For complexity of the heterogeneous minimum spanning forest problem has not been determined, we reduce 3-SAT which is NP-complete to 2-heterogeneous minimum spanning forest problem to prove this problem is NP-hard and spread result to general problem, which determines complexity of this problem. It provides a theoretical basis for the future designing of approximation algorithms for the problem.
Heterogeneous minimum spanning forest problem(HMSF) was introduced by Yadlapalli et al. [1], and approximation algorithm was designed for heterogeneous vehicle routing problem in [1]. Heterogeneity of an undirected weighted complete graph refers to that each edge in the graph possesses a number of different costs. The goal of HMSF is to search the minimum cost spanning forest in a heterogeneous graph. An approximate algorithm of HMSF was proposed in [1], but complexity of this problem is not clear [1] [2] . The main contribution of this paper is to prove HMSF is NP-hard by reducing a well known NP-complete problem to HMSF in which each edge in graph possesses two costs, which determines the complexity of the problem.
This section describes the notation used in the whole paper.
Undirected complete graph G=(V,E) is heterogeneous if each edge in the graph possesses more than one non-negative integer costs. If each edge e in graph G possesses exactly two non-negative integer costs then G is 2-heterogeneous. Let w 1 (e) and w 2 (e) denote the costs, w 1 and w 2 are cost functions on edge set E. For any node
Spanning forest F in the graph G consists of two disjoint trees T 1 and T 2 , where T 1 and T 2 contain all nodes in the graph. The cost of edge in T 1 (T 2 ) is defined by function w 1 ( w 2 ). The cost of tree T 1 (T 2 ) is the sum of costs of edges in tree T 1 (T 2 ). The cost of spanning forest F is the sum of the costs of T 1 and T 2 .
2-Heterogeneous minimum spanning forest problem(2-HMSF) refers to search a minimum cost spanning forest in a 2-heterogeneous graph with given two nodes as tree roots. Determination form of 2-HMSF refers to that given a 2heterogeneous graph, two nodes r 1 ,r 2 and a integer k, determinate whether there exists a spanning forest F such that nodes r 1 and r 2 are roots of tree T 1 and T 2 respectively and the cost of F is no larger than k.
3-SAT is a classical NP-complete problem, and it will be used in section 3. A formula is in 3-conjunctive normal form (3-CNF) if it is a conjunction of clauses, where a clause is a disjunction of three literals. For example,
two clauses and uses five variables. 3-SAT refers to determine whether a given formula in 3-CNF could be satisfied.
We first use reduction technique to prove 2-HMSF in general graph is NP-hard, and then explains how to use the same method in complete graph which satisfies triangle inequality. 3-SAT will be used as reduction problem. For any instance of 3-SAT, we construct a heterogeneous graph G, and specify the two nodes r 1 ,r 2 and integer k, then prove that the instance could be satisfied if and only if there exists a spanning forest F in graph G such that cost of F is no larger than k and nodes r 1 ,r 2 are tree roots.
Assume that the instance of 3-SAT contains m clauses C 1 ,C 2 ,…,C m and uses n variables x 1 ,x 2 ,…,x n . Construct a 2heterogeneous graph G as follows: For each variable x i in the instance of 3-SAT construct nodes x i and ¬x i , and each clause C j construct a node C j , then construct two nodes t and f represent true and false respectively; For each pair of nodes x i and ¬x i , construct a edge (x i ,¬x i ), define the cost w 1 (x i ,¬x i )=w 2 (x i ,¬x i )=1, call these edges type x edges; Construct edge (t,x i ) between node t and each x i , define the cost w 1 (t,x i )=n+1, w 2 (t,x i )=(n+1) 2 , call these edges type t edges; Construct edge (f, ¬ x i ) between node f and each ¬x i , define the cost w 1 (f,¬x i )=(n+1) 2 , w 2 (f,¬x i )=n+1, call these edges type f edges; For each clause C j , construct three edges between the clause node and nodes corresponding three literals in C j , for the edges of form (C j ,x i ) define the cost w 1 (C j ,x i )= (n+1) 2 , w 2 (C j ,x i )=2(n+1) 2 , for the edges of form (C j ,¬x i ) define the cost w 1 (C j ,¬x i )=2(n+1) 2 , w 2 (C j , ¬x i )=(n+1) 2 , call these edges type C edges; Let node t be root of tree T 1 and node f be root of tree T 2 , then let k=m(n+1) 2 +n(n+1)+n. Proof: (necessity) Assume τ is a satisfying assignment of the formula. Construct a spanning forest F with cost no larger than k, and at first F does not contain any edges. For each variable x i in formula, if τ(xi)=true than add edge (t,x i ) to F, otherwise add edge (f,¬x i ) to F; Since the formula is satisfied under the assignment τ, for each clause C j , one could choose a literal in C j where true value of the literal is true, if literal x i is chosen than add edge (C j ,x i ) to F, and if literal ¬x i is chosen than add edge (C j ,¬x i ) to F; Finally add all type x edges to F. It is easy to verify that F is a spanning forest. Total cost of type C edges in F is m(n+1) 2 . Total cost costs of type t and type f edges in F is n(n+1). Total cost costs of type x edges in F is n. Therefore the cost of forest F is exactly k. Fig. 1(b) shows a spanning forest in a 2heterogeneous graph.
(sufficiency) Assume F is a spanning forest such that nodes t and f are tree roots and cos
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