Cost-Constrained Matchings and Disjoint Paths

Transcription

1 Cost-Constrained Matchings and Disjoint Paths Kenneth A. Berman 1 Department of ECE & Computer Science University of Cincinnati, Cincinnati, OH Abstract Let G = (V, E) be a graph, where the edges are weighted with elements from a finite commutative semigroup (S, +), and consider the problem of finding a matching M that covers a given set U of vertices whose cost c(m) (where c(m) is the sum of the weights on the edges of M) satisfies a given constraint, i.e., for a given constraint function Φ mapping S to {0, 1}, Φ(c(M)) = 1. This problem is NP-complete for exponential size semigroups. However, in this paper for small semigroups S, i.e., the size of S is bounded above by a polynomial in the input size, that are 2-divisible, i.e., for each element x there exists a unique element x/2 S such that 2(x/2) = x, we present an NC 2 algorithm (on an EREW PRAM) for computing the parity of the number of such matchings and an RNC 2 for finding one. In the case where the cost-constraint is monotonic, i.e., for every pair of elements x, y S, x + y satisfies the constraint implies that x also does, this yields a solution to the multiple constrained path selection problem, and more generally to the cost-constrained disjoint path problem. Finally, we present generalizations of our results to binary matroids. Key words. graph algorithms, matchings, disjoint paths, shortest paths, parallel algorithms, randomized algorithms, cost-constrained problems, matroids 1 This research is partially supported by NSF Grant No. CCR

2 1. Introduction There are myriad applications of matchings and disjoint paths in mathematics and computer science, and there is an extensive literature on these subjects. See [11, 5, 1, 10, 8, 12]. Computing vertex disjoint paths joining a set of source vertices to a set of sink vertices have applications to routing and reliability. In practice many such applications involve multiple weightings of the edges of the graph or network, e.g., cost, length, latency, congestion, bandwidth, capacity, demand, utility, monetary cost, etc., where we wish to find a set of disjoint paths or a matching that satisfies some constraint conditions with respect to these costs. Multiple weightings can be viewed as a single vector weighting, or more generally a weighting over a commutative semigroup. In this paper we consider the general problem of finding a cost-constrained matching for weightings of the edges over a commutative semigroup. This has applications to the multiple constrained path selection (MCP) problem (see [6]), and more generally to an extension of the latter problem to disjoint paths. Consider a graph G = (V, E) vertex set V of cardinality n and edge set E of cardinality m, where multiple edges permitted. A matching M is a set of independent edges, i.e., no two edges of M have a vertex in common. Given a subset U of V, we say that M covers U if every vertex in U belongs to at least one edge of M. A perfect matching is a matching that covers all the vertices. Let (S, +) be a commutative semigroup, i.e., a set S together with an additive operation +, such that + is commutative and associative and S is closed under +. We will say that S is 2-divisible if, for each element x there exists a unique element x/2 S such that 2(x/2) = x. A (cost) weighting c of E over (S, +) is a mapping from E to S. The cost of M is given by c(m) = e M c(e). In this paper we consider the following very general framework for cost constraining the problem. We define a constraint function Φ to be any mapping from S to {0, 1}. Given a constraint function Φ a cost-constrained matching M is one that satisfies the constraint Φ(c(M)) = 1. We will refer to this problem as the cost-constrained matching problem. The cost-constrained matching problem is NP-complete for exponential size semigroups S. To see this take S to be the set 2 E of all subsets of E and + to be the operation of symmetric difference. Then, taking c(e) = {e} for each e E, c(m) equals the edge set E(M) of M. Thus, Φ can be chosen to place any constraint we want on the matching M, and it is easy to find a constraint, which makes the problem NP-complete. Thus, in this paper we restrict our attention to the cost-constrained matching problem in the case where the semigroup S is small, i.e., is bounded above by a polynomial in the input size. In section 2, subject to this restriction, we given an NC 2 algorithm on an EREW PRAM for computing the parity of the number of cost-constrained matchings covering a given set U and an RNC 2 algorithm for finding one. let M(n) denote the number of arithmetic operations used by the best known sequential algorithm for multiplying two n n matrices: currently M(n) = O(n ). Theorem 1.1 Let G be a graph with vertex set V of size n whose edge set E is weighted with 2

3 elements of a commutative 2-divisible semigroup (S, +) and let U be a subset of the vertex set V. Then, for a given constraint function Φ, there exists an O(nM(n) S 2 ) sequential algorithm and an O(log 2 n+log n log S ) algorithm on an EREW PRAM with O(nM(n) S 2 ) processors for computing the parity of the number of a cost-constrained matchings that cover U. In the special case of a perfect matching (U = V) in a bipartite graph the 2-divisibility condition can be dropped. In the case of small semigroups S, it follows from Theorem 1.1 that there exists an NC 2 algorithms for computing the parity of the number of cost-constrained matchings covering a given set U. Further, utilizing the Isolating Lemma we obtain an RNC 2 algorithm for finding such a matching. Theorem 1.2 The cost-constrained matching problem for small 2-divisible semigroups is in RNC 2. In the special case of a perfect matching in a bipartite graph the 2-divisibility condition can be dropped. Now consider the problem of finding a set P of k (vertex) disjoint paths in a digraph having a given set X of initial vertices and a given set Y of terminal vertices, where X = Y = k. The cost-constrained version of this problem, where the cost of a set of disjoint paths is the sum of the weights over all the edges in the union of all the paths in the set, is NP-complete, even in the special case where k = 1 (and S is small). To see this consider a unit weighting of the edges over the integers modulo n, and let Φ(i) = 1 if i = n 1 and 0, otherwise. Then, a cost-constrained path joining a vertex x to a vertex y is a hamiltonian path joining x to y. However, in section 3 we utilize the algorithm from section 2 to solve the cost-constrained disjoint path problem for a monotonic constraint function Φ, i.e., a constraint function Φ with the property that, for every pair of elements s, t S, Φ(s + t) = 1 implies that Φ(s) = 1. We obtain the following result. Theorem 1.3 Consider a graph G whose edge set E is weighted with elements of a small commutative semigroup (S, +), and let X and Y be disjoint subsets of the vertex set of cardinality k. Then, given a monotonic constraint function Φ, there exists a RNC 2 algorithm for finding a cost-constrained set of k disjoint paths from X to Y. In the case of directed acyclic graphs (dags) the monotonic condition on the constraint function can be dropped. Now suppose we are given j edge weightings c 1, c 2,..., c j over the positive integers, and j positive integers b 1, b 2,..., b j. Now consider the problem of finding k disjoint paths from X to Y whose cost with respect to c i is at most b i, i = 1,..., j. We will refer to this problem as the multiple constrained disjoint path problem. The special case of this problem where k = 1 is the multiple constrained path selection problem (MCP), which is known to be NP-complete even for j = 2 (see [6]). The multiple constrained disjoint path problem is a the special case of the monotonic cost-constrained disjoint path problem where S and Φ are 3

4 defined as follows: given an integer b, let (S(b), ) denote the semigroup consisting of the integers {1,..., b} such that for any two integers x, y S(b), x y = x + y if x + y < b and x y = b, otherwise. Let S be the semigroup given by S = S(b 1 ) S(b 2 )... S(b j ), and let Φ be the constraint function given by Φ(a 1, a 2,..., a j ) = 1 if, and only if, a i b i, i = 1,..., j, (a 1, a 2,..., a j ) S. Observing that an RNC 2 algorithm involving integer weights bounded above by a polynomial in the input size can be viewed as an RNC 2 algorithm where the integer weights are input in unary, the following corollary is an immediate consequence of the above discussion and Theorem 1.3. Corollary 1.1 Consider a graph G and let X and Y be disjoint subsets of the vertex set of cardinality k. For j a constant, let c 1, c 2,..., c j be unary positive integer weightings of the edges, and let b 1, b 2,..., b j be unary positive integers. Then, the problem of finding a set of k disjoint paths from X to Y whose cost with respect c i is at most b i, i = 1,..., j, is in RNC 2. Binary matroids, which include graphic matroids as a special case, is an important class of matroids. A binary matroid M = (E, I) on ground set E = {e 1,..., e n } having independent set I is a matroid that can be represented by a matrix A over GF (2) whose columns are indexed by E, such that the independent sets in I correspond to the independent columns of A, i.e., I I if, and only if, the columns indexed by the elements in I are independent. The rank r of the matroid is the maximum size of an independent set. A base of M is a maximum independent set, i.e., an independent set of size r. Given a partition Π of E into pairs a matroid matching is an independent set that, for each pair in Π, either contains both elements of the pair or neither. Binary matroid matchings generalize perfect matchings in graphs (see Gondran and Minoux [5]). The first polynomial-time algorithm for the matroid matching problem for linearly-representable matroids was given by Lovász [?]. Using a determinant formula given by Lovász [10] and the Isolating Lemma, Narayanan et. al. [13] obtained an RNC 2 algorithm for the matroid matching problem. In this paper we extend the notion of a matroid matching to a general partition Π by defining a generalized matching to be an independent set M of M that intersects each class in Π an even number of times. If M is a base, then we refer to M as a generalized perfect matching. In section 4, we prove the following theorems. Theorem 1.4 Let M = (E, I) be a binary matroid whose base set E is weighted with elements of a 2-divisible commutative semigroup (S, +). Then, for a given constraint function Φ, there exists an O(nM(n) S 2 ) sequential algorithm and an O(log 2 n + log n log S ) algorithm on an EREW PRAM with O(nM(n) S 2 ) processors for computing the parity of the number of a cost-constrained generalized perfect matchings. Theorem 1.5 Let M = (E, I) be a binary matroid whose base set E is weighted with elements of a small 2-divisible commutative semigroup (S, +). Then, there exists and RNC 2 algorithm for finding a cost-constrained generalized perfect matching for a given constraint function Φ. 4

5 2. Matchings in graphs We prove Theorem 1.1 with the aid of the following lemma. Let G = (V, E) be a graph. For U V, let M U denote the set of all matching that cover U Given a weighting ω of the edges over a commutative ring R, let A ω = (a ij ) be the n n matrix given by: a ij = e k ={i,j} ω(e k) if i j, 1 if i = j & v i V U, 0 if i = j & v i U, where by convention the empty summation equals zero, i.e., a ij = 0 if there is no edge joining i and j. Let µ(ω) be defined by: µ(ω) = M M U e M ω(e). Lemma 2.1 If ω is a weighing over a commutative ring R of characteristic 2, then, µ 2 (ω) = det A ω. Proof. Let G = (V, E ) be the digraph on vertex set V obtained from G by replacing each edge e i E with two edges e + i and e i joining the same two vertices as e but having opposite orientation. Assign both e + i and e i the weight ω(e i ), i = 1,..., m. We will refer to e i and e + i as mates The mate of a directed circuit C of size at least 3 is the directed circuit C obtained from C by replacing each edge with its mate. Note that a directed circuit C of size 2 consisting of an edge and its mate does not have a mate, since replacing each edge of C with its mate does not yield a new circuit. A circuit set C is a collection of k vertex disjoint directed circuits. Let π(c) denote product of the weights on the edges of C. We will say that two circuit sets are equivalent if one can be obtained from the other by replacing some subset of circuits with their mates. Let C denote the equivalence class of C. Note that π(c) is the same for any circuit set in C. If λ(c) denotes the number of circuits of C that have a mate. Then C = 2 λ(c). Note that λ(h) = 0 if, and only if, C consists of the set of 2-circuits {{e + i, e i } e i M} for some matching M. Let K U denote the collection of all circuit sets C that cover U and let K U denote the collection of all circuit set equivalence classes. Then, it follows directly from the definition of the matrix A U and the determinant that deta ω = C K U π(c) = C K U 2 λ(c) π(c) 5

6 Using the fact that the elements are over a ring R of characteristic 2, we have that deta ω = λ(c)=0 π(c) = M M U e i M ω 2 (e i ) = µ 2 (ω). We are now ready to prove Theorem 1.1. Extend the semigroup S to include the 0 element (if it doesn t include it already). We associate with the power set 2 S of S, a ring [S] = (2 S,,, 0, 1), which we call the symmetric difference ring, where 0 =, 1 = {0}, is symmetric difference and is defined by S 1 S 2 = Σ s1 S 1,s 2 S 2 {s 1 + s 2 }, where the summation is with respect to the operation. Clearly, the symmetric difference ring [S] has characteristic 2. Letting µ(c) = µ(ω) and A c = A ω, respectively, where ω(e i ) is the singleton set containing c(e i ), i.e., ω(e i ) = {c(e i )}, i = 1,..., m, we have that µ(c) = {c(e i )} = c(m). M M U e i M M M U Since the summation is with respect to symmetric difference, it is immediate that µ(c) is precisely the set of all costs s S for which there is an odd number of matchings covering U that have that cost. Thus, the parity of the number of cost-constrained matchings covering U equals the parity of the number of elements of µ(c) that satisfy the constraint Φ. Utilizing Lemma 2.1, and the fact that µ 2 (c) = 2µ(c), it follows that µ(c) = (deta c )/2, where (deta c )/2 is obtained from deta c by dividing every element by 2 (this can be done since S is 2-divisible). It has been shown (see [2, 3]) that a matrix over a commutative ring with multiplicative identity can be computed using O(nM(n)) arithmetic operations on a sequential machine and using O(log 2 n) parallel additions and O(log n) multiplications on an EREW PRAM with O(nM(n))processors. Further, two elements of the symmetric difference ring [S] can be added in time O( S ) and multiplied in time O( S 2 ) on a sequential machine and can be added in constant time on an EREW PRAM using O( S ) processors and multiplied in time O(log S ) using O( S 2 ) processors. Theorem 1.1 follows. The associated algorithm is given below. Parity of Number of Cost-Constrained Matchings Algorithm. Input: a graph G = (V, E) a subset U of V a weighting c of E over a small commutative 2-divisible semigroup (S,+) 6

7 a constraint function Φ Output: 0 if G contains an odd number of cost-constrained matchings that cover U. 1 if G contains an even number of cost-constrained matchings that cover U. Step 1. Compute D = (deta c )/2. Step 2. If an even number of the elements of D satisfy the constraint Φ then OUTPUT: 0 Otherwise OUTPUT: 1 We are now ready to prove Theorem 1.2. We utilize the following result of Mulmully et. al. ([12]) known as the Isolating Lemma. Lemma 2.2 (Isolating Lemma). Let F be a family of subsets of a set E. Choose integers r 1, r 2,..., r n randomly and independently from [1,..., 2m]. Then, the probability that there is a unique minimum weight set in F is at least 1/2. We first compute a random weighting r of E, i.e., assign e i E the integer weight r(e i ) chosen randomly and independently from [1,..., 2m]. Let c denote the weighting over the semigroup S Z p given by c (e i ) = (c(e i ), r(e i )), i = 1,..., m, where Z p denote the integers modulo p and p is an odd number that is larger than the sum of the r-weights of any matching, e.g., p mn. Now suppose there exist a cost-constrained matching covering U (with respect to the original weighting c and constraint function Φ). Then, by the Isolating Lemma, the minimum-weight with respect to r of such matching is unique with probability at least 1/2. Letting k denote this minimum weight and observing that 1 is an odd number, it follows that, with high probability (at least 1/2), µ(c ) contains a unique pair (s, k ), such that Φ(s) = 1. Hence, it follows that if Φ(s) = 0 for all pairs (s, k) D then with probability (at least 1/2), there is no cost-constrained matching that covers U. On the other hand, if Φ(s) = 1 for some pair (s, k) D, then we know with 100% certainty that there exists a cost-constrained matching that covers U. Further, we can construct such a matching (with high probability) as follows. Over all the pairs (s, k) D such that Φ(s) = 1, choose a pair that minimizes k, and let (s, k ) denote this pair. Then, with high probability (at least 1/2) there exists a unique cost-constrained matching M covering U of cost k (with respect to r), and cost s (with respect to c). Thus, if for a given edge e j we replace the r-weight of e j with 0, to obtain the new weighting r j and a new cost weighting c j (i.e., c j (e i) = (c(e i ), r j (e i )), i = 1,..., m), then there will be zero (i.e., an even number of) matchings covering U of cost s (with respect to c) and weight k (with respect to r j ) if e j belongs to M and one (i.e., and odd number of) matchings covering U of cost s (with respect to c) and weight k (with respect to r j ), otherwise. Thus, if we compute the set D j 7

8 the same as we computed D, except with r j replacing r, then e j belongs to M if, and only if, the pair (s, k ) does not belong to D j, j = 1,..., m. The algorithm described above is summarized below: Cost-Constrained Matching Algorithm. Input: a graph G = (V, E) a subset U of V a weighting c of E over a small commutative 2-divisible semigroup (S,+) a constraint function Φ Output: a cost-constrained matching M that covers the vertices in U. With high probability (at least 1/2) a correct matching M is returned if such a matching exists. Step 1. Compute a random weighting r of E, i.e., assign e i E the unary integer weight r(e i ) chosen randomly and independently from [1,..., 2m] and the weighting c over the semigroup S Z p, given by c (e i ) = (c(e i ), r(e i )), i = 1,..., m, where Z p denotes the integers modulo p and p is an odd integer greater than or equal to mn. Step Compute D = (deta c )/2. 2. If none of the elements of D satisfy constraint Φ then OUTPUT: There is no cost-constrained matching that covers U. and STOP. Step Over all pairs (s, k) D such that Φ(s) = 1 choose a pair (s, k ) such that k is minimized. 2. For each element e j, j = 1,..., m, compute the weighting r j, such that r j (e i ) = r(e i ) for all edge e i different from e j and r j (e j ) = 0, and the weighting c j such that c j (e i) = (c(e i ), r j (e i )), i = 1,..., m. 3. Compute D j = (deta c j )/2 4. Compute the set M of all edges e j such that (s, k ) does not belong to D j, j = 1,..., m. 5. OUTPUT: M. It follows from the earlier discussion that D j can be computed in time O(nM(n) S Z p 2 ) = O(n 3 M(n)m 2 S 2 ) on a sequential machine and in time O(log 2 n + log n log S Z p ) = O(log 2 n+log n log S ) on an EREW PRAM with O(nM(n) S 2 ) = O(n 3 M(n)m 2 S 2 ) processors. Thus the above algorithm can be executed in time O(n 3 M(n)m 3 S 2 ) on a sequential machine, and in time O(log 2 n + log n log S ) on an EREW PRAM with O(n 3 M(n)m 3 S 2 ) processors. 8

9 In the special case of a perfect matching (i.e., where U = V ) in a bipartite graph G = (V, E), we have the following determinant formula for µ(ω). Let V = P Q denote the associated bipartition of V, where P = {p 1,..., p n/2 } and Q = {q 1,..., q n/2 }, and let B ω = (b ij ) be the n/2 n/2 matrix given by: b ij = ω(e k ), e k ={p i,q j } i, j {1,..., n/2}, where the empty sum equals 0. It is immediate from the definition of a determinant that µ(ω) = det B ω. Using this formula for µ(ω) in place of the one for µ 2 (ω) we can drop the 2-divisibility condition in the results and algorithms above. 3. Disjoint paths In this section, for a monotonic constraint function, we show how the cost-constrained disjoint path problem can be reduced to the cost-constrained perfect matching problem in a bipartite graph, proving Theorem 1.3. We first prove the following lemma. A circuit set C in a digraph G = (V, E) is a collection of pairwise vertex disjoint directed circuits. Given a subset U V, we say that C covers U if each vertex u U is contained in at least one circuit of C. Lemma 3.3 There exists a NC 0 time computable reduction from the problem of finding a cost-constrained circuit set covering a given set U of vertices in a digraph G to the problem of finding a cost-constrained perfect matching in a bipartite graph. Proof. Let Ĝ = ( ˆV, Ê) be the bipartite graph constructed from the digraph G = (V, E) as follows. The vertex set ˆV consists of two copies V + and V of V, i.e., ˆV = V + V. The edge set Ê is obtained by joining v+ and v with an edge e v for each vertex v V U and joining u + and v with an edge e for each (directed) edge e in G joining u to v. Assign each edge e u, u U the weight 0 and assign the each edge e the weight c(e). It is immediate that a perfect matching P in Ĝ corresponds to a collection C of pairwise disjoint circuits in G that cover U, where the set V C of vertices not belonging to any circuit in C, corresponds to the set of edges {e v v V C } of P. Conversely, given any circuit set C covering U, the corresponding edges in Ĝ together with the edges {e v v V C } form a perfect matching in {e v v V C }. Further, P and C have the same cost. To prove Theorem 1.3 augment the digraph G as follows: add k new vertices u 1,..., u k to G, and join y i to u i with an edge directed towards u i and u i to x i with an edge directed 9

10 towards x i, i = 1,..., k. Assign all the new edges a cost of 0. It is easily verified that a circuit set C in this augmented digraph that the covers the new vertices u 1,..., u k corresponds to a set of disjoint paths joining X to Y together with a set of circuits in the original digraph. Since C satisfies the constraint Φ and Φ is monotonic, the cost of this set of disjoint paths satisfies Φ. In the case of dags G contains no directed circuits, so that the result holds for any constraint function Φ.. 4. Matroid matching Let M = (E, I) be a binary matroid on the ground set E whose elements are weighted with a commutative semigroup S. Let B denote the set of all bases of M. Consider a partition Π = (π 1,..., π k ) of E. We prove Theorem 1.4 with the aid of the following lemma. A Π- bounded base is a base that has at most one edge in common with each class π i, i = 1,....k. Let B Π denote the set of Π-bounded bases. For ω a weighting of the ground set E over a ring R, let β(ω) be given by: β(ω) = B B Π e B ω(e). Let r be the rank of M and let A = a ij be an r E matrix representation of M. Assume the rows are indexed by {1,..., r} and the columns by E. Construct the bipartite graph G A = (V A, E A ) as follows, with associated bipartition V A = X Y, where X = {x 1,..., x r } and Y = {y 1,..., y k }. The vertices in X will be associated with the rows of A and the vertices of Y with the sets in Π. For each i, j, i {1,..., r}, j {1,..., m} such that a ij = 1, join x i to y j, where j is the index of the set in Π that contains e j, i.e., e j π j, with and edge e of weight ω A (e ) = ω(e j ). Let E A denote the set of all such edges e. Letting M X denote the set of all matchings of G A that cover the set X define: Lemma 4.4 β(ω) = µ A (ω A ) µ A (ω A ) = M M X e M ω A (e). Proof. Let P = (p ij ) be the m k matrix given by p ij = 1 if, and only if, edge e i π j, i = 1,..., m, j = 1,..., k. Let W be the m m diagonal matrix whose i th diagonal entry is ω(e i ), i = 1,..., m. Consider the matrix Q = AW P. It follows that Q has dimensions r k and the ij th entry q ij satisfies: ω A (x i y j ) if x i y j E A, q ij = 0 otherwise. 10

11 Let I r (k) denote all subsets I = {i 1,..., i r }, where i 1,..., i r {1,..., k}. For I I r, let P I and Q I denote the r r submatrices of P and Q, respectively, obtained by deleting all columns not indexed by elements in I, and let Y I = {y i1,..., y ir }. Then, Q I = AW P I. Further, letting M I denote the set of matchings of G A having vertex set X Y I it follows directly from the definition of the determinant that detaw P I = detq I = M M I e M ω A(e). Thus, we have µ A (ω A ) = I I r(k) detaw P I. For J = (j 1,..., j m ) I r (m), let A J denote the submatrix consisting of all the rows of A and all columns indexed by J, W J,J the submatrix of W consisting of all rows and columns indexed by J, and P J,I the submatrix of P consisting of rows indexed by J and columns indexed by I. It follows from the Binet-Cauchy Formula (see [5, 13]) that: detaw P I = deta J W J,J P J,I = deta J Thus, we have that: J J r(m) J J r(m) i=1 m ω(e ji )detp J,I. µ A (ω A ) = r deta J ω(e ji )( detp J,I ). J J r (m) i=1 I I r (k) Let e J = {e j1,..., e jm }. It follows from the definition of a base that A J is nonsingular if, and only if, e J is a base of M, i.e., e J B. Equivalently, Now deta J equals 1 if e J B and will equals 0, otherwise. Therefore, we have that: µ A (ω A ) = r ω(e ji )( detp J,I ). e J B i=1 I I r(k) If e J is not a Π-bounded base, then P J,I will contain a zero row for every I I r (k), so that detp J,I ) = 0 for every I I r (k). On the other hand if e J is a Π-bounded base with e ji belonging to π j i, i = 1,..., m, then P J,I will be a permutation matrix for I = {j 1,..., j r} and will contain a zero row otherwise, i.e., detp J,I ) = 1 if I = {j 1,..., j r} and will equal 0, otherwise. It follows that I I r (k) detp J,I = 1 if e J is a Π-bounded base and equals 0 otherwise. Thus, we have that: µ A (ω A ) = r ω(e ji ) = β(ω). e J B Π i=1 This completes the proof of the Lemma. 11

12 Let P Π denote the set of all generalized perfect matchings of the matroid M with respect to partition Π and let π(ω) = M P Π e B ω(e). Now let E be a copy of the ground set E, i.e., E = {e 1,..., e m}. Let A = (A A), and M be the matroid on ground set E E, represented by the matrix A, where the first set of r columns correspond to E and the next set of r columns correspond to E. Let ω be the weighting of E E, given by ω (e i ) = ω (e i ) = ω(e i ), i = 1,..., m. Let Π be the partition of E E consisting of the classes from Π together with the singleton classes {e i }, i = 1,..., m, i.e., Π = {π 1,..., π k, {e 1 },..., {e m}}. Consider the bipartite graph G A and µ A (ω A ) (defined as before with A replaced with A and ω replaced with ω ). Lemma 4.5 π(ω) = µ A (ω A ) Proof. Given a base b B, let h i (b) denote the number of elements of b that lie in class π i, i = 1,..., k. Note that a generalized perfect matching is a base b such that h i (b) is even for all i {1,..., k}. It is easily verified that a base b of M corresponds to (h 1 (b) + 1) (h k (b) + 1) Π -bounded bases of M, each having weight ω(b). Thus, we have that: β M (ω) = k i (b) + 1) b B i=1(h ω(e) = π(ω). e b The last equality follows from the fact that the only terms for which k i=1 (h i (b) + 1) is odd are those for which h i (b) is even for each i {1,..., k}, i.e., when b is a generalized perfect matching. To prove Theorem 1.4, let ω be the weighting such that ω(e i ) is the singleton set contain the element c(e i ) of the symmetric difference ring [S], i.e., ω(e i ) = {c(e i )}. Then, it follows from Lemma 4.5 that the parity of the number of cost-constrained generalized perfect matchings in M is the same as the parity of the number of cost-constrained matchings in G A covering X. Theorem 1.4 now follows immediately from Theorem 1.1. Theorem 1.5 follows from Theorem 1.4 and the Isolating Lemma in the same way the Theorem 1.2 followed from Theorem 1.1 and the Isolating Lemma. 5. Further research In the theorems and algorithms for cost-constrained matchings covering a given set U and cost-constrained generalized perfect matchings in binary matroids, we assumed that the 12

13 semigroup (S, +) was 2-divisible. In the case of perfect matchings in a bipartite graph we showed that this condition could be dropped. Can this condition be dropped in all cases? We now show that the latter question is equivalent to asking whether the 2-divisibility condition can be dropped for cost-constrained perfect matchings in a graph by proving the following result. Theorem 5.6 For a fixed semigroup S, the problem of computing the parity of the number of cost-constrained solutions to each of the following problems are computationally equivalent (i.e., there exists a NC 0 reduction from any one problem to any other): 1) Perfect matchings in a graph 2) Matchings covering a given set of vertices in a bipartite graph 3) Matchings covering a given set of vertices in a graph 4) Generalized perfect matchings in a binary matroid Proof. In the previous section we showed (4) can be reduced to (2) and (1) is a special case of (4). Clearly, (2) is a special case of (3). Thus, to prove the Theorem it is sufficient to show that (3) can be reduced to (1). Let G = (V, E ) be the graph constructed from G = (V, E) as follows. Take V to be the same as V if V U is even; otherwise take V = V {ˆv} for some new vertex ˆv. Take E to consist of E together with a new edge u v for every pair of distinct vertices u, v V U, giving this new edge a weight of 0. Then a perfect matching P in G corresponds to a matching M in G that covers U. Let k = V U 2 M. Then, it is clear that the number perfect matchings P that correspond to each given matching M M U equals the number of perfect matchings in the complete graph K k on k vertices, which in turn equals: (k 3) (k 1). Since this number is odd, it follows that the number of cost-constrained matchings covering U in G has the same parity as the number of cost-constrained perfect matchings in G. We ask the following questions: Is the problem of computing the parity of the number of cost-constrained perfect matchings in a graph whose edges are weighted with a small semigroup S in P? Is it in NC? We have answered these questions in the affirmative for (small) 2-divisible semigroups S and for general (small) semigroups S in the case of bipartite graphs. In this paper, for small semigroups S, we presented random Monte Carlo algorithms for finding a cost-constrained matching covering U in a graph and a cost-constrained generalized perfect matching in a binary matroid M. An interesting research direction would be to find deterministic polynomial-time algorithms for the these problems. The problem of finding 13

14 deterministic NC algorithms for these problems would be hard, since it is not even known whether the problem of finding an unconstrained perfect matching in a graph is in NC. Another research direction would be to extend the RNC algorithm given here for finding a cost-constrained perfect matroid matching in a binary matroid to a linear-representable (or matric) matroid and more generally to any matroid. References [1] M. Aigner, Combinatorial Theory, Springer-Verlag, New York, [2] S.J. Berkowitz, On computing the determinant in small parallel time using a small number of processors, Information Processing Letters 12 (1984), pp [3] A. Borodin, S.A. Cook and N. Pippinger, Parallel computation for well-endowed rings and space bounded probabilistic machines, Inform. and Control, 58 (1983), pp [4] D. Coppersmith and S. Winograd: Matrix multiplication via arithmetic progressions, J. Symbolic Comp., 9 (1990), [5] M. Gondran and M. Minoux, Graphs and Algorithms, John Wiley & Sons, Chichester, [6] J.M. Jaffe,Algorithms for finding paths with multiple constraints, Networks, 14 (1984), [7] R.M. Karp and V. Ramachandran, A survey of parallel algorithms for shared-memory machines, Handbook of Theoretical Computer Science, col A, MIT Press/Elsevier, New York, 1990, pp [8] R.M. Karp, E. Upfal and A. Wigderson, Constructing a maximum matching is in random N C, Combinatorica, 6 (1986), pp [9] F. T. Leighton, Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes, Morgan Kaufmann Pub., [10] L. Lovász, On determinants, matchings, and random algorithms, Fundamentals of Computation Theory, FCT 79, L. Budach, ed., Math Research 2, Akademie-Verlag, Berlin, 1979, pp [11] L. Lovász and M. Plummer, Matching Theory, Academic Press, Budepest, [12] K. Mulmuley, U.V. Vazirini and V.V. Vazirini, Matching is as easy as matrix inversion, Combinatorica, 7 (1987), pp

15 [13] H. Narayanan, H. Saran and V.V. Vazirani, Randomized parallel algorithms for matroid union and intersection, with applications to arboresences and edge-disjoint spanning trees, SIAM J. Comput. 23 (1994), pp