Matroid Intersection Problem Research Articles

Multi-objective matroid optimization with ordinal weights

Bi-objective optimization problems on matroids are in general intractable and their corresponding decision problems are in general NP-hard. However, if one of the objective functions is restricted to binary cost coefficients the problem becomes efficiently solvable by an efficient swap algorithm. Binary cost coefficients often represent two categories and are thus a special case of ordinal coefficients that are in general non-additive. In this paper we consider ordinal objective functions with more than two categories in the context of matroid optimization. We introduce several problem variants that can be distinguished w.r.t. their respective optimization goals, analyze their interrelations, and derive a polynomial time solution method that is based on the repeated solution of matroid intersection problems. Numerical tests on minimum spanning tree problems and on partition matroids confirm the efficiency of the approach.

Matroid Intersection under Restricted Oracles

Matroid intersection is one of the most powerful frameworks of matroid theory that generalizes various problems in combinatorial optimization. Edmonds’ fundamental theorem provides a min-max characterization for the unweighted setting, while Frank’s weight-splitting theorem provides one for the weighted case. Several efficient algorithms were developed for these problems, all relying on the usage of one of the conventional oracles for both matroids. In the present paper, we consider the tractability of the matroid intersection problem under restricted oracles. In particular, we focus on the rank sum, common independence, and maximum rank oracles. We give a strongly polynomial-time algorithm for weighted matroid intersection under the rank sum oracle. In the common independence oracle model, we prove that the unweighted matroid intersection problem is tractable when one of the matroids is a partition matroid and that even the weighted case is solvable when one of the matroids is an elementary split matroid. Finally, we show that the common independence and maximum rank oracles together are strong enough to realize the steps of our algorithm under the rank sum oracle.

Pfaffian Pairs and Parities: Counting on Linear Matroid Intersection and Parity Problems

Spanning trees are a representative example of linear matroid bases that are efficiently countable. Perfect matchings of Pfaffian bipartite graphs are a countable example of common bases of two matrices. Generalizing these two, Webb Counting Bases, Ph.D. thesis, University of Waterloo, 2004 introduced the notion of Pfaffian pairs as a pair of matrices for which counting of their common bases is tractable via the Cauchy--Binet formula. This paper studies counting on linear matroid problems extending Webb's work. We first introduce “Pfaffian parities” as an extension of Pfaffian pairs to the linear matroid parity problem, which is a common generalization of the linear matroid intersection problem and the matching problem. We show that a large number of efficiently countable discrete structures are interpretable as special cases of Pfaffian pairs and parities. Our study then turns to algorithmic aspects. We observe that the fastest randomized algorithms for the linear matroid intersection and parity problems by Harvey SIAM J. Comput., 39 (2009), pp. 679--702 and Cheung, Lau, and Leung ACM Trans. Algorithms, 10 (2014), pp. 1--26 can be derandomized for Pfaffian pairs and parities. We further present polynomial-time algorithms to count the number of minimum-weight solutions on weighted Pfaffian pairs and parities. Our algorithms make use of Frank's weight-splitting lemma for the weighted matroid intersection problem and the algebraic optimality criterion of the weighted linear matroid parity problem given by Iwata and Kobayashi SIAM J. Comput., 51 (2002) pp. STOC17-238--STOC17-280.

Approximation by lexicographically maximal solutions in matching and matroid intersection problems

We study how good a lexicographically maximal solution is in the weighted matching and matroid intersection problems. A solution is lexicographically maximal if it takes as many heaviest elements as possible, and subject to this, it takes as many second heaviest elements as possible, and so on. If the distinct weight values are sufficiently dispersed, e.g., the minimum ratio of two distinct weight values is at least the ground set size, then the lexicographical maximality and the usual weighted optimality are equivalent. We show that the threshold of the ratio for this equivalence to hold is exactly 2. Furthermore, we prove that if the ratio is less than 2, say α, then a lexicographically maximal solution achieves (α/2)-approximation, and this bound is tight.

A Weighted Linear Matroid Parity Algorithm

The matroid parity (or matroid matching) problem, introduced as a common generalization of matching and matroid intersection problems, is so general that it requires an exponential number of oracle...

Linear Matroid Intersection is in Quasi-NC

Given two matroids on the same ground set, the matroid intersection problem asks to find a common independent set of maximum size. In case of linear matroids, the problem had a randomized parallel algorithm but no deterministic one. We give an almost complete derandomization of this algorithm, which implies that the linear matroid intersection problem is in quasi-NC. That is, it has uniform circuits of quasi-polynomial size $$n^{O(\log n)}$$ and O(polylog(n)) depth. Moreover, the depth of the circuit can be reduced to O(log2 n) in case of zero characteristic fields. This generalizes a similar result for the bipartite perfect matching problem. Our main technical contribution is to derandomize the Isolation lemma for the family of common bases of two matroids. We use our isolation result to give a quasi-polynomial time blackbox algorithm for a special case of Edmonds' problem, i.e., singularity testing of a symbolic matrix, when the given matrix is of the form $$A_{0} + A_{1 }x_{1} + \cdots + A_{m} x_{m}$$ , for an arbitrary matrix A0 and rank-1 matrices $$A_{1}, A_{2}, \dots, A_{m}$$ . This can also be viewed as a blackbox polynomial identity testing algorithm for the corresponding determinant polynomial. Another consequence of this result is a deterministic solution to the maximum rank matrix completion problem. Finally, we use our result to find a deterministic representation for the union of linear matroids in quasi-NC.

Strongly Possible Keys for SQL

Missing data value is an extensive problem in both research and industrial developers. Two general approaches are there to deal with the problem of missing values in databases; they could be either ignored (removed) or imputed (filled in) with new values (Farhangfar et al. in IEEE Trans Syst Man Cybern-Part A: Syst Hum 37(5):692–709, 2007). For some SQL tables, it is possible that some candidate key of the table is not null-free and this needs to be handled. Possible keys and certain keys to deal with this situation were introduced in Köhler et al. (VLDB J 25(4):571–596, 2016). In the present paper, we introduce an intermediate concept called strongly possible keys that is based on a data mining approach using only information already contained in the SQL table. A strongly possible key is a key that holds for some possible world which is obtained by replacing any occurrences of nulls with some values already appearing in the corresponding attributes. Implication among strongly possible keys is characterized, and Armstrong tables are constructed. An algorithm to verify a strongly possible key is given applying bipartite matching. Connection between matroid intersection problem and system of strongly possible keys is established. For the cases when no strongly possible keys hold, an approximation notion is introduced to calculate the closeness of any given set of attributes to be considered as a strongly possible key using the g_3 measure, and we derive its component version g_4. Analytical comparisons are given between the two measures.

On a weighted linear matroid intersection algorithm by Deg-Det computation

In this paper, we address the weighted linear matroid intersection problem from the computation of the degree of the determinants of a symbolic matrix. We show that a generic algorithm computing the degree of noncommutative determinants, proposed by the second author, becomes an $O(mn^3 \log n)$ time algorithm for the weighted linear matroid intersection problem, where two matroids are given by column vectors $n \times m$ matrices $A,B$. We reveal that our algorithm is viewed as a "nonstandard" implementation of Frank's weight splitting algorithm for linear matroids. This gives a linear algebraic reasoning to Frank's algorithm. Although our algorithm is slower than existing algorithms in the worst case estimate, it has a notable feature: Contrary to existing algorithms, our algorithm works on different matroids represented by another "sparse" matrices $A^0,B^0$, which skips unnecessary Gaussian eliminations for constructing residual graphs.

On resolvability of a graph associated to a finite vector space

In this paper, the resolving parameters such as metric dimension and partition dimension for the nonzero component graph, associated to a finite vector space, are discussed. The exact values of these parameters are determined. It is derived that the notions of metric dimension and locating-domination number coincide in the graph. Independent sets, introduced by Boutin [Determining sets, resolving set, and the exchange property, Graphs Combin. 25 (2009) 789–806], are studied in the graph. It is shown that the exchange property holds in the graph for minimal resolving sets with some exceptions. Consequently, a minimal resolving set of the graph is a basis for a matroid with the set [Formula: see text] of nonzero vectors of the vector space as the ground set. The matroid intersection problem for two matroids with [Formula: see text] as the ground set is also solved.

Exact and approximation algorithms for weighted matroid intersection

In this paper, we propose new exact and approximation algorithms for the weighted matroid intersection problem. Our exact algorithm is faster than previous algorithms when the largest weight is relatively small. Our approximation algorithm delivers a $$(1-\epsilon )$$-approximate solution with a running time significantly faster than most known exact algorithms. The core of our algorithms is a decomposition technique: we decompose an instance of the weighted matroid intersection problem into a set of instances of the unweighted matroid intersection problem. The computational advantage of this approach is that we can make use of fast unweighted matroid intersection algorithms as a black box for designing algorithms. More precisely, we show that we can solve the weighted matroid intersection problem via solving W instances of the unweighted matroid intersection problem, where W is the largest given weight, assuming that all given weights are integral. Furthermore, we can find a $$(1-\epsilon )$$-approximate solution via solving $$O(\epsilon ^{-1} \log r)$$ instances of the unweighted matroid intersection problem, where r is the smaller rank of the two given matroids. Our algorithms make use of the weight-splitting approach of Frank (J Algorithms 2(4):328–336, 1981) and the geometric scaling scheme of Duan and Pettie (J ACM 61(1):1, 2014). Our algorithms are simple and flexible: they can be adapted to special cases of the weighted matroid intersection problem, using specialized unweighted matroid intersection algorithms. In addition, we give a further application of our decomposition technique: we solve efficiently the rank-maximal matroid intersection problem, a problem motivated by matching problems under preferences.

Chain-constrained spanning trees

We consider the problem of finding a spanning tree satisfying a family of additional constraints. Several settings have been considered previously, the most famous being the problem of finding a spanning tree with degree constraints. Since the problem is hard, the goal is typically to find a spanning tree that violates the constraints as little as possible. Iterative rounding has become the tool of choice for constrained spanning tree problems. However, iterative rounding approaches are very hard to adapt to settings where an edge can be part of more than a constant number of constraints. We consider a natural constrained spanning tree problem of this type, namely where upper bounds are imposed on a family of cuts forming a chain. Our approach reduces the problem to a family of independent matroid intersection problems, leading to a spanning tree that violates each constraint by a factor of at most 9. We also present strong hardness results: among other implications, these are the first to show, in the setting of a basic constrained spanning tree problem, a qualitative difference between what can be achieved when allowing multiplicative as opposed to additive constraint violations.

Polynomial-Time Approximation Schemes for Maximizing Gross Substitutes Utility Under Budget Constraints

We consider the maximization of a gross substitutes utility function under budget constraints. This problem naturally arises in applications such as exchange economies in mathematical economics and combinatorial auctions in (algorithmic) game theory. We show that this problem admits a polynomial-time approximation scheme (PTAS). More generally, we present a PTAS for maximizing a discrete concave function called an M♮-concave function under budget constraints. Our PTAS is based on rounding an optimal solution of a continuous relaxation problem, which is shown to be solvable in polynomial time by the ellipsoid method. We also consider the maximization of the sum of two M♮-concave functions under a single budget constraint. This problem is a generalization of the budgeted max-weight matroid intersection problem to the one with certain nonlinear objective functions. We show that this problem also admits a PTAS.

MATROID INTERSECTION WITH PRIORITY CONSTRAINTS

In this paper, we consider the following varian to f the matroid intersection problem. We are given two matroids M1,M2 on the same ground set E and a subset A of E. Our goal is to find a common independent set I of M1,M2 such that |I ∩A| is maximum among all common independent sets of M1,M2 and such that (secondly) |I| is maximum among all common independent sets of M1,M2 satisfying the first condition. This problem is a matroid-generalization of the simplest case of the rank-maximal matching problem introduced by Irving, Kavitha, Mehlhorn, Michail and Paluch (2006). In this paper, we extend the combinatorial algorithm of Irving et al. for the rank-maximal matching problem to our problem by using a Dulmage-Mendelsohn type decomposition for the matroid intersection problem.

Robust Matchings and Matroid Intersections

In a weighted independence system, an independent set is said to be $\alpha$-robust if, for all $p$, the total weight of its heaviest $p$ elements is at least $\alpha$ times the maximum weight of a $p$-independent set. Here a $p$-independent set is an independent set with at most $p$ elements. The set of matchings in a weighted graph is a typical example of a weighted independence system, and Hassin and Rubinstein [SIAM J. Discrete Math., 15 (2002), pp. 530--537] showed that every graph has a $\frac{1}{\sqrt2}$-robust matching and it can be found by a $k$th power algorithm in polynomial time. In this paper, we show that it can be extended to the matroid intersection problem; i.e., there always exists a $\frac{1}{\sqrt2}$-robust matroid intersection, which is polynomially computable. We also study the time complexity of the robust matching problem. We show that a 1-robust matching can be computed in polynomial time (if one exists), and, for any fixed number $\alpha$ with $\frac{1}{\sqrt2}<\alpha<1$, the problem to determine whether a given weighted graph has an $\alpha$-robust matching is NP-complete. These together with the positive result for $\alpha=\frac{1}{\sqrt2}$ in [R. Hassin and S. Rubinstein, SIAM J. Discrete Math., 15 (2002), pp. 530--537] give us a sharp border for the complexity for the robust matching problem. Moreover, we show that the problem is strongly NP-complete when $\alpha$ is a part of the input. Finally, we show the limitations of the $k$th power algorithm for robust matchings; i.e., for any $\epsilon>0$, there exists a weighted graph such that no $k$th power algorithm outputs a $(\frac{1}{\sqrt2}+\epsilon)$-approximation for computing the most robust matching.

Shortest bibranchings and valuated matroid intersection

For a digraph D = (V, A) and a partition {S, T} of V, an arc set \({B \subseteq A}\) is called an S−T if each vertex in T is reachable from S and each vertex in S reaches T in the subgraph (V, B). Bibranchings commonly generalize bipartite edge covers and arborescences. A totally dual integral linear system determining the S−T polytope is provided by Schrijver, and the shortest S−T problem, whose objective is to find an S−T of minimum total arc weight, can be solved in polynomial time by the ellipsoid method or a faster combinatorial algorithm due to Keijsper and Pendavingh. The valuated matroid intersection problem, introduced by Murota, is a weighted generalization of the independent matching problem, including the independent assignment problem and the weighted matroid intersection problem. The valuated matroid intersection problem can be solved efficiently with polynomially many value oracles by extending classical combinatorial algorithms for the weighted matroid intersection problem. In this paper, we show that the shortest S−T problem is polynomially reducible to the valuated matroid intersection problem. This reduction suggests one answer to why the shortest S−T problem is tractable, and implies new combinatorial algorithms for the shortest S−T problem based on the valuated matroid intersection algorithm, where a value oracle corresponds to computing a minimum-weight arborescence.

A weighted independent even factor algorithm

An even factor in a digraph is a vertex-disjoint collection of directed cycles of even length and directed paths. An even factor is called independent if it satisfies a certain matroid constraint. The problem of finding an independent even factor of maximum size is a common generalization of the nonbipartite matching and matroid intersection problems. In this paper, we present a primal-dual algorithm for the weighted independent even factor problem in odd-cycle-symmetric weighted digraphs. Cunningham and Geelen have shown that this problem is solvable via valuated matroid intersection. Their method yields a combinatorial algorithm running in O(n 3 γ + n 6 m) time, where n and m are the number of vertices and edges, respectively, and γ is the time for an independence test. In contrast, combining the weighted even factor and independent even factor algorithms, our algorithm works more directly and runs in O(n 4 γ + n 5) time. The algorithm is fully combinatorial, and thus provides a new dual integrality theorem which commonly extends the total dual integrality theorems for matching and matroid intersection.

Covering directed graphs by in-trees

Given a directed graph D=(V,A) with a set of d specified vertices S={s 1,?,s d }?V and a function f : S?? where ? denotes the set of positive integers, we consider the problem which asks whether there exist ? i=1 d f(s i ) in-trees denoted by $T_{i,1},T_{i,2},\ldots,T_{i,f(s_{i})}$ for every i=1,?,d such that $T_{i,1},\ldots,T_{i,f(s_{i})}$ are rooted at s i , each T i,j spans vertices from which s i is reachable and the union of all arc sets of T i,j for i=1,?,d and j=1,?,f(s i ) covers A. In this paper, we prove that such set of in-trees covering A can be found by using an algorithm for the weighted matroid intersection problem in time bounded by a polynomial in ? i=1 d f(s i ) and the size of D. Furthermore, for the case where D is acyclic, we present another characterization of the existence of in-trees covering A, and then we prove that in-trees covering A can be computed more efficiently than the general case by finding maximum matchings in a series of bipartite graphs.

Facets of the independent path-matching polytope

Cunningham and Geelen introduced the independent path-matching problem as a common generalization of the weighted matching problem and the weighted matroid intersection problem. Associated with an independent path-matching is an independent path-matching vector. The independent path-matching polytope of an instance of the independent path-matching problem is the convex hull of all the independent path-matching vectors. Cunningham and Geelen described a system of linear inequalities defining the independent path-matching polytope. In this paper, we characterize which inequalities in this system induce facets of the independent path-matching polytope, generalizing previous results on the matching polytope and the common independent set polytope.

Matching, matroids, and extensions

Perhaps the two most fundamental well-solved models in combinatorial optimization are the optimal matching problem and the optimal matroid intersection problem. We review the basic results for both, and describe some more recent advances. Then we discuss extensions of these models, in particular, two recent ones--jump systems and path-matchings.

Conjugate Scaling Algorithm for Fenchel-Type Duality in Discrete Convex Optimization

This paper presents a polynomial time algorithm for solving submodular flow problems with a class of discrete convex cost functions. This class of problems is a common generalization of the submodular flow and valuated matroid intersection problems. The algorithm adopts a new scaling technique that scales the discrete convex cost functions via the conjugacy relation. The algorithm can be used to find a pair of optima in the form of the Fenchel-type duality theorem in discrete convex analysis.