Elsevier

Theoretical Computer Science

Volume 818, 24 May 2020, Pages 51-59
Theoretical Computer Science

Linear representation of transversal matroids and gammoids parameterized by rank

https://doi.org/10.1016/j.tcs.2018.02.029Get rights and content

Abstract

Given a bipartite graph G=(UV,E), a linear representation of the transversal matroid associated with G on the ground set U, can be constructed in randomized polynomial time. In fact one can get a linear representation deterministically in time 2O(m2n), where m=|U| and n=|V|, by looping through all the choices made in the randomized algorithm. Other important matroids for which one can obtain linear representation deterministically in time similar to the one for transversal matroids include gammoids and strict gammoids. Strict gammoids are duals of transversal matroids and gammoids are restrictions of strict gammoids. We give faster deterministic algorithms to construct linear representations of transversal matroids, gammoids and strict gammoids. All our algorithms run in time (mr)mO(1), where m is the cardinality of the ground set and r is the rank of the matroid. In the language of parameterized complexity, we give an XP algorithm for finding linear representations of transversal matroids, gammoids and strict gammoids parameterized by the rank of the given matroid.

Introduction

Matroids are important mathematical objects in the theory of algorithms and combinatorial optimization. Often an algorithm for a class of matroids gives us an algorithmic meta theorem, which gives a unified solution to several problems. For example, it is known that any problem which admits a greedy algorithm can be embedded into a matroid and finding minimum (maximum) weighted independent sets in this matroid corresponds to finding a solution to the problem. Other important examples are the Matroid Intersection and Matroid Parity problems, which encompass several combinatorial optimization problems such as Bipartite Matching, 2-Edge Disjoint Spanning Trees and Arborescence. A matroid M is defined as a pair (E,I), where E is called the ground set and I is a family of subsets of E, called independent sets, with the following three properties: (i) I, (ii) if AI and AA, then AI and (iii) if A,BI and |A|<|B|, then there is a eBA such that A{e}I. As the cardinality of I could be exponential in |E|, as it is in many applications, explicitly listing I for algorithms is highly inefficient both in terms of time complexity as well as space complexity. Several matroid based algorithms are designed in the oracle model. An independence oracle for a matroid is a black box algorithm which takes as input a subset of the ground set and returns Yes if the set is independent in the matroid and No otherwise. Many algorithms are designed using these oracles. These oracle based algorithms lead to efficient algorithms for problems where we have good algorithms that can act as oracles. A few matroids for which we have efficient oracles include, but are not limited to, graphic matroids, co-graphic matroids, transversal matroids and linear matroids.

Another way of representing a matroid succinctly is by encoding the information about the family of independent sets in a matrix. A matrix A over a field F is called a linear representation of a matroid M=(E,I), if there is a bijection between the columns of A and E and a subset SE is independent in M if and only if the corresponding columns in A are linearly independent over the field F. Note that while not all matroids admit a linear representation, a number of important classes of matroids do. Recently, several algorithmic results have been obtained in the fields of Parameterized Complexity and Exact Algorithms, which require linear representations of certain classes of matroids [1], [2], [3], [4], [5], [6], [7], [8], [9]. This naturally motivates the question of constructing linear representations for various classes of matroids efficiently. Deterministic polynomial time algorithms were known for linear representations of many important classes of matroids such as uniform matroids, partition matroids, graphic matroids and co-graphic matroids. In all these algorithms the running time is polynomial in the size of the ground set. However, for transversal matroids and gammoids, only randomized polynomial time algorithms are known for constructing its linear representations. These matroids feature in many of the results mentioned above, and deterministic algorithms to compute linear representations for them will derandomize several algorithms in literature. In this paper we give a modest improvement over the naïve algorithm for constructing deterministic linear representations for transversal matroids and gammoids.

Let G=(UV,E) be a bipartite graph, the transversal matroid MG on the ground set U has the following family of independent sets: UU is independent in MG if and only if there is a matching in G saturating U. Furthermore assume that |U|=|V|=m and G has a perfect matching. A natural question in this direction is as follows.

Question 1

Could we exploit the fact that G has a perfect matching to design a deterministic polynomial time algorithm to find a linear representation for MG, where we can test whether a subset is independent or not in deterministic polynomial time?

The answer to this question is of course, Yes! An m×m identity matrix is a linear representation of MG. This naturally leads to the following question.

Question 2

Suppose G has a matching of size m, where is a constant. Can we design a deterministic polynomial time algorithm to find a linear representation for MG, where we can test whether a subset is independent or not in deterministic polynomial time?

This question is the starting point of the present work. As mentioned earlier, there is a randomized polynomial time algorithm to obtain a linear representation of a transversal matroid for any bipartite graph. Let G=(UV,E) be a bipartite graph, where U={u1,,um} and V={v1,,vn}. Let X={xi,j|i[n],j[m]}. Define an n×m matrix A as follows: for each i[n],j[m],A[i,j]=0 if viujE and xi,j otherwise. Then for any R[n],C[m],|R|=|C|, det(A[R,C])0 if and only if there is a perfect matching in G[{vi|iR}{uj|jC}]. This implies that A is in fact a linear representation of the transversal matroid MG on the ground set U over the field of fractions F(X), where F is any field.1 Notice that the above construction can be done in deterministic polynomial time.

However, in the representation above, to check whether a set is linearly independent we need to test if the corresponding determinant polynomial, which is a multivariate polynomial, is identically non-zero. This is a case of the well known polynomial identity testing (PIT) problem, and we do not know of a deterministic polynomial time algorithm for this problem. Hence, this representation is difficult to use in deterministic algorithms for many applications [3], [1] (as they require a deterministic test of independence). Furthermore, it is rather difficult to carry out field operations over the field of fractions F(X) in polynomial time.2 As a result, even though we get the above linear representation in deterministic polynomial time, we are not able to use it for deterministic algorithms efficiently. We can obtain another representation, by substituting random values for each xi,jX from a field of size at least 2pm2m, where pN, and succeed with probability at least (112p). This leads to a randomized polynomial time algorithm [1], to obtain a representation over a finite field or a field such as R, where field operations can be carried out efficiently. It appears that derandomizing the above approach has some obstacles, as this will have some important consequences on lower bounds in complexity theory [10]. Observe that the above approach implies a deterministic algorithm of running time 2O(m2n), that tests all possible assignments of at most mn variables from a field of size O(m2m) (setting p=O(1)), since one of them will certainly be a linear representation of MG. Although we have not been able to obtain polynomial time deterministic algorithms for computing linear representations of transversal matroids and gammoids, our results do imply an affirmative answer to Question 2. Our main theorem is the following.

Theorem 1

There is a deterministic algorithm that, given a bipartite graph G=(UV,E) with a maximum matching of size r, outputs a linear representation of the transversal matroid MG=(U,I) over a field F of size >(|U|r), in time O((|U|r)|E||V|+N), where N is the time required to perform O((|U|r)|V||U|) operations over F.

An XP algorithm for a parameterized problem ΠΣ×N, takes as input (I,k)Σ×N and decides whether (I,k)Π in time |I|g(k), where g(k) is a computable function in k alone. Thus, in the language of parameterized complexity, Theorem 1 gives an XP algorithm for finding a linear representation of transversal matroids parameterized by the rank of the given matroid. Observe that if r is the rank of the matroid then the maximum matching of the graph is r. That is, r=|U| and hence =|U|r. This together with the fact that (ma)=(mma) implies that Theorem 1 gives a polynomial time algorithm for Question 2, whenever is a constant.

Transversal matroids are closely related to the class of gammoids. A gammoid is defined by a digraph D and two vertex subsets S and T of V(D). Here, T is the ground set and a subset X of T is independent if and only if X is reachable from S by a collection of |X| vertex disjoint paths. It was shown by Ingleton and Piff [11], that a subclass of gammoids, called strict gammoids where T=V(D), are the duals of transversal matroids. Thus, one can also view gammoids as matroids obtained from strict gammoids by deleting some of the elements from the ground set. Therefore the task of designing an algorithm to construct a linear representation for gammoids is at least as hard as constructing one for transversal matroids. In this work we prove the following theorem.

Theorem 2

There is a deterministic algorithm that, given an n-vertex digraph D and S,TV(D) such that |S|=r and |T|=n, outputs a linear representation of the gammoid defined in D with ground set T, over a field F of size strictly greater than (nr) in time O((nr)n3+N), where N is the time required to perform O((nr)n3) operations over F.

Section snippets

Preliminaries

For nN, we use [n] to denote the set {1,2,,n}. Let U be a set. We use |U| to denote the cardinality of U and 2U to denote the set of subsets of U. For i[|U|], (Ui) denotes the set {SU||S|=i}.

Graphs. We use G to denote a graph and D to denote a digraph. The vertex set and edge (arc) set of a graph (digraph) are denoted as V(G) (V(D)) and E(G) (A(D)) respectively. We also use G=(V,E) (or D=(V,A)) to denote a graph (digraph) with vertex set V and edge set E (arc set A). We use standard

The algorithm for representing transversal matroids

In this section, we first give a deterministic algorithm to compute a linear representation of transversal matroids. In the next section, we show that this algorithm may be modified to obtain more efficient algorithms for other classes of matroids that are related to transversal matroids.

Let G=(UV,E) be a bipartite graph such that U and V contain m and n vertices, respectively. Let U={u1,,um} and V={v1,,vn}. Let MG=(U,I) be the transversal matroid associated with G where the ground set is U.

Representing matroids related to transversal matroids

In this section, we give deterministic algorithms for constructing linear representations of gammoids and strict gammoids. These algorithms utilize the algorithm for constructing linear representation of transversal matroids.

Truncations of transversal matroids. Several algorithmic applications require a linear representation of the k-truncation of matroids [14], [7], [8]. While we can obtain a representation of the k-truncation of a transversal matroid MG, by applying Theorem 3 to a

Conclusion

In this work we gave algorithms to construct linear representations of transversal matroids and gammoids with running time (mr)mO(1), where m is the cardinality of ground set and r is the rank of the matroid. That is we give an XP algorithm, when parameterized by the rank of the matroid. The natural direction forward is to resolve the existence of deterministic FPT algorithms to construct linear representations of transversal matroids and gammoids when parameterized by the rank of the matroid.

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