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Exact Weight Perfect Matching of Bipartite Graph Is NP-Complete ∗ Proceedings of the World Congress on Engineering 2008 Vol II WCE 2008, July 2 - 4, 2008, London, U.K. Exact Weight Perfect Matching of Bipartite Graph is NP-Complete ∗ Guohun Zhu, Xiangyu Luo, and Yuqing Miao Abstract—This paper proves that the complexity Question. Does there exist a perfect matching M with of exact weight perfect matching problem is NP- W (M) = α ? complete by reduction from the good perfect match- ing problem. Following this result, the other two open problems DNA sequence analysis and discrete The EW P M becomes a more and more famous open min-max assignment problems are proven to be NP- problem in recent years. For instance, Jacek proved that complete. it is equivalent to the DNA sequencing analysis problem in polynomial time [2]. Vladimir proved that it is equiv- Keywords: computational complexity, color coding, ex- alent to the discrete mini-max assignment problem with act perfect matching, good perfect matching, knapsack a fixed number of scenarios in polynomial time [4]. problem Input. A complete bipartite graph with bipartition X U 1 Introduction Y; a set of k cost scenarios that are specified by non- Throughout this paper we only consider the finite simple negative integer edges costs b1, . , bk : X × Y → N; undirected graph G = (V, E), i.e. the graph has no multi- a bound B. edges and self loops. If each edge of G is assigned a color, Question. Find a perfect matching M ⊆ X × Y with then G is an edge-colored graph G(C), where C is a color bi(M) ≤ B for i = 1, . , k. partition C = {c1, c2, . , ck}. Let ci(ej) denotes the edge ej assigned in color ci and ci(E) means the subset of edges all in color ci.A matching in G is a subset of pairwise In this paper, we study this problem by two known NP- non-adjacent edges; that is, no two edges share a common complete problems. The first one is the good perfect vertex. A perfect matching M is that every vertex meets matching problem. exactly one memeber of M.A good matching is that all the edges in M are of different colors. Let w(e ) denotes i Name. Good Perfect Matching (GP M for short) the weight edge ei, ei ∈ E. And W (S) = w(e1)+w(e2)+ ... + w(es) denotes the sum of weights of all the edges Input. A edge colored bipartite graph. S ⊆ E. Question. Does there exist a perfect matching M with It is well known that the perfect matching or assignment each edge in M has different color? problem in weighted/unweighted bipartite graphs could be solved efficiently [7]. But for the following question, it becomes more difficult: Good prefect matching problem is a special case of maximum labeled perfect matching problem in [9]. And Cameron [3] proved that Name. Exact Weight Perfect Matching (EW P M for short) Theorem 1 [3] The good prefect matching problem in Input. An edge weight bipartite graph and a positive bipartite graph is NP-complete. integer α. But Cameron showed that the restricted GP M problem ∗Manuscript received March 22, 2008. This work is partially 2 supported by the National Natural Science Foundation of China un- could be solved in O(n ) when the graph is complete der Grant No.60763004 and Guangxi Natural Science Foundation bipartite graph and there are no more than two edges in of China under Grant No.0310006. Guohun Zhu, Xiangyu Luo and the same color existing in this complete bipartite graph Yuqing Miao are with the Department of Computer Science, Guilin University of Electronic Technology, Guilin 541004, China (Guohun [3]. Zhu’s phone: +86-773-560-1330; Guohun Zhu’s e-mail: zhuguo- [email protected]; Xiangyu Luo’s e-mail: [email protected]; Another related NP-complete problem is 0-1 knapsack Yuqing Miao’s e-mail: [email protected]). problem, which can be expressed as follows: ISBN:978-988-17012-3-7 WCE 2008 Proceedings of the World Congress on Engineering 2008 Vol II WCE 2008, July 2 - 4, 2008, London, U.K. Name. 0 − 1 Knapsack Problem Finally, let us taken an example to show how to distin- guish the color from a weight perfect matching. Given a Input. A vector of positive integers A = {a1, a2, . , an} red-blue edge color K3,3 graph, let er be an edge in color and an integer T . red and eb an edge in color blue. Then according to Equa- Question. Does there exist a binary vector X satisfying tion (1), we have w(er) = 1 and w(eb) = 3, suppose the Pn W (M) = 3, then the matching M includes 3 red edges. i=1 aixi = T ? 3 Exact weight perfect matching prob- In this paper, we only consider a simple 0 − 1 Knap- sack problem: the sequence of A is superincreasing if lem is NP-complete Pi−1 a ≤ a − 1 (i = 2, . , n). This case is known j=1 j i In this section the EW P M problem is proven to be NP- to be solvable in polynomial time [10]. complete by reducing the good perfect matching problem. All graph theoretical terms that do not defined in this In other words, EW P M problem is harder than GP M paper can be found in [5]. We refer to [6] for definitions problem. The main theorem is as follows: linked to complexity and knapsack problem. Theorem 2 The exact weight perfect matching problem 2 Color coding of bipartite graph is NP-complete. Itai studied the restricted matching problem with a single restriction |M ∩ c1(E)| ≤ r[8]. His approach is to encode Proof : Given a perfect matching M of bipartite graph the edges with binary coding, thus the restricted match- and an integer k, it is easy to check that the sum of ing problem can be solved in polynomial time by solving weight edges in M equals to the value of k, thus EW P M the minimum cost maximum flow problem. is NP. But the binary coding is so greedy that it lose a lot of Now let us reduce the good perfect matching problem to useful information in this solution. For instance, if there EW P M of a bipartite graph G. is a K graph with 1 red edge, 1 green edge and 2 blue 2,2 Suppose that an edge colored bipartite graph G has 2n colors and we encode red edge as 1, and other edges value vertices, and the colors set C is {c , c , . , c }, where as 0. Since the perfect matching M of K only include 1 2 k 2,2 (k ≥ n). A good perfect matching M of G consists of a two edges, if W (M) = 1, it shows that there must be a set of edges {e , e , . , e }. red edge in M. However, it could not tell us the color of 1 2 n the another edge in M. Let’s encode each edge with color coding according to Equation (1). Thus the set of edges in M are labeled Therefore, it is necessary to distinguish all colors from with a set of weights {w(e ), w(e ), . , w(e )}. The sum the perfect matching efficiently. Let us consider the nat- 1 2 n of weights of M is ural primary colors: Red, Blue and Green. For any given color, it is always composed of these primary colors; Con- W (M) = w(e ) + w(e ) + ... + w(e ). (2) versely, if any value of primary colors is given, an unique 1 2 n color can be obtained. Similarly, if each edge in a perfect where w(ei) satisfies the superincreasing condition. matching M is colored by the primary colors, then color of M can be viewed as a mixed color; on the other hand, Thus Equation (2) could be rewritten as the following if the mixed color of a M is given, we can get the color equation: of every edge in M. W (M) = x1w(c1) + x2w(c2) + ... + xkw(ck), (3) Now suppose that there exists k different colors in graph G. Let w(ei) equals to the weight of edge ei and c(ej) where xi ∈ {0, 1} and w(ci) satisfies the superincreasing the color of edge ej, we encode each weight of edge as a condition. superincreasing number by the following approach Obviously, Equation (3) shows that it is a 0-1 knapsack 1, if c(ei) = c1; problem . Thus if there exists a good perfect matching M |c1(E)| + 1, if c(ei) = c2; in G, then there exists an exact weight perfect matching wei = ...,... (1) satisfying Equation (3) and if an edge in M is labeled Pk−1 with color c , then we can get x = 1, otherwise x = 0. j=1 |Cj(E)| + 1, if c(ei) = ck; i i i Conversely, if there exists an exact weight perfect match- It is obvious that the color coding is an one-to-one map. ing W (M) in G, then it could solve Equation (3) in poly- We can thus obtain the color of an edge if the weight of nomial time according to the complexity of superincreas- the edge is given. ing knapsack problem [10]. 2 ISBN:978-988-17012-3-7 WCE 2008 Proceedings of the World Congress on Engineering 2008 Vol II WCE 2008, July 2 - 4, 2008, London, U.K. According to Theorem 8 in [2], we can give the following [9] Monnot, J., “The labeled perfect matching in bipar- corollaries: tite graphs“, Information Processing Letters, V96, N3, pp.
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