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2_ 
 
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 UIUCDCS-R-79-967 
 
 UILU-ENG 79 1713 
 
 An NP-Complete Matching Problem 
 
 by 
 
 David A. Plaisted 
 
 April 1979 
 
 »n 
 
 DEPARTMENT OF COMPUTER SCIENCE 
 UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 
 
 URBANA, ILLINOIS 
 
Corrections to University of Illinois Departmental Report "An NP- 
 Complete Matching Problem" by David A. Plaisted 
 
 page 1 line 13 "vertex of El" should be "edge of El." 
 page 9 line -2 "q edges" should be "q triples" 
 

 
 An NP-Complete Matching Problem 
 
 by 
 
 David A. Plaisted 
 
 222 Digital Computer Lab 
 
 Department of Computer Science 
 
 University of Illinois 
 
 Urbana, Illinois 61801 
 
 This research was partially supported by the National Science Foundation 
 under grant MCS 77-22830. 
 

 Abstract 
 
 We show that a restricted form of the perfect matching problem 
 for bipartite graphs is NP-complete. The restriction involves partitions 
 of the vertices of the graph. This problem is still NP-complete if the 
 degrees of the vertices are restricted to be 3 or less. For degrees 
 restricted to 2 or less, a polynomial time algorithm exists. 
 
 
-1- 
 
 1. Introduction 
 
 The general perfect matching problem can be solved in 
 polynomial time [4], but certain related problems are NP-complete [2, 3]. 
 We show that a more restricted version of the perfect matching problem is 
 still NP-complete. Much of this work was done together with Shmuel Zaks. 
 
 Suppose G = (V, E) is an undirected graph, where V is the 
 set of vertices and E is the set of edges. We represent an edge joining 
 v.| and v^ by the set {v-j, v^} for vertices v, and v~. Suppose G is a 
 bipartite graph. That is, V = V, u \l V, n \l = 0, and every edge joins 
 a vertex in V, with a vertex in \L. We represent such a bipartite graph 
 by G = (V r V 2 , E). 
 
 The perfect matching problem is to determine if there is a 
 subset El of E such that every vertex of G is incident with exactly one 
 vertex of El. We are interested in the following problem, which we call 
 "perfect matching with node partitions." 
 
 Given an undirected bipartite graph G = (V, , V~, E) and given 
 partitions P, and P~ of V, and V« s respectively, to determine 
 if there is a subset El of E such that El is a perfect matching 
 for G and such that no distinct edges {v,, v ? } and {w, , w~} of 
 El have the property that v, and w, are in the same block of 
 P, and v ? and w ? are in the same block of P ? . 
 We say that such a matching El is a perfect matching consistent with the 
 node partitions P, and P ? . This problem is a specialization of the 
 3-dimensional matching problem [3] and of the restricted matching problem 
 treated in [2]. Both of these latter problems are known to be NP-complete 
 
-2- 
 
 A motivation for the perfect matching problem with node partitions is 
 the following: Let V ] be a set of people and let V 2 be a set of offices 
 in various committees. Let E be a set of pairs {v, s v 2 > indicating that 
 person v 1 is eligible for office v 2< We may partition V 1 into families 
 and V 2 into committees. The problem is to assign people to committees 
 so that no two people from the same family are on the same committee. 
 It is easy to see that this is an instance of the perfect matching 
 problem with node partitions. 
 
 2. NP-Completeness 
 
 We now show that this restricted perfect matching problem is 
 NP-complete. Clearly it is in NP. We show that it is NP-hard by reducing 
 from the one-in-three 3-SAT problem, known to be NP-complete [5 ]. The 
 final form of this reduction is mostly due to S. Zaks. 
 
 The one-in-three 3SAT problem is as follows: Given a set X 
 of variables and a set S of clauses over X such that for all C e S, C 
 has 3 literals, to determine if there is a 1 -truth assignment (a truth 
 assignment making exactly one literal in each clause true). 
 
 First we reduce the one-in-three 3SAT problem to a related 
 version of the one-in-three 3SAT problem. The difference is that for 
 all variables x in X, at most 3 clauses in S contain either x or x. 
 However, some clauses in S may have only 2 literals. This reduction proceeds 
 as follows: Suppose x or x appears in C, , C ? , . . . , C , for clauses C e S, 
 and x or x appears in no other clauses of S. Suppose m > 3. Introduce 
 new variables yj , y 2 , ..., y m . Let C| ' be C i with x replaced by y. . Let S' be 
 with C i replaced by C^ ' , for 1 < i < m, and with the additional clauses 
 
-3- 
 
 y-\ v y 2 » y 2 v y 3' y 3 v y 4' ■•" y m-l v y m' y m v y l added - Now n is 
 
 clear that S' has a 1-truth assignment iff S does, since any interpretation 
 
 satisfying the clauses y^ v y 2 , ^2 v y 3' • * " ' y m v y l must give a11 y i 
 the same truth value. Also, each y. appears in exactly three clauses. 
 Hence by repeating this step, we obtain a set T of clauses which has a 
 1-truth assignment iff S does, and such that for each variable x, at most 
 3 clauses contain x or x. 
 
 We now reduce the restricted one-in-three 3SAT problem to the 
 restricted matching problem. Suppose S is a set of clauses over the 
 variables X. Suppose each clause C in S has 3 or fewer literals, and 
 for each x in X, at most 3 clauses of S contain either x or x. We construct 
 a bipartite graph G = (v^, V 2 , E) and partitions P ] of V, and P„ of V 2 
 such that G has a matching as desired iff S has a 1-truth assignment. 
 This construction of G from S can be done in polynomial time, as will be 
 clear. In fact, the construction can be done in linear time. 
 
 Let L be X u {x: x e X}. Thus L is the set of literals of S. 
 We say L e C if L is a literal appearing in clause C. Thus x e {x v y v z} 
 but x t {x v y v z). For literals L in L we define L in the following way: 
 If L is x then L is x; if L is x then L is x. We call L the complement 
 of L. Let S x be {C e S: x e C or x e C} for x e X. 
 
 The graph G is the "union" of graphs obtained from the clauses 
 
 and variables in S. We define the union of graphs G 1 = (V, ' , V ? ', E') and 
 
 G" = (V^', V 2 ", E") to be the graph G' u G" = (V ] ' u V " V 2 ' u V 2 ", E 1 u E") 
 
 For each variable x in X, we define a graph G(x) in a manner to be described, 
 
 and for each clause C in S we define a graph G(C) in a manner to be 
 
 described. The final graph G is then ( U G(x) ) u ( r U c G(C) ) . Also, the 
 
 XcA Leo 
 
-4- 
 
 partitions P, and ? 2 are defined by P. = ( x y x P i^ x ^ u ^c U S P l^ C ^ and 
 
 P 2 = ^xeX P 2^ X ^ U ^CeS P 2^ C ^ Where V*)' p -|( C )' p 2 ^' and P 2^ remain 
 to be defined. 
 
 We define G(C) and G(x) by cases. Suppose x e X and |S | = 3. 
 
 Let {A, B, C} be the set of three (distinct) clauses in which x or x 
 
 appears. Then G(x) = (V^x), V 2 (x), E(x)) where V-,(x) = «x, 1), (x, 2), 
 
 <x, 3>}, V„(x) = {<x, A), <x, A), <x, B), <x, B>, <x, C>, <x, C)}, and 
 
 E(x) = {{<x, 1>, <x, A>},«x, 1>, <x, B>}, {<x, 2>, <x, B>},{(x, 2), (x, C>}, 
 
 {<x, 3>,<x, C>},«x, 3>, <x, A)}}. Note that G(x) may differ depending 
 
 on which order we take A, B, and C. Any order will do. Also, P,(x) = 
 
 {V^x)} and pJx) = {{<x, A>, <x, A>}, {<x, B>, < x, B>},{<x, C>, <x, C>}}. 
 
 Diagram: 
 
 V^x) 
 
 V 2 (x) 
 
 G(x) 
 
 The ovals illustrate the blocks of the partition. Note that there 
 
 are only two restricted matchings of this graph. One of them leaves 
 
 the vertices <x, A>, <x, B>, and <x, C) free, and the other one leaves 
 
 <x, A>, <x, B>, and < x, C> free. (For example, if < x, 1 > matches 
 
 (X, B> then < x, 2) can't match <x, B> so < x, 2> matches <x, C> and so on.) 
 
 The former matching corresponds to an interpretation making x true and 
 
 the latter matching corresponds to an interpretation making x false. 
 
-5- 
 
 If IS I = 2 or IS i = 1 , similar constructions for G(x) are used. 
 
 1 x 1 'x 1 
 
 Suppose |S | = 2 and S = {A, B}. We then have V,(x) = 
 
 A A ' 
 
 {<x, 1), (x, 2)} and V 2 (x) = «x, A>, <x, A>, < x, B>, < x, B>} and 
 
 P^x) = (V^x)} and P 2 (x) = {{<x, A), <x, A>},{(x, B>, < x, B>}}. Also, 
 
 E(x) = {«x, 1>, (X, A)},«x, 1), <x, B)},{<x, 2>, <x, B>},«x, 2>, <x, A>}}. 
 
 Suppose |S | = 1 and S = {A}. We then have V,(x) = {< x, 1 >} and 
 xx i 
 
 V 2 (x) = {(x, A>, <x, A>) and P^x) = (V^x)} and P 2 (x) = (V 2 (x)}. 
 Also, E(x) = {{<x, 1>, <x, A)}, {<x, 1>, <x, A)}}. 
 
 We define G(C) for C e S as follows: Suppose C = LI v L2 v L3 
 for LI, L2, L3 e L. Then V^C) = {<C, 1>, (C, 2), <C, 3>} and 
 V 2 (C) = «L1, C>, <fl, C>, <L2, C>, <L2, C), < L3, C>, <L3, C)}. Also, 
 P^C) = {{<C, 1>}, «C, 2>}, «C, 3>}} and P 2 (C) = {{<L1, C), < U, C)}, 
 {<L2, C>, <L2, C>}, «L3, C>, <L3, C)}}. Finally, E(C) = {{<C, 1>, 
 (LI, C>}, {<C, 1>, <L2, C>}, «C, 1>, (L3, C>}, {<C, 2>, < 11 , C>}, {<C, 2>, <L2, C>}, 
 {<C, 3), <L2, C>}, {(C, 3>, <L3, C>}}. 
 Diagram: 
 
 v 2 (c) 
 
 The vertex <C, 1) can match any one of the vertices < LI , C>, <L2, C), 
 <L3, C). The vertices <C, 2> and <C, 3> can then match (L, C> for all 
 L e C such that < C, 1 ) does not match < L, C>. We cannot have ( C, 1 > 
 
-6- 
 
 matching < LI , C> and < C, 2> matching < LI , C) because only one of < LI , C) 
 and < LI , C> will be left free, by allowable matchings on G(x) for x = L 
 or x = L. If <C, 1) matches (LI, C>, this corresponds to LI being true 
 in an interpretation and L2 and L3 being false (as required by one-in-three 
 3-satisfi ability), and similarly if < C , 1> matches (L2, C) or<L3, C>. 
 The construction for C = LI v L2 or for C = LI is similar. 
 
 Finally, G = ( U x G(x)) u ( U S G(C)) as stated earlier. We now 
 show that G has a matching consistent with the constraints represented 
 by P, and ? ? iff S has a 1-truth assignment. Assume G has such a matching. 
 Then each G(x) has a matching consisting of |S | edges. Also, this 
 matching is consistent with the node partitions P-,(x) and P 2 (x) of G(x). 
 Each such matching corresponds to a truth value for x as indicated above. 
 Consider the interpretation I of X giving all x e X truth values as 
 specified by these matchings of G(x) , for x e X. Since each G(C) has 
 a matching, at least one literal of C must be true in I. By the con- 
 struction of G(C), if such a matching exists then the other two literals 
 of C must be false in I. Hence I is a 1-truth assignment for S. Conversely, 
 if a 1-truth assignment I for S exists, a matching for G(x) exists 
 corresponding to the truth value of x in I , for x e X. This matching 
 will be consistent with the node partitions P, (x) and P 2 (x) of G(x). 
 Also, matchings for G(C) can be obtained since exactly one literal of 
 C is true in I. These matchings can be combined to obtain a matching for 
 G consistent with the node partitions. This completes the proof. 
 
 3. Further restrictions 
 
 Note that the above graph G has the property that no node has 
 
-7- 
 
 degree larger than three and that no block of P, or P~ has more than 
 three elements. We now indicate how this result can still be obtained 
 if we restrict the blocks of P-. and Pp to have no more than two elements, 
 while maintaining the bound of three on node degrees. The idea is to 
 define G as above except that G(x) is defined differently for x e X. 
 In particular, if |S | = 3, let A, B, and C be the three (distinct) 
 
 A 
 
 clauses in which x or x appears. Then G(x) = (V^(x), V 2 (x), E(x)) where 
 V^x) = {x, x} x {1, 2, 3} and 
 V 2 (x) = ({x, x} x {A, B, C}) u {x} x {4, 5, 6}. 
 Here x indicates Cartesian product. Also, 
 E(x) = {{<x, 1), <x, A>}, {<x, 1>, <x, A)}, {<x, 2>, < x, B>}, 
 {<x, 2>, <x, B>}, {<x, 3>, <x, C>}, {<x, 3>, < x, C>}} 
 u {{<x, 1>, (x, 4>}, {<x, 2), <x, 4)}, «x, 2>, < x, 5>}, 
 {<x, 3), <x, 5)}, «x, 3>, <x, 6>}, {<x, 1>, <x, 6>}} 
 and P^x) = {{<x, 1), <x, 1>}, «x, 2>, <x, 2>}, {< x, 3>, < x, 3>}} 
 and P 2 (x) = {{<x, A), < x, A>}, {<x, B>, <x, B>}, «x, C>, <x, C>}, 
 
 «x, 4>}, «x, 5>}, {<x, 6>}} 
 
 Diagram: 
 
 G(x) 
 
■8- 
 
 The idea is that if <x, 1 > matches <x, A) then < x, 1 > must match <x, 4>. 
 Therefore < x, 2> must match < x, B> and so on. If < x, 1 > matches < x, 6> 
 then < £, 3> must match <x, C> so <x, 3> must match <x, 5> and so on. In 
 the former case, the nodes <x, A), <x, B>, and <x, C> of V ? (x) are free, 
 and in the latter case <x, A>, (x, B>, and <x, C) are free. Therefore 
 the same argument as before holds; the former case corresponds to an 
 interpretation making x false and the latter case corresponds to an 
 interpretation making x true. However, with G(x) as defined here, no 
 block of P, or P has more than 2 vertices. If IS I = 2 or IS I = 1 
 a similar construction is used. 
 
 Reducing the blocks of P, and P~ to contain only one vertex 
 yields the usual perfect matching problem for which a polynomial time 
 algorithm is known [4] regardless of the degrees of the vertices. 
 We now show that reducing the degrees of the vertices to 2 permits a 
 polynomial time solution regardless of the sizes of the blocks of P, 
 and P ? . 
 
 Let G = (V,, V„, E) be a bipartite graph in which the degree 
 of each vertex is 2 or less. Let P, and P 2 be arbitrary partitions of 
 V-j and V2, respectively. We show how to construct in polynomial time 
 a set S of 2-literal clauses such that S is satisfiable iff G has a 
 matching consistent with the node partitions. Since 2-satisfiability 
 has a polynomial time solution [ 1 ], the given problem also has a 
 polynomial time solution if the degrees are restricted to be 2 or less. 
 
 The set S of clauses has a propositional variable e for each 
 edge e of G. Also, if distinct edges e-, and e ? meet a common vertex, 
 the clauses e-| v e~ and i, v e~ are in S. This signifies that exactly 
 
-9- 
 
 one of e^ and e 2 must be included in a perfect matching. If distinct 
 edges e 1 and e^ join vertices in the same block of P, with vertices 
 in the same block of P, then e-j v e 2 is in S. This signifies that at 
 most one of e-j and e 2 can be in a matching consistent with the node 
 partitions. No other clauses are in S. It should be clear that S is 
 satisfiable iff G has a perfect matching consistent with the node 
 partitions P, and P ? . 
 
 4. Comparison with other matching problems 
 
 The general problem of perfect matching with node partitions 
 is a restriction of multiple choice matching [2] because the sets of 
 edges in multiple choice matching are arbitrary, whereas for the former 
 problem the sets of edges must correspond to node partitions. (Multiple 
 choice matching specifies subsets of the edges, each of which can contain 
 at most one edge in the matching). In fact, for node partitions with 
 at most two nodes per block of the partition, our construction yields 
 edge sets containing at most two edges. (Two edges are in the same 
 edge set if they join nodes in the same block of P, with nodes in the 
 same block of P ? . ) 
 
 We now show that perfect matching with node partitions is a 
 restriction of 3-dimensional matching. The 3-dimensional matching problem 
 is the following: Given a set M c W x Y x Z where W, Y, and Z are 
 disjoint sets having the same number q of elements, does M contain a 
 matching containing q edges? A matching is a subset M' of M such that no two 
 elements of M' agree in any co-ordinate. 
 
-10- 
 
 Given a graph G = (V ] , V 2> E), with |V ] | = |V 2 | and given 
 partitions P ] and ? 2 of V ] and V 2 , respectively, we construct a 
 problem closely related to 3-dimensional matching. Let W be V, , let 
 Y be V 2 , and let Z be P ] x ? r Let M be {< V] , v 2 , < p ] , p 2 >> : V] e V,, 
 v 2 e V 2 , {v r v 2 ) e E, and p ] E P ] , p £ E P 2 and V] e p ] , v 2 e p 2 ). That 
 is, p 1 and p 2 are the blocks of P 1 and P 2 containing v, and v ? , 
 respectively. Now, a matching M' of M corresponds to a matching of G 
 in which no two distinct edges join vertices in the same block of P, 
 with vertices in the same block of ? r Hence a matching M 1 exists with 
 l M ' I = l V ]l = ! V 2l iff G nas a perfect matching consistent with the 
 node partitions. However, we do not have | M " J = |Z| since it may be that 
 |V-, | f |Z| and |V 2 | + |Z|. Despite this, it is easy to use this modified 
 form of 3-dimensional matching to show that the general 3-dimensional 
 matching problem is NP-complete. 
 
■11 
 
 References 
 
 [1] Cook, S. A. The complexity of theorem proving procedures, Proc . 
 3rd Annual ACM Symposium on Theory of Computing (1971) 151-158. 
 
 [2] Itai, A. Rodeh, M. and Tanimoto, S. L. Some matching problems 
 for bipartite graphs, J. ACM 25(1978) 517-525. 
 
 [3] Karp, R. M. Reducibility among combinatorial problems, in Miller, 
 R. E. and Thatcher, J. W. , Complexity of Computer Computations 
 (Plenum Press, New York, 1972) 85-104. 
 
 [4] Lawler, E. Combinatorial Optimization (Holt, Rinehart, and Winston, 
 New York, 1976^ 
 
 [5] Schaefer, T. J. The complexity of satisfiability problems, Proc . 
 10th Annual ACM Symposium on Theory of Computing (1978) 216-226. 
 
BIBLIOGRAPHIC DATA 
 SHEET 
 
 1. Report No. 
 
 UIUCDCS-R-79-967 
 
 4 T • > ->nj Subt it le 
 
 An NP-Complete Matching Problem 
 
 3. Recipient's Accession No. 
 
 5. Report Date 
 
 April 1979 
 
 7. Author(s ) 
 
 David A. Plaisted 
 
 8. Performing Organization Rept. 
 No. 
 
 rtorming Organization Name and Address 
 
 Dept. of Computer Science 
 222 Digital Computer Lab 
 University of Illinois 
 Urbana, Illinois 61801 
 
 10. Project/Task/Worlc Unit No. 
 
 11. Contract /Grant No. 
 
 NSF MCS 77-22830 
 
 nsoring Organization Name and Address 
 
 National Science Foundation 
 Washington, D.C. 
 
 13. Type of Report & Period 
 Covered 
 
 14. 
 
 '5. .pplementary Notes 
 
 :racts 
 
 We show that a restricted form of the perfect matching problem 
 for bipartite graphs is NP-complete. The restriction involves partitions 
 of the vertices of the graph. This problem is still NP-complete if the 
 degrees of the vertices are restricted to be 3 or less. For degrees 
 restricted to 2 or less, a polynomial time algorithm exists. 
 
 v ttords and Document Analysis. 17a. Descriptors 
 
 perfect matching, 3-dimensional matching, graph algorithms, NP-complete. 
 
 17k Identif iers/Opcn-Ended Terms 
 
 '7c iA ri Field/Group 
 
 m Availability Statement 
 
 19. Security Class (This 
 Report) 
 
 unclassified 
 
 20. Security Class (This 
 Page 
 
 UNCLASSIFIED 
 
 21. No. of Pages 
 
 22. Price 
 
 • NT IS- 35 ( 1( 
 
 USCOMM-DC 40329-P71 
 
1979 
 
AUb 1 b N