(qi,9 , qi,6 ), (qi,6 , qi,3 )}, E (Ti,5 ) = {(qi,1 , qi,2 ), (qi,2 , qi,3 )}, E (Ti,6 ) = {(qi,4 , qi,5 ), (qi,5 , qi,6 )}, E (Ti,7 ) = {(qi,9 , qi,8 ), (qi,8 , qi,7 )}, E (Ti,8 ) = {(qi,9 , qi,8 ), (qi,9 , qi,6 )}, E (Ti,9 ) = {(qi,2 , qi,3 ), (qi,3 , qi,6 )}. It can be seen that {Ti,j : 1 ≤ j ≤ 9} ⊆ F (2, 3). We are now going to show that the 3DM instance has an exact matching if and only if the (3/2 − )-approximation algorithm for the k-TCP returns a feasible solution to I with cost at most 2.
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On one hand, if the 3DM instance has an exact matching M 0 , then we construct a subtree set T by including subtrees Ti,j for 1 ≤ j ≤ 4, for all Mi ∈ M 0 , and including subtrees Ti,j for 5 ≤ j ≤ 7, for all Mi ∈ M \ M 0 . Thus, |T | = 4|M 0 | + 3(|M | − |M 0 |) = k. It is easy to verify that T covers all qi,j for 1 ≤ i ≤ m and 1 ≤ j ≤ 9. Moreover, for each w ∈ W , since M 0 is an exact matching, there must exist a unique element Mi ∈ M 0 such that Mi = (w, x, y) for some x ∈ X and y ∈ Y . Thus, {Ti,j : 1 ≤ j ≤ 4} ⊆ T . Since
w∈
j=1 V (Ti,j ), we have that T covers all vertices in W . Similarly, we can have that T covers all vertices in X and in Y . Thus, T covers all vertices in V . Notice that each subtree in T has a total edge weight of 2. As defined in (1), cost (T ) equals the maximum total edge weight of any subtree in T . We thus obtain that T is a feasible solution to I with cost (T ) = 2. Therefore, since the edge weights are all integers, and since > 0, the (3/2 − )-approximation algorithm must return a feasible solution to I with cost at most 2. On the other hand, if the (3/2 −)-approximation algorithm returns a feasible solution T to the k-TCP instance I with cost (T ) ≤ 2, then we define M 0 by including all elements Mi ∈ M that have Ti,3 ∈ T , where 1 ≤ i ≤ m. To prove that M 0 is an exact matching for the 3DM instance, consider each subtree T ∈ T . We know d(E (T )) ≤ 2. Since d(e) ≥ 1 for all e ∈ E, we obtain |E (T )| ≤ 2, implying |V (T )| ≤ 3. Since T covers all vertices in V , and since |V | = 9m + 3n and k = 3m + n, we obtain |V (T )| = 3, and d(e) = 1 for all e ∈ E (T ), which implies T ∈ F (2, 3). Thus, T ⊆ F (2, 3), and no two different subtrees in T share the same vertex in V . Consider each element Mi ∈ M , where Mi = (w, x, y). If Mi ∈ M 0 , then Ti,3 ∈ T . Since Ti,4 , Ti,8 , and Ti,9 are the only subtrees in F (2, 3) that cover qi,6 but no vertex in V (Ti,3 ), exact one of them must be in T . However, Ti,9 cannot be in T , because qi,1 cannot be covered by any subtree in F (2, 3) that covers no vertex in V (Ti,3 ) ∪ V (Ti,9 ). By a similar argument, Ti,8 cannot be in T , because otherwise qi,7 cannot be covered by T . Thus, we obtain Ti,4 ∈ T . Since Ti,1 (or Ti,2 ) is the only subtree in F (2, 3) that covers qi,1 (or qi,7 , respectively) but no vertex in E (Ti,3 ) ∪ E (Ti,4 ), we obtain Ti,1 ∈ T and Ti,2 ∈ T . Thus, {Ti,j : 1 ≤ j ≤ 4} ⊆ T . Since no two different subtrees in T share the same vertex in V , no elements of M 0 \ {Mi } can match w , x, or y, which implies that no two elements of M 0 can agree in any coordinate. Otherwise, Mi ∈ M \ M 0 , which implies Ti,3 6∈ T . Since Ti,6 is the only subtree in F (2, 3) \ {Ti,3 } that covers qi,4 , we obtain Ti,6 ∈ T . Notice that Ti,2 and Ti,7 are the only subtrees in F (2, 3) that cover qi,7 . However, Ti,2 cannot be in T , because qi,9 cannot be covered by any subtree in F (2, 3) that contains no vertex in V (Ti,6 ) ∪ V (Ti,2 ). We obtain Ti,7 ∈ T . Similarly, we can obtain Ti,5 ∈ T to cover qi,1 and qi,3 . Therefore, {Ti,j : 5 ≤ j ≤ 7} ⊆ T . Hence, 4|M 0 | + 3(|M | − |M 0 |) ≤ |T |, implying |M 0 | ≤ |T | − 3|M | = k − 3m = n. Moreover, for each element y ∈ Y , let T (y) denote the subtree in T that covers vertex y. We know d(e) = 1 for all e ∈ E (T (y)), and |V (T (y))| = 3. Let e(y) denote an edge in T (y) that is incident on y. Since d(e(y)) = 1, from Fig. 1 we know that e(y) must be incident on qi,5 for an element Mi ∈ M with Mi = (w, x, y) for some w ∈ W and x ∈ X . Therefore, Mi ∈ M0 , because otherwise, we know {Ti,j : 5 ≤ j ≤ 7} ⊆ T , which implies that two different subtrees in T , i.e., T (y) (which contains y), and Ti,6 (which does not contain y), share the same vertex qi,5 , leading to a contradiction. Hence, |M 0 | ≥ |Y | = n, implying |M 0 | = n. Therefore, M 0 is an exact matching for the 3DM instance.
S4
4. Hardness of R-RTCP The following theorem states an inapproximability bound of 3/2 for the R-RTCP.
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Theorem 2. Unless NP = P, there is no polynomial time (3/2 − )approximation algorithm for the R-RTCP, for any > 0. Proof. The proof is similar to that of Theorem 1, by reducing any 3DM instance, (W , X , Y , n, M , m), to an R-RTCP instance, I = (S G, k, R, d), where G and d are defined the same way, and R = X ∪ m i=1 {qi,1 , qi,4 , qi,9 } is defined as the root set with k = |R| = 3m+n. (See Fig. 1.) We then redefine F (2, 3) as the set of subtrees T of G such that |V (T )| = 3 and d(E (T )) ≤ 2, and that T contains a root in R. Notice that for each element Mi ∈ M , every subtree E (Ti,j ), defined in the proof of Theorem 1 for 1 ≤ j ≤ 8, contains a root in R. Thus, by following an argument similar to that in the proof of Theorem 1, we can obtain that the 3DM instance has an exact matching if and only if the (3/2 − )-approximation algorithm for the R-RTCP, if it exists, returns a feasible solution to I with cost at most 2. This contradicts to NP 6= P. 5. Hardness of RTCP The following theorem states an inapproximability bound of 10/9 for the RTCP. Theorem 3. Unless NP = P, there is no polynomial time (10/9 − )approximation algorithm for the RTCP, for any > 0. Proof. Suppose there exists a (10/9 −)-approximation algorithm for the RTCP with > 0. We are going to show that this algorithm can be used to solve the 3DM in polynomial time, which contradicts to NP 6= P. Given any 3DM instance (W , X , Y , n, M , m), consider an RTCP instance I = (G, k, r , d), defined as follows. Let r be the given root. Let G = (V , E ) be a complete Sm undirected graph on V , where V = {r } ∪ W ∪ X ∪ Y ∪ i=1 {qi,j : 1 ≤ j ≤ 6}. Let e E = Sm e e E , where E , for each element M ∈ M where M = (w, x , y), i i i i i =1 denotes the set of the 11 edges shown in Fig. 2. Accordingly, for each Mi ∈ M , where Mi = (w, x, y), we define d(x, qi,3 ) = 3, and d(r , x) = d(r , qi,1 ) = d(r , qi,4 ) = 4, and define the edge weights of remainders in e Ei to be 2. Moreover, for each (u, v) ∈ E \e E, define d(u, v) as the total edge weight of the shortest path from u to v in the subgraph (V , e E ). Finally, let k = 2m + n. It can be verified that d forms a metric. Due to the triangle inequality, we have d(e) ≥ 2 for all e ∈ E, and d(e) ≥ 4 for each e ∈ E \ e E. Moreover, we have d(r , v) ≥ 6 for each v ∈ V \ {r } \ X \ {qi,1 , qi,4 : 1 ≤ i ≤ m}. Define F (4, 9) as the set of subtrees T of G such that |V (T )| = 4 and d(E (T )) ≤ 9, and that T contains the root r. Consider each subtree T ∈ F (4, 9). We can denote V (T ) by {r , v1 , v2 , v3 }, where r and vj for 1 ≤ j ≤ 3 are four different vertices in V . Thus, T must contain at least one edge incident on r, which can be denoted by (r , v1 ) without loss of generality. Since |V (T )| = 4, we have |E (T )| = 3. Since d(e) ≥ 2 for all e ∈ E, and since d(E (T )) ≤ 9, subtree T can contain at most one edge that has a weight greater than or equal to 4, and no edge that has a weight greater than or equal to 6. Therefore, since d(r , v) = 4 for each v ∈ X ∪ {qi,1 , qi,4 : 1 ≤ i ≤ m}, and since d(r , v) ≥ 6 for each v ∈ V \ {r } \ X \ {qi,1 , qi,4 : 1 ≤ i ≤ m}, we obtain d(r , v1 ) = 4 and (r , v1 ) ∈ e E. Thus, no edge in E (T ) \ {(r , v1 )} can be incident on r. Moreover, since d(e) ≥ 4 for each e ∈ E \ e E, subtree T cannot contain any edge in E \ e E. Since d(E (T )) ≤ 9, d(r , v1 ) = 4, |E (T )| = 3, and d(e) ≥ 2 for all e ∈ E, we obtain that E (T ) \ {(r , v1 )} can contain at most one edge that has a weight greater than or equal to 3. Accordingly, it can be verified (in Fig. 2)
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6. Applications
Fig. 2. A component for each Mi ∈ M , where Mi = (w, x, y), used to transform the 3DM to the RTCP.
that T must be one of the following six subtrees, Ti,j for 1 ≤ j ≤ 6, which all contain r: E (Ti,1 ) = {(r , qi,1 ), (qi,1 , qi,2 ), (qi,2 , w)}, E (Ti,2 ) = {(r , x), (x, qi,3 ), (qi,3 , qi,6 )}, E (Ti,3 ) = {(r , qi,4 ), (qi,4 , qi,5 ), (qi,5 , y)}, E (Ti,4 ) = {(r , qi,1 ), (qi,1 , qi,2 ), (qi,2 , qi,3 )}, E (Ti,5 ) = {(r , qi,4 ), (qi,4 , qi,5 ), (qi,5 , qi,6 )}, E (Ti,6 ) = {(r , x), (x, qi,3 ), (qi,3 , qi,2 )}. Moreover, since each subtree Ti,j for 1 ≤ i ≤ m and 1 ≤ j ≤ 6 is in F (4, 9), we obtain F (4, 9) = {Ti,j : 1 ≤ i ≤ m, 1 ≤ j ≤ 6}. We are now going to show that the 3DM instance has an exact matching if and only if the (10/9 −)-approximation algorithm for the RTCP returns a feasible solution to I with cost at most 9. On one hand, if the 3DM instance has an exact matching M 0 , then we construct a subtree set T by including subtrees Ti,j for 1 ≤ j ≤ 3, for all Mi ∈ M 0 , and including subtrees Ti,j for 4 ≤ j ≤ 5, for all Mi ∈ M \ M 0 . Thus, |T | = 3|M 0 | + 2(|M | − |M 0 |) = k. Similarly to the proof of Theorem 1, we can obtain that T covers all vertices in V . Since each subtree in T contains r, and has a total edge weight at most 9, we obtain that T is a feasible solution to I with cost (T ) ≤ 9 due to (1). Therefore, since the edge weights are all integers, and since > 0, the (10/9 − )-approximation algorithm must return a feasible solution to I with cost at most 9. On the other hand, if the (10/9 − )-approximation algorithm returns a feasible solution T to the RTCP instance I with cost (T ) ≤ 9, then we construct M 0 by including all elements Mi ∈ M that have Ti,3 ∈ T , where 1 ≤ i ≤ m. To prove that M 0 is an exact matching for the 3DM instance, consider each subtree T ∈ T . We know r ∈ V (T ) and d(E (T )) ≤ 9. Since d(r , v) ≥ 4 for all v ∈ V , and since d(e) ≥ 2 for all e ∈ E, we obtain |E (T )| ≤ 3, which implies |V (T ) \ {r }| ≤ 3. Thus, noticing |V \ {r }| = 6m + 3n and k = 2m + n, we obtain |V (T ) \ {r }| = 3, implying T ∈ F (4, 9). Thus, T ⊆ F (4, 9), and no two different subtrees in T can share the same vertex in V \ {r }. Consider each element Mi ∈ M , where Mi = (w, x, y). If Mi ∈ M 0 , then Ti,3 ∈ T . Since Ti,2 is the only subtree in F (4, 9) that covers qi,6 but no vertex in V (Ti,3 ), we obtain Ti,2 ∈ T . Since Ti,1 is the only subtree in F (4, 9) that covers qi,1 but no vertex in V (Ti,2 ) ∪ V (Ti,3 ), we obtain Ti,1 ∈ T , and so {Ti,j : 1 ≤ j ≤ 3} ⊆ T . Since no two different subtrees in T share the same vertex in V \{r }, no elements of M 0 \ {Mi } can match w , x, or y, which implies that no two elements of M 0 can agree in any coordinate. Otherwise, Mi ∈ M \ M 0 , which implies Ti,3 6∈ T . Since Ti,5 is the only subtree in F (4, 9) \ {Ti,3 } that covers qi,4 , we obtain Ti,5 ∈ T . Notice that Ti,4 and Ti,6 are the only two subtrees in F (4, 9) that cover qi,3 but no vertex in V (Ti,5 ). However, Ti,6 cannot be in T , because qi,1 cannot be covered by any subtree in F (4, 9) that contains no vertex in V (Ti,5 ) ∪ V (Ti,6 ). Hence, Ti,4 ∈ T , and so {Ti,j : 4 ≤ j ≤ 5} ⊆ T . Thus, 3|M 0 | + 2(|M | − |M 0 |) ≤ |T |, implying |M 0 | ≤ n. Moreover, by following an argument similar to that in the proof of Theorem 1, we can obtain |M 0 | ≥ n. Hence |M 0 | = n, and so M 0 is an exact matching for the 3DM instance.
Techniques, used in reducing the 3DM to the min–max k-TCP and its rooted variants, can be applied to derive inapproximability bounds for min–max problems that use other covering objects. For example, given any instance of the min–max k-TCP, one can use stars or paths (instead of trees) to cover vertices of the given graph. This leads to the min–max k-star, and the min–max k-path problems, for which [1] proved a bicriteria approximation ratio of (3, 3), and an approximation ratio of 4, respectively. The following theorem states their first inapproximability bounds of 3/2. Theorem 4. Unless NP = P, there is no polynomial time (3/2 − )approximation algorithm for the min–max k-star cover problem or for the min–max k-path cover problem, for any > 0. Proof. The proof of Theorem 1 for the k-TCP is applicable to Theorem 4 for the min–max star cover and the min–max path cover problems. This is because for any subgraph H of the given graph G, to say that H is a tree of three vertices, it is equivalent to saying that H is a path of three vertices, and is equivalent to saying that H is a star of three vertices. Therefore, all of the subtrees in F (2, 3), including {Ti,j : 1 ≤ i ≤ m, 1 ≤ j ≤ 9}, which are used in the proof of Theorem 1, are paths and stars as well. Accordingly, it can be directly verified that the arguments in the proof of Theorem 1 for the k-TCP, which are based on F (2, 3) and {Ti,j : 1 ≤ i ≤ m, 1 ≤ j ≤ 9}, are also valid for problems of path cover and star cover. Thus, we can obtain that the (3/2 − )approximation algorithm for the min–max star cover problem or for the min–max path cover problem, if it exists, can be used to solve the 3DM in polynomial time, contradicting to NP 6= P. Finally, we derive an inapproximability bound of 20/17 as follows for the k-TSP, which significantly improves its best bound of 117/116 known in the literature [10]. Given any instance of the RTCP, the k-TSP is to determine k tours to cover vertices of the given graph under the min–max objective. Its best known approximation ratio is 5/2 [6]. Theorem 5. Unless NP = P, there is no polynomial time (20/17−)approximation algorithm for the k-TSP, for any > 0. Proof. Suppose there exists a (20/17 − )-approximation algorithm for the k-TSP with > 0. We are going to show that this algorithm can be used to solve the 3DM in polynomial time, which contradicts to NP 6= P. Given any 3DM instance (W , X , Y , n, M , m), consider a k-TSP instance I = (G, k, r , d) as follows, where G = (V , E ), r, and k are the same as that of the RTCP Sm instance used in the proof of Theorem 3. Let e E still denote i=1 e Ei , however, e Ei here, for each element Mi ∈ M where Mi = (w, x, y), represents the set of the 15 edges shown in Fig. 3. For each Mi ∈ M , where Mi = (w, x, y), we define d(r , qi,3 ) = d(x, qi,3 ) = 5, and define the edge weights of remainders in e Ei to be 4. For each (u, v) ∈ E \ e E, we define d(u, v) as the total edge weight of the shortest path from u to v in the subgraph (V , e E ). Finally, we let k = 2m + n. It can be easily verified that d forms a metric. Due to the triangle inequality, we have d(e) ≥ 4 for all e ∈ E, and d(u, v) ≥ 8 for each (u, v) ∈ E \ e E. Define F (4, 19) as the set of tours C such that |V (C )| = 4 and d(E (C )) ≤ 19, and that C is a subgraph of G and contains r. Consider each tour C ∈ F (4, 19). Since |V (C )| = 4, we can denote C by (r v1 v2 v3 r ), where r and vj for 1 ≤ j ≤ 3 are four different vertices in V . Therefore, |E (C )| = 4. Since d(e) ≥ 4 for all e ∈ E, and since d(E (C )) ≤ 19, tour C can contain at most 3 edges that all have a weight greater than or equal to 5, and contain no edge that has a weight greater than or equal to 8, implying E (C ) ⊆ e E. Moreover, since r ∈ V (C ), |V (C )| = 4, and d(E (C )) ≤ 19, from Fig. 3 we
Z. Xu, Q. Wen / Operations Research Letters 38 (2010) 169–173
Fig. 3. A component for each Mi ∈ M , where Mi = (w, x, y), used to transform the 3DM to the k-TSP.
can see that C cannot contain two or three edges that all have a weight greater than or equal to 5. Thus, C either consists of four edges with each having a weight equal to 4, or consists of three edges with each having a weight equal to 4 and one edge having a weight equal to 5. Accordingly, it can be verified (in Fig. 3) that C must be one of the following five tours, Ci,j for 1 ≤ j ≤ 7, which all contain r: Ci,1 = (rqi,1 qi,2 w r ),
Ci,2 = (rxqi,3 qi,6 r ),
Ci,3 = (rqi,4 qi,5 yr ),
Ci,4 = (rqi,1 qi,2 qi,3 r ),
Ci,5 = (rqi,4 qi,5 qi,6 r ),
Ci,6 = (rqi,2 qi,3 w r ),
Ci,7 = (rqi,6 qi,5 yr ). Moreover, since each tour Ci,j , for 1 ≤ i ≤ m and 1 ≤ j ≤ 7, is in F (4, 19), we obtain F (4, 19) = {Ci,j : 1 ≤ i ≤ m, 1 ≤ j ≤ 7}. We are now going to show that the 3DM instance has an exact matching if and only if the (20/17 − )-approximation algorithm for the k-TSP returns a feasible solution to I with cost at most 19. On one hand, if the 3DM instance has an exact matching M 0 , then we construct a tour set C , by including Ci,j for 1 ≤ j ≤ 3, for all Mi ∈ M 0 , and including Ci,j for 4 ≤ j ≤ 5, for all Mi ∈ M \ M 0 . Thus, |C | = 3|M 0 | + 2(|M | − |M 0 |) = k. Similarly to the proof of Theorem 3, we can obtain that C covers all vertices in V . Since each tour in C contains r, and has a total edge weight at most 17, we obtain that C is a feasible solution to I with cost (C ) ≤ 17 due to (1). Therefore, since the edge weights are all integers, and since > 0, the (20/17 − )-approximation algorithm must return a feasible solution to I with cost at most 19. On the other hand, if the (20/17 − )-approximation algorithm returns a feasible solution C to the k-TSP instance I with cost (C ) ≤
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19, then we construct M 0 by including all elements Mi ∈ M that have Ci,3 ∈ C , where 1 ≤ i ≤ m. To prove that M 0 is an exact matching for the 3DM instance, consider each tour C ∈ C . We know r ∈ V (C ) and d(E (C )) ≤ 19. Since d(e) ≥ 4 for all e ∈ E, we obtain |E (C )| ≤ 4, implying |V (C ) \ {r }| ≤ 3. Thus, from |V \ {r }| = 6m + 3n and k = 2m + n, we obtain |V (C ) \ {r }| = 3, implying C ∈ F (4, 19). Thus, C ⊆ F (4, 19), and no two different tours in C can share the same vertex in V \ {r }. Accordingly, by following an argument similar to that in the proof of Theorem 3, and using F (4, 19) (instead of F (4, 9)), and Ci,j (instead of Ti,j ), we can obtain that no two elements of M 0 can agree in any coordinate, and that |M 0 | = n. Therefore, M 0 is an exact matching for the 3DM instance. Acknowledgements The authors are grateful to the anonymous referees for their valuable comments. They are also grateful to Mr. Liang Xu and Mr. Xiaofan Lai for their constructive discussions. This work is partially supported by the Internal Competitive Research Grant A-PD0W and the Niche Area Grant J-BB7C of the Hong Kong Polytechnic University. References [1] E.M. Arkin, R. Hassin, A. Levin, Approximations for minimum and min–max vehicle routing problems, Journal of Algorithms 59 (1) (2006) 1–18. [2] I. Averbakh, O. Berman, (p − 1)/(p + 1)-approximate algorithms for ptraveling salesmen problems on a tree with minmax objective, Discrete Applied Mathematics 75 (3) (1997) 201–216. [3] I. Averbakh, O. Berman, A heuristic with worst-case analysis for minimax routing of two travelling salesmen on a tree, Discrete Applied Mathematics 68 (1–2) (1996) 17–32. [4] A.M. Campbell, D. Vandenbussche, W. Hermann, Routing for relief efforts, Transportation Science 42 (2) (2008) 127–145. [5] G. Even, N. Garg, J. Könemann, R. Ravi, A. Sinha, Minmax tree covers of graphs, Operations Research Letters 32 (4) (2004) 309–315. [6] G.N. Frederickson, M.S. Hecht, C.E. Kim, Approximation algorithms for some routing problems, SIAM Journal on Computing 7 (1978) 178–193. [7] Michael R. Garey, David S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, WH Freeman & Co, New York, USA, 1979. [8] H. Nagamochi, Approximating the minmax rooted-subtree cover problem, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E88-A (5) (2005) 1335–1338. [9] K. Okada, H. Nagamochi, Polynomial time 2-approximation algorithms for the minmax subtree cover problem, Lecture Notes in Computer Science 2906 (2003) 138–147. [10] C.H. Papadimitriou, S. Vempala, On the approximability of the traveling salesman problem, Combinatorica 26 (1) (2006) 101–120.
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Operations Research Letters journal homepage: www.elsevier.com/locate/orl
Approximation hardness of min–max tree covers Zhou Xu a,∗ , Qi Wen b a
Department of Logistics and Maritime Studies, The Hong Kong Polytechnic University, Hong Kong
b
Department of Management Sciences, The City University of Hong Kong, Hong Kong
article
info
Article history: Received 4 August 2009 Accepted 29 January 2010 Available online 10 February 2010 Keywords: Inapproximability bound Min–max Vehicle routing Tree covers
abstract We prove the first inapproximability bounds to study approximation hardness for a min–max k-tree cover problem and its variants. The problem is to find a set of k trees to cover vertices of a given graph with metric edge weights, so as to minimize the maximum total edge weight of any of the k trees. Our technique can also be applied to improve inapproximability bounds for min–max problems that use other covering objectives, such as stars, paths, and tours. © 2010 Elsevier B.V. All rights reserved.
1. Introduction This paper follows a growing body of research on the min–max k-tree cover problem [5] and its variants [1,8,2,9]. The problem is to find a set of k trees, for k ≥ 1, to cover all vertices in a given graph, whose edge weights form a metric, so as to minimize the maximum total edge weight of any of the k trees. It has many applications, especially in routing multiple vehicles to service customers when the latest service time is critical [4,5]. Depending on different applications, each of the k subtrees may have to contain a given root, or to contain a distinct root in a given set R of k roots. This leads to two rooted variants of the problem that have been studied in the literature, which are called the min–max rooted tree cover problem [8], and the min–max R-rooted tree cover problem [5]. In this paper, we present the first results on approximation hardness of the min–max k-tree cover problem and its rooted variants. Related work. The problems studied in this paper have been shown to be NP-hard [3,5]. Therefore, it is of great practical value to develop ρ -approximation algorithms for them, where ρ are some constant approximation ratios. [8] proposed a 3-approximation algorithm for the rooted tree cover problem. [5] developed 4approximation algorithms for the k-tree cover problem, and for the R-rooted tree cover problem. For some special cases, where the given graph is a tree, approximation ratios can be further improved [2,9].
∗
Corresponding author. E-mail address: [email protected] (Z. Xu).
0167-6377/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2010.02.004
Besides trees, other covering objects, such as stars, paths, and tours, have also been studied in the literature [1,5]. The earliest such work was [6] in the 1970s. It presented a 5/2-approximation algorithm for the k-traveling salesman problem (k-TSP), which is under the min–max objective to find a set of k tours to cover vertices of a given graph with each tour containing a given root. Moreover, it is very common in the literature [1,5] to approximate a path or a tour cover from a tree cover, because by doubling each edge of the k trees that cover all vertices, one can obtain k Eulerian subgraphs, which embed k paths or k tours that cover all vertices. In contrast to approximation algorithms, very few results on approximation hardness are known in the literature for problems mentioned above. It is only known that the k-TSP is at least as hard as one of its special cases, the classic traveling salesman problem (TSP), which cannot be approximated with a ratio less than 117/116 unless NP = P [10]. Notice that when k = 1, the min–max k-tree cover problem and its rooted variants are all reduced to the problem of finding a minimum spanning tree of a given graph, which is well known for its tractability. However, when k is an arbitrary positive integer, the existing best approximation ratios for these problems are either 3 or 4 [8,5]. Thus, questions about their inapproximability bounds, i.e., the lower bounds on their best possible approximation ratios, becomes interesting to answer. Our results and techniques. We have obtained the first inapproximability bounds for the min–max k-tree cover problem and its rooted variants. We prove that unless NP = P, the k-rooted tree cover problem and the min–max rooted tree cover problem cannot be approximated with a ratio less than 3/2, and the min–max rooted tree cover problem cannot be approximated with a ratio less than 10/9.
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Our technique to derive inapproximability bounds for min–max problems of tree covers is based on reductions from the problem of three dimensional matching (3DM), a well known NP-complete problem [7]. The technique can also be applied to derive inapproximability bounds for problems that use other covering objects. For example, we have derived the first inapproximability bounds of 3/2 for two problems studied in [1] which use paths and stars to cover vertices of a given graph. Furthermore, we have shown that unless NP = P, the k-TSP cannot be approximated with a ratio less than 20/17, which enhances its best known inapproximability bound of 117/116 by more than an order of magnitude. Organization. We formulate the min–max problems of k-tree cover, R-rooted tree cover, and rooted tree cover in Section 2. By reductions from the 3DM, we prove inapproximability bounds for these three problems, in Sections 3–5, respectively. Techniques used in these proofs are applied, in Section 6, to derive inapproximability bounds for other related problems, such as the k-TSP. 2. Problem formulations Let G = (V , E ) be a complete undirected graph on a vertex set V . Each edge e ∈ E is weighted by d(e), where d forms a metric, and so satisfies the triangle inequality. A subtree of G is an acyclic connected subgraph of G. For any edge subset Q ⊆ E, let d(Q ) = P e∈Q d(e) denote the total weight of edges in Q . For any subgraph H of G, let V (H ) and E (H ) denote its vertex and its edge sets. Thus, d(E (Q )) indicates the total edge weight of Q . For any set H of subgraphs of G, let cost (H ) denote the cost of H , which is defined in (1) as the maximum total edge weight of any subgraph in H : cost (H ) = max d(E (H )).
Fig. 1. A component for each Mi ∈ M , where Mi = (w, x, y), used to transform the 3DM to the k-TCP.
Finally, we introduce as follows the 3DM, a well known NPcomplete problem, which is extensively used in the remainder of this paper. Three Dimensional Matching (3DM) [7] Instance: (W , X , Y , n, M , m), where W , X and Y are disjoint sets with |W | = |X | = |Y | = n, and M = {Mi : 1 ≤ i ≤ m} is a subset of W × X × Y with |M | = m. Objective: to decide whether M contains an exact matching M 0 ⊆ M such that |M 0 | = n and no two elements of M0 agree in any coordinate. 3. Hardness of k-TCP
(1)
The following theorem states an inapproximability bound of 3/2 for the k-TCP.
The followings formulate the min–max k-tree cover problem and its rooted variants, and present their approximation ratios known in the literature, and our inapproximability bounds.
Theorem 1. Unless NP = P, there is no polynomial time (3/2 − )approximation algorithm for the k-TCP, for any > 0.
H ∈H
Min–max k-Tree Cover Problem (k-TCP) Instance: (G, k, d), where G = (V , E ) is a complete undirected graph, k is a positive integer, d : E → R+ is a metric function representing edge weights. Feasible solution: A set of k subtrees T = {T1 , T2 , . . . , Tk } such Sk that V = i=1 V (Ti ). Objective: minT cost (T ). Approximation ratio: 4. [5,1] Our inapproximability bound: 3/2. Min–max R-Rooted Tree Cover Problem (R-RTCP) Instance: (G, k, R, d), where G = (V , E ) is a complete undirected graph, R ⊆ V is a given set of k roots (with |R| = k), d : E → R+ is a metric function representing edge weights. Feasible solution: A set of k subtrees T = {T1 , T2 , . . . , Tk } such that each subtree Ti for 1 ≤ i ≤ k contains a distinct root in R, Sk and that V = i=1 V (Ti ). Objective: minT cost (T ). Approximation ratio: 4. [5] Our inapproximability bound: 3/2. Min–max Rooted Tree Cover Problem (RTCP) Instance: (G, k, r , d), where G = (V , E ) is a complete undirected graph, k is a positive integer, r ∈ V is a given root, d : E → R+ is a metric function representing edge weights. Feasible solution: A set of k subtrees T = {T1 , T2 , . . . , Tk } such Sk that Ti contains root r for all 1 ≤ i ≤ k, and that V = i=1 V (Ti ). Objective: minT cost (T ). Approximation ratio: 3. [8] Our inapproximability bound: 10/9.
Proof. Suppose there exists a (3/2 − )-approximation algorithm for the k-TCP with > 0. We are going to show that this algorithm can be used to solve the 3DM in polynomial time, which contradicts to NP 6= P. Given any 3DM instance (W , X , Y , n, M , m), consider a k-TCP instance I = (G, k, d), defined as follows. Let G = (V , E ) be a complete graph on V , where V = W ∪ X ∪ Y ∪m i=1 {qi,j : 1 ≤ j ≤ 9}. Sm Let e E = i =1 e Ei , where e Ei , for each element Mi ∈ M where Mi = (w, x, y), denotes the set of the 11 edges shown in Fig. 1. Accordingly, we define d(e) = 1 for each e ∈ e E, and d(e) = 2 for each e ∈ E \e E. It can be verified that d forms a metric. Finally, let k = 3m + n. Define F (2, 3) as the set of subtrees T of G such that |V (T )| = 3 and d(E (T )) ≤ 2. For each Mi ∈ M , where Mi = (w, x, y), consider the following nine subtrees, denoted by Ti,j for 1 ≤ j ≤ 9: E (Ti,1 ) = {(qi,1 , qi,2 ), (qi,2 , w)}, E (Ti,2 ) = {(x, qi,8 ), (qi,8 , qi,7 )}, E (Ti,3 ) = {(qi,4 , qi,5 ), (qi,5 , y)}, E (Ti,4 ) = {(qi,9 , qi,6 ), (qi,6 , qi,3 )}, E (Ti,5 ) = {(qi,1 , qi,2 ), (qi,2 , qi,3 )}, E (Ti,6 ) = {(qi,4 , qi,5 ), (qi,5 , qi,6 )}, E (Ti,7 ) = {(qi,9 , qi,8 ), (qi,8 , qi,7 )}, E (Ti,8 ) = {(qi,9 , qi,8 ), (qi,9 , qi,6 )}, E (Ti,9 ) = {(qi,2 , qi,3 ), (qi,3 , qi,6 )}. It can be seen that {Ti,j : 1 ≤ j ≤ 9} ⊆ F (2, 3). We are now going to show that the 3DM instance has an exact matching if and only if the (3/2 − )-approximation algorithm for the k-TCP returns a feasible solution to I with cost at most 2.
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On one hand, if the 3DM instance has an exact matching M 0 , then we construct a subtree set T by including subtrees Ti,j for 1 ≤ j ≤ 4, for all Mi ∈ M 0 , and including subtrees Ti,j for 5 ≤ j ≤ 7, for all Mi ∈ M \ M 0 . Thus, |T | = 4|M 0 | + 3(|M | − |M 0 |) = k. It is easy to verify that T covers all qi,j for 1 ≤ i ≤ m and 1 ≤ j ≤ 9. Moreover, for each w ∈ W , since M 0 is an exact matching, there must exist a unique element Mi ∈ M 0 such that Mi = (w, x, y) for some x ∈ X and y ∈ Y . Thus, {Ti,j : 1 ≤ j ≤ 4} ⊆ T . Since
w∈
j=1 V (Ti,j ), we have that T covers all vertices in W . Similarly, we can have that T covers all vertices in X and in Y . Thus, T covers all vertices in V . Notice that each subtree in T has a total edge weight of 2. As defined in (1), cost (T ) equals the maximum total edge weight of any subtree in T . We thus obtain that T is a feasible solution to I with cost (T ) = 2. Therefore, since the edge weights are all integers, and since > 0, the (3/2 − )-approximation algorithm must return a feasible solution to I with cost at most 2. On the other hand, if the (3/2 −)-approximation algorithm returns a feasible solution T to the k-TCP instance I with cost (T ) ≤ 2, then we define M 0 by including all elements Mi ∈ M that have Ti,3 ∈ T , where 1 ≤ i ≤ m. To prove that M 0 is an exact matching for the 3DM instance, consider each subtree T ∈ T . We know d(E (T )) ≤ 2. Since d(e) ≥ 1 for all e ∈ E, we obtain |E (T )| ≤ 2, implying |V (T )| ≤ 3. Since T covers all vertices in V , and since |V | = 9m + 3n and k = 3m + n, we obtain |V (T )| = 3, and d(e) = 1 for all e ∈ E (T ), which implies T ∈ F (2, 3). Thus, T ⊆ F (2, 3), and no two different subtrees in T share the same vertex in V . Consider each element Mi ∈ M , where Mi = (w, x, y). If Mi ∈ M 0 , then Ti,3 ∈ T . Since Ti,4 , Ti,8 , and Ti,9 are the only subtrees in F (2, 3) that cover qi,6 but no vertex in V (Ti,3 ), exact one of them must be in T . However, Ti,9 cannot be in T , because qi,1 cannot be covered by any subtree in F (2, 3) that covers no vertex in V (Ti,3 ) ∪ V (Ti,9 ). By a similar argument, Ti,8 cannot be in T , because otherwise qi,7 cannot be covered by T . Thus, we obtain Ti,4 ∈ T . Since Ti,1 (or Ti,2 ) is the only subtree in F (2, 3) that covers qi,1 (or qi,7 , respectively) but no vertex in E (Ti,3 ) ∪ E (Ti,4 ), we obtain Ti,1 ∈ T and Ti,2 ∈ T . Thus, {Ti,j : 1 ≤ j ≤ 4} ⊆ T . Since no two different subtrees in T share the same vertex in V , no elements of M 0 \ {Mi } can match w , x, or y, which implies that no two elements of M 0 can agree in any coordinate. Otherwise, Mi ∈ M \ M 0 , which implies Ti,3 6∈ T . Since Ti,6 is the only subtree in F (2, 3) \ {Ti,3 } that covers qi,4 , we obtain Ti,6 ∈ T . Notice that Ti,2 and Ti,7 are the only subtrees in F (2, 3) that cover qi,7 . However, Ti,2 cannot be in T , because qi,9 cannot be covered by any subtree in F (2, 3) that contains no vertex in V (Ti,6 ) ∪ V (Ti,2 ). We obtain Ti,7 ∈ T . Similarly, we can obtain Ti,5 ∈ T to cover qi,1 and qi,3 . Therefore, {Ti,j : 5 ≤ j ≤ 7} ⊆ T . Hence, 4|M 0 | + 3(|M | − |M 0 |) ≤ |T |, implying |M 0 | ≤ |T | − 3|M | = k − 3m = n. Moreover, for each element y ∈ Y , let T (y) denote the subtree in T that covers vertex y. We know d(e) = 1 for all e ∈ E (T (y)), and |V (T (y))| = 3. Let e(y) denote an edge in T (y) that is incident on y. Since d(e(y)) = 1, from Fig. 1 we know that e(y) must be incident on qi,5 for an element Mi ∈ M with Mi = (w, x, y) for some w ∈ W and x ∈ X . Therefore, Mi ∈ M0 , because otherwise, we know {Ti,j : 5 ≤ j ≤ 7} ⊆ T , which implies that two different subtrees in T , i.e., T (y) (which contains y), and Ti,6 (which does not contain y), share the same vertex qi,5 , leading to a contradiction. Hence, |M 0 | ≥ |Y | = n, implying |M 0 | = n. Therefore, M 0 is an exact matching for the 3DM instance.
S4
4. Hardness of R-RTCP The following theorem states an inapproximability bound of 3/2 for the R-RTCP.
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Theorem 2. Unless NP = P, there is no polynomial time (3/2 − )approximation algorithm for the R-RTCP, for any > 0. Proof. The proof is similar to that of Theorem 1, by reducing any 3DM instance, (W , X , Y , n, M , m), to an R-RTCP instance, I = (S G, k, R, d), where G and d are defined the same way, and R = X ∪ m i=1 {qi,1 , qi,4 , qi,9 } is defined as the root set with k = |R| = 3m+n. (See Fig. 1.) We then redefine F (2, 3) as the set of subtrees T of G such that |V (T )| = 3 and d(E (T )) ≤ 2, and that T contains a root in R. Notice that for each element Mi ∈ M , every subtree E (Ti,j ), defined in the proof of Theorem 1 for 1 ≤ j ≤ 8, contains a root in R. Thus, by following an argument similar to that in the proof of Theorem 1, we can obtain that the 3DM instance has an exact matching if and only if the (3/2 − )-approximation algorithm for the R-RTCP, if it exists, returns a feasible solution to I with cost at most 2. This contradicts to NP 6= P. 5. Hardness of RTCP The following theorem states an inapproximability bound of 10/9 for the RTCP. Theorem 3. Unless NP = P, there is no polynomial time (10/9 − )approximation algorithm for the RTCP, for any > 0. Proof. Suppose there exists a (10/9 −)-approximation algorithm for the RTCP with > 0. We are going to show that this algorithm can be used to solve the 3DM in polynomial time, which contradicts to NP 6= P. Given any 3DM instance (W , X , Y , n, M , m), consider an RTCP instance I = (G, k, r , d), defined as follows. Let r be the given root. Let G = (V , E ) be a complete Sm undirected graph on V , where V = {r } ∪ W ∪ X ∪ Y ∪ i=1 {qi,j : 1 ≤ j ≤ 6}. Let e E = Sm e e E , where E , for each element M ∈ M where M = (w, x , y), i i i i i =1 denotes the set of the 11 edges shown in Fig. 2. Accordingly, for each Mi ∈ M , where Mi = (w, x, y), we define d(x, qi,3 ) = 3, and d(r , x) = d(r , qi,1 ) = d(r , qi,4 ) = 4, and define the edge weights of remainders in e Ei to be 2. Moreover, for each (u, v) ∈ E \e E, define d(u, v) as the total edge weight of the shortest path from u to v in the subgraph (V , e E ). Finally, let k = 2m + n. It can be verified that d forms a metric. Due to the triangle inequality, we have d(e) ≥ 2 for all e ∈ E, and d(e) ≥ 4 for each e ∈ E \ e E. Moreover, we have d(r , v) ≥ 6 for each v ∈ V \ {r } \ X \ {qi,1 , qi,4 : 1 ≤ i ≤ m}. Define F (4, 9) as the set of subtrees T of G such that |V (T )| = 4 and d(E (T )) ≤ 9, and that T contains the root r. Consider each subtree T ∈ F (4, 9). We can denote V (T ) by {r , v1 , v2 , v3 }, where r and vj for 1 ≤ j ≤ 3 are four different vertices in V . Thus, T must contain at least one edge incident on r, which can be denoted by (r , v1 ) without loss of generality. Since |V (T )| = 4, we have |E (T )| = 3. Since d(e) ≥ 2 for all e ∈ E, and since d(E (T )) ≤ 9, subtree T can contain at most one edge that has a weight greater than or equal to 4, and no edge that has a weight greater than or equal to 6. Therefore, since d(r , v) = 4 for each v ∈ X ∪ {qi,1 , qi,4 : 1 ≤ i ≤ m}, and since d(r , v) ≥ 6 for each v ∈ V \ {r } \ X \ {qi,1 , qi,4 : 1 ≤ i ≤ m}, we obtain d(r , v1 ) = 4 and (r , v1 ) ∈ e E. Thus, no edge in E (T ) \ {(r , v1 )} can be incident on r. Moreover, since d(e) ≥ 4 for each e ∈ E \ e E, subtree T cannot contain any edge in E \ e E. Since d(E (T )) ≤ 9, d(r , v1 ) = 4, |E (T )| = 3, and d(e) ≥ 2 for all e ∈ E, we obtain that E (T ) \ {(r , v1 )} can contain at most one edge that has a weight greater than or equal to 3. Accordingly, it can be verified (in Fig. 2)
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6. Applications
Fig. 2. A component for each Mi ∈ M , where Mi = (w, x, y), used to transform the 3DM to the RTCP.
that T must be one of the following six subtrees, Ti,j for 1 ≤ j ≤ 6, which all contain r: E (Ti,1 ) = {(r , qi,1 ), (qi,1 , qi,2 ), (qi,2 , w)}, E (Ti,2 ) = {(r , x), (x, qi,3 ), (qi,3 , qi,6 )}, E (Ti,3 ) = {(r , qi,4 ), (qi,4 , qi,5 ), (qi,5 , y)}, E (Ti,4 ) = {(r , qi,1 ), (qi,1 , qi,2 ), (qi,2 , qi,3 )}, E (Ti,5 ) = {(r , qi,4 ), (qi,4 , qi,5 ), (qi,5 , qi,6 )}, E (Ti,6 ) = {(r , x), (x, qi,3 ), (qi,3 , qi,2 )}. Moreover, since each subtree Ti,j for 1 ≤ i ≤ m and 1 ≤ j ≤ 6 is in F (4, 9), we obtain F (4, 9) = {Ti,j : 1 ≤ i ≤ m, 1 ≤ j ≤ 6}. We are now going to show that the 3DM instance has an exact matching if and only if the (10/9 −)-approximation algorithm for the RTCP returns a feasible solution to I with cost at most 9. On one hand, if the 3DM instance has an exact matching M 0 , then we construct a subtree set T by including subtrees Ti,j for 1 ≤ j ≤ 3, for all Mi ∈ M 0 , and including subtrees Ti,j for 4 ≤ j ≤ 5, for all Mi ∈ M \ M 0 . Thus, |T | = 3|M 0 | + 2(|M | − |M 0 |) = k. Similarly to the proof of Theorem 1, we can obtain that T covers all vertices in V . Since each subtree in T contains r, and has a total edge weight at most 9, we obtain that T is a feasible solution to I with cost (T ) ≤ 9 due to (1). Therefore, since the edge weights are all integers, and since > 0, the (10/9 − )-approximation algorithm must return a feasible solution to I with cost at most 9. On the other hand, if the (10/9 − )-approximation algorithm returns a feasible solution T to the RTCP instance I with cost (T ) ≤ 9, then we construct M 0 by including all elements Mi ∈ M that have Ti,3 ∈ T , where 1 ≤ i ≤ m. To prove that M 0 is an exact matching for the 3DM instance, consider each subtree T ∈ T . We know r ∈ V (T ) and d(E (T )) ≤ 9. Since d(r , v) ≥ 4 for all v ∈ V , and since d(e) ≥ 2 for all e ∈ E, we obtain |E (T )| ≤ 3, which implies |V (T ) \ {r }| ≤ 3. Thus, noticing |V \ {r }| = 6m + 3n and k = 2m + n, we obtain |V (T ) \ {r }| = 3, implying T ∈ F (4, 9). Thus, T ⊆ F (4, 9), and no two different subtrees in T can share the same vertex in V \ {r }. Consider each element Mi ∈ M , where Mi = (w, x, y). If Mi ∈ M 0 , then Ti,3 ∈ T . Since Ti,2 is the only subtree in F (4, 9) that covers qi,6 but no vertex in V (Ti,3 ), we obtain Ti,2 ∈ T . Since Ti,1 is the only subtree in F (4, 9) that covers qi,1 but no vertex in V (Ti,2 ) ∪ V (Ti,3 ), we obtain Ti,1 ∈ T , and so {Ti,j : 1 ≤ j ≤ 3} ⊆ T . Since no two different subtrees in T share the same vertex in V \{r }, no elements of M 0 \ {Mi } can match w , x, or y, which implies that no two elements of M 0 can agree in any coordinate. Otherwise, Mi ∈ M \ M 0 , which implies Ti,3 6∈ T . Since Ti,5 is the only subtree in F (4, 9) \ {Ti,3 } that covers qi,4 , we obtain Ti,5 ∈ T . Notice that Ti,4 and Ti,6 are the only two subtrees in F (4, 9) that cover qi,3 but no vertex in V (Ti,5 ). However, Ti,6 cannot be in T , because qi,1 cannot be covered by any subtree in F (4, 9) that contains no vertex in V (Ti,5 ) ∪ V (Ti,6 ). Hence, Ti,4 ∈ T , and so {Ti,j : 4 ≤ j ≤ 5} ⊆ T . Thus, 3|M 0 | + 2(|M | − |M 0 |) ≤ |T |, implying |M 0 | ≤ n. Moreover, by following an argument similar to that in the proof of Theorem 1, we can obtain |M 0 | ≥ n. Hence |M 0 | = n, and so M 0 is an exact matching for the 3DM instance.
Techniques, used in reducing the 3DM to the min–max k-TCP and its rooted variants, can be applied to derive inapproximability bounds for min–max problems that use other covering objects. For example, given any instance of the min–max k-TCP, one can use stars or paths (instead of trees) to cover vertices of the given graph. This leads to the min–max k-star, and the min–max k-path problems, for which [1] proved a bicriteria approximation ratio of (3, 3), and an approximation ratio of 4, respectively. The following theorem states their first inapproximability bounds of 3/2. Theorem 4. Unless NP = P, there is no polynomial time (3/2 − )approximation algorithm for the min–max k-star cover problem or for the min–max k-path cover problem, for any > 0. Proof. The proof of Theorem 1 for the k-TCP is applicable to Theorem 4 for the min–max star cover and the min–max path cover problems. This is because for any subgraph H of the given graph G, to say that H is a tree of three vertices, it is equivalent to saying that H is a path of three vertices, and is equivalent to saying that H is a star of three vertices. Therefore, all of the subtrees in F (2, 3), including {Ti,j : 1 ≤ i ≤ m, 1 ≤ j ≤ 9}, which are used in the proof of Theorem 1, are paths and stars as well. Accordingly, it can be directly verified that the arguments in the proof of Theorem 1 for the k-TCP, which are based on F (2, 3) and {Ti,j : 1 ≤ i ≤ m, 1 ≤ j ≤ 9}, are also valid for problems of path cover and star cover. Thus, we can obtain that the (3/2 − )approximation algorithm for the min–max star cover problem or for the min–max path cover problem, if it exists, can be used to solve the 3DM in polynomial time, contradicting to NP 6= P. Finally, we derive an inapproximability bound of 20/17 as follows for the k-TSP, which significantly improves its best bound of 117/116 known in the literature [10]. Given any instance of the RTCP, the k-TSP is to determine k tours to cover vertices of the given graph under the min–max objective. Its best known approximation ratio is 5/2 [6]. Theorem 5. Unless NP = P, there is no polynomial time (20/17−)approximation algorithm for the k-TSP, for any > 0. Proof. Suppose there exists a (20/17 − )-approximation algorithm for the k-TSP with > 0. We are going to show that this algorithm can be used to solve the 3DM in polynomial time, which contradicts to NP 6= P. Given any 3DM instance (W , X , Y , n, M , m), consider a k-TSP instance I = (G, k, r , d) as follows, where G = (V , E ), r, and k are the same as that of the RTCP Sm instance used in the proof of Theorem 3. Let e E still denote i=1 e Ei , however, e Ei here, for each element Mi ∈ M where Mi = (w, x, y), represents the set of the 15 edges shown in Fig. 3. For each Mi ∈ M , where Mi = (w, x, y), we define d(r , qi,3 ) = d(x, qi,3 ) = 5, and define the edge weights of remainders in e Ei to be 4. For each (u, v) ∈ E \ e E, we define d(u, v) as the total edge weight of the shortest path from u to v in the subgraph (V , e E ). Finally, we let k = 2m + n. It can be easily verified that d forms a metric. Due to the triangle inequality, we have d(e) ≥ 4 for all e ∈ E, and d(u, v) ≥ 8 for each (u, v) ∈ E \ e E. Define F (4, 19) as the set of tours C such that |V (C )| = 4 and d(E (C )) ≤ 19, and that C is a subgraph of G and contains r. Consider each tour C ∈ F (4, 19). Since |V (C )| = 4, we can denote C by (r v1 v2 v3 r ), where r and vj for 1 ≤ j ≤ 3 are four different vertices in V . Therefore, |E (C )| = 4. Since d(e) ≥ 4 for all e ∈ E, and since d(E (C )) ≤ 19, tour C can contain at most 3 edges that all have a weight greater than or equal to 5, and contain no edge that has a weight greater than or equal to 8, implying E (C ) ⊆ e E. Moreover, since r ∈ V (C ), |V (C )| = 4, and d(E (C )) ≤ 19, from Fig. 3 we
Z. Xu, Q. Wen / Operations Research Letters 38 (2010) 169–173
Fig. 3. A component for each Mi ∈ M , where Mi = (w, x, y), used to transform the 3DM to the k-TSP.
can see that C cannot contain two or three edges that all have a weight greater than or equal to 5. Thus, C either consists of four edges with each having a weight equal to 4, or consists of three edges with each having a weight equal to 4 and one edge having a weight equal to 5. Accordingly, it can be verified (in Fig. 3) that C must be one of the following five tours, Ci,j for 1 ≤ j ≤ 7, which all contain r: Ci,1 = (rqi,1 qi,2 w r ),
Ci,2 = (rxqi,3 qi,6 r ),
Ci,3 = (rqi,4 qi,5 yr ),
Ci,4 = (rqi,1 qi,2 qi,3 r ),
Ci,5 = (rqi,4 qi,5 qi,6 r ),
Ci,6 = (rqi,2 qi,3 w r ),
Ci,7 = (rqi,6 qi,5 yr ). Moreover, since each tour Ci,j , for 1 ≤ i ≤ m and 1 ≤ j ≤ 7, is in F (4, 19), we obtain F (4, 19) = {Ci,j : 1 ≤ i ≤ m, 1 ≤ j ≤ 7}. We are now going to show that the 3DM instance has an exact matching if and only if the (20/17 − )-approximation algorithm for the k-TSP returns a feasible solution to I with cost at most 19. On one hand, if the 3DM instance has an exact matching M 0 , then we construct a tour set C , by including Ci,j for 1 ≤ j ≤ 3, for all Mi ∈ M 0 , and including Ci,j for 4 ≤ j ≤ 5, for all Mi ∈ M \ M 0 . Thus, |C | = 3|M 0 | + 2(|M | − |M 0 |) = k. Similarly to the proof of Theorem 3, we can obtain that C covers all vertices in V . Since each tour in C contains r, and has a total edge weight at most 17, we obtain that C is a feasible solution to I with cost (C ) ≤ 17 due to (1). Therefore, since the edge weights are all integers, and since > 0, the (20/17 − )-approximation algorithm must return a feasible solution to I with cost at most 19. On the other hand, if the (20/17 − )-approximation algorithm returns a feasible solution C to the k-TSP instance I with cost (C ) ≤
173
19, then we construct M 0 by including all elements Mi ∈ M that have Ci,3 ∈ C , where 1 ≤ i ≤ m. To prove that M 0 is an exact matching for the 3DM instance, consider each tour C ∈ C . We know r ∈ V (C ) and d(E (C )) ≤ 19. Since d(e) ≥ 4 for all e ∈ E, we obtain |E (C )| ≤ 4, implying |V (C ) \ {r }| ≤ 3. Thus, from |V \ {r }| = 6m + 3n and k = 2m + n, we obtain |V (C ) \ {r }| = 3, implying C ∈ F (4, 19). Thus, C ⊆ F (4, 19), and no two different tours in C can share the same vertex in V \ {r }. Accordingly, by following an argument similar to that in the proof of Theorem 3, and using F (4, 19) (instead of F (4, 9)), and Ci,j (instead of Ti,j ), we can obtain that no two elements of M 0 can agree in any coordinate, and that |M 0 | = n. Therefore, M 0 is an exact matching for the 3DM instance. Acknowledgements The authors are grateful to the anonymous referees for their valuable comments. They are also grateful to Mr. Liang Xu and Mr. Xiaofan Lai for their constructive discussions. This work is partially supported by the Internal Competitive Research Grant A-PD0W and the Niche Area Grant J-BB7C of the Hong Kong Polytechnic University. References [1] E.M. Arkin, R. Hassin, A. Levin, Approximations for minimum and min–max vehicle routing problems, Journal of Algorithms 59 (1) (2006) 1–18. [2] I. Averbakh, O. Berman, (p − 1)/(p + 1)-approximate algorithms for ptraveling salesmen problems on a tree with minmax objective, Discrete Applied Mathematics 75 (3) (1997) 201–216. [3] I. Averbakh, O. Berman, A heuristic with worst-case analysis for minimax routing of two travelling salesmen on a tree, Discrete Applied Mathematics 68 (1–2) (1996) 17–32. [4] A.M. Campbell, D. Vandenbussche, W. Hermann, Routing for relief efforts, Transportation Science 42 (2) (2008) 127–145. [5] G. Even, N. Garg, J. Könemann, R. Ravi, A. Sinha, Minmax tree covers of graphs, Operations Research Letters 32 (4) (2004) 309–315. [6] G.N. Frederickson, M.S. Hecht, C.E. Kim, Approximation algorithms for some routing problems, SIAM Journal on Computing 7 (1978) 178–193. [7] Michael R. Garey, David S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, WH Freeman & Co, New York, USA, 1979. [8] H. Nagamochi, Approximating the minmax rooted-subtree cover problem, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E88-A (5) (2005) 1335–1338. [9] K. Okada, H. Nagamochi, Polynomial time 2-approximation algorithms for the minmax subtree cover problem, Lecture Notes in Computer Science 2906 (2003) 138–147. [10] C.H. Papadimitriou, S. Vempala, On the approximability of the traveling salesman problem, Combinatorica 26 (1) (2006) 101–120.