Linear representation of transversal matroids and gammoids parameterized by rank
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 , where E is called the ground set and is a family of subsets of E, called independent sets, with the following three properties: (i) , (ii) if and , then and (iii) if and , then there is a such that . As the cardinality of could be exponential in , as it is in many applications, explicitly listing 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 is called a linear representation of a matroid , if there is a bijection between the columns of A and E and a subset is independent in M if and only if the corresponding columns in A are linearly independent over the field . 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 be a bipartite graph, the transversal matroid on the ground set U has the following family of independent sets: is independent in if and only if there is a matching in G saturating . Furthermore assume that 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 , 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 identity matrix is a linear representation of . This naturally leads to the following question. Question 2 Suppose G has a matching of size , where ℓ is a constant. Can we design a deterministic polynomial time algorithm to find a linear representation for , 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 be a bipartite graph, where and . Let . Define an matrix A as follows: for each if and otherwise. Then for any , if and only if there is a perfect matching in . This implies that A is in fact a linear representation of the transversal matroid on the ground set U over the field of fractions , where 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 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 from a field of size at least , where , and succeed with probability at least . This leads to a randomized polynomial time algorithm [1], to obtain a representation over a finite field or a field such as , 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 , that tests all possible assignments of at most mn variables from a field of size (setting ), since one of them will certainly be a linear representation of . 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 with a maximum matching of size r, outputs a linear representation of the transversal matroid over a field of size , in time , where N is the time required to perform operations over .
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 . 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 vertex disjoint paths. It was shown by Ingleton and Piff [11], that a subclass of gammoids, called strict gammoids where , 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 such that and , outputs a linear representation of the gammoid defined in D with ground set T, over a field of size strictly greater than in time , where N is the time required to perform operations over .
Section snippets
Preliminaries
For , we use to denote the set . Let U be a set. We use to denote the cardinality of U and to denote the set of subsets of U. For , denotes the set .
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 () and () respectively. We also use (or ) 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 be a bipartite graph such that U and V contain m and n vertices, respectively. Let and . Let 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 , 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 , 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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