Decomposable multi-parameter matroid optimization problems (original) (raw)
On the generality of the greedy algorithm for solving matroid base problems
Discrete Applied Mathematics, 2014
It is well known that the greedy algorithm solves matroid base problems for all linear cost functions and is, in fact, correct if and only if the underlying combinatorial structure of the problem is a matroid. Moreover, the algorithm can be applied to problems with sum, bottleneck, algebraic sum or k-sum objective functions. In this paper, we address matroid base problems with a more general-"universal"-objective function which contains the previous ones as special cases. This universal objective function is of the sum type and associates multiplicative weights with the ordered cost coefficients of the elements of matroid bases such that, by choosing appropriate weights, many different-classical and new-objectives can be modeled. We show that the greedy algorithm is applicable to a larger class of objective functions than commonly known and, as such, it solves universal matroid base problems with non-negative or non-positive weight coefficients. Based on problems with mixed weights and a single (−, +)-sign change in the universal weight vector, we give a characterization of uniform matroids. In case of multiple sign changes, we use partition matroids. For non-uniform matroids, single sign change problems can be reduced to problems in minors obtained by deletion and contraction. Finally, we discuss how special instances of universal bipartite matching and shortest
SIAM Journal on Discrete Mathematics, 2003
We consider a problem of optimizing convex functionals over matroid bases. It is richly expressive and captures certain quadratic assignment and clustering problems. While generally NP-hard, we show it is polynomial time solvable when a suitable parameter is restricted.
Parametric nonlinear discrete optimization over well-described sets and matroid intersections
Mathematical Programming, 2010
We address optimization of parametric nonlinear functions of the form 1 f (W x), where f : R d → R is a nonlinear function, W is a d × n matrix, and feasible 2 x are in some large finite set F of integer points in R n . Generally, such problems 3 are intractable, so we obtain positive algorithmic results by looking at broad natural 4 classes of f, W and F. One of our main motivations is multi-objective discrete opti-5 mization, where f trades off the linear functions given by the rows of W . Another 6 motivation is that we want to extend as much as possible the known results about 7 polynomial-time linear optimization over trees, assignments, matroids, polymatroids, 8
Branch Decomposition Heuristics for Linear Matroids
Manuscript in preparation
This paper presents two new heuristics which utilize classification and max-flow algorithms respectively to derive near-optimal branch decompositions for linear matroids. In the literature, there are already excellent heuristics for graphs, however, no practical branch decomposition methods for general linear matroids have been addressed yet. Introducing a "measure" which compares the "similarity" of elements of a linear matroid, this work reforms the linear matroid into a similarity graph. Then, the classification method and the max-flow method, both based on the similarity graph, are utilized on the similarity graph to derive separations for a near-optimal branch decomposition. Computational results using the classification method and the max-flow method on linear matroid instances are shown respectively. * 1 techniques on input graphs with bounded branchwidth. The original result is about bounded treewidth, the invariant associated with tree decompositions of graphs-another byproduct of Robertson and Seymour's proof of Wagner's conjecture. In contrast, the result is equivalent for bounded branchwidth, since the branchwidth and treewidth of a graph bound each other by constants .
Linear-time algorithms for parametric minimum spanning tree problems on planar graphs
Theoretical Computer Science, 1997
A linear-time algorithm for the minimum-ratio spanning tree problem on planar graphs is presented. The algorithm is based on a new planar minimum spanning tree algorithm. The approach extends to other parametric minimum spanning tree problems on planar graphs and to other families of graphs having small separators. ' Corresponding author. '
Matroid Secretary Problem in the Random Assignment Model
2010
In the Matroid Secretary Problem, introduced by Babaioff et al. [SODA 2007], the elements of a given matroid are presented to an online algorithm in random order. When an element is revealed, the algorithm learns its weight and decides whether or not to select it under the restriction that the selected elements form an independent set in the matroid. The objective is to maximize the total weight of the chosen elements. In the most studied version of this problem, the algorithm has no information about the weights beforehand. We refer to this as the zero information model. In this paper we study a different model, also proposed by Babaioff et al., in which the relative order of the weights is random in the matroid. To be precise, in the random assignment model, an adversary selects a collection of weights that are randomly assigned to the elements of the matroid. Later, the elements are revealed to the algorithm in a random order independent of the assignment.
The 2-quasi-greedy algorithm for cardinality constrained matroid bases
Discrete Applied Mathematics, 1986
The quasi-greedy algorithm, as proposed by Glover and Klingman [8], efficiently solves minimum weight spanning tree problems with a fixed (or bounded) number of edges incident to a specified vertex. As observed in [8], the results carry through to general matroid problems (where a base contains a bounded number of elements from a specified set). We extend this work to provide an efficient 2-quasi-greedy algorithm where a minimum weight base is constrained to have a fixed number of elements from two disjoint sets.
A Deterministic Parallel Reduction from Weighted Matroid Intersection Search to Decision
Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 2022
Given two matroids on the same ground set, the matroid intersection problem asks for a common base, i.e., a subset of the ground set that is a base in both the matroids. The weighted version of the problem asks for a common base with maximum weight. In the general case, when the two matroids are given via rank oracles, the question of its parallel complexity is completely open. In the case of linearly representable matroids, the problem is known to have randomized parallel (RNC) algorithms, when the given weights are polynomially bounded. Finding a deterministic parallel (NC) algorithm in this case, even for the decision question, has been a long standing open question. We make some progress towards understanding the parallel complexity of matroid intersection by showing that the weighted matroid intersection (WMI) search problem is equivalent to its decision version, in a parallel model of computation. More precisely, we give an NC algorithm for WMI-search using an oracle access to WMI-decision. This resolves an open question posed by Anari and Vazirani (ITCS 2020).
Using sparsification for parametric minimum spanning tree problems
Lecture Notes in Computer Science, 1996
Two applications of sparsification to parametric computing are given. The first is a fast algorithm for enumerating all distinct minimum spanning trees in a graph whose edge weights vary linearly with a parameter. The second is an asymptotically optimal algorithm for the minimum ratio spanning tree problem, as well as other search problems, on dense graphs.
A general model for matroids and the greedy algorithm
Mathematical Programming, 2008
We present a general model for set systems to be independence families with respect to set families which determine classes of proper weight functions on a ground set. Within this model, matroids arise from a natural subclass and can be characterized by the optimality of the greedy algorithm. This model includes and extends many of the models for generalized matroid-type greedy algorithms proposed in the literature and, in particular, integral polymatroids. We discuss the relationship between these general matroids and classical matroids and provide a Dilworth embedding that allows us to represent matroids with underlying partial order structures within classical matroids. Whether a similar representation is possible for matroids on convex geometries is an open question.
On the Connection of Matroids and Greedy Algorithms
IJEER, 2022
Matroids are the combinatorial structure and Greedy algorithmic methods always produces optimal solutions for these mathematical models. A greedy method always selects the option that looks best at each step of process of finding optimal solution. In other words, it selects a choice which is optimal choice locally in such a strategy that this locally chosen option may direct to a solution that will be globally optimal. It is true that while selecting locally optimal solution at each stage, Greedy algorithms may not always yield optimal solutions, but if we can transform an unknown problem into matroid structure, then there must be a greedy algorithm that will always lead optimal solution for that unknown problem. The range of solutions provided by Greedy is large as compared to the applicability of the Matroid structure. In other words the problems that can be translated into Matroid structure is proper subset of set of all problems whether Greedy algorithm produces optimal solution. Matroid structure thus ensures the global optimal solution one can obtain with help of Greedy approach. We study various logarithmic and linear hierarchical based mathematical models from divergence sources to maximize our information for research purposes. We analyze the time complexity and provide constrains over the upper/lower bounds in correspondence with the optimal (maximum/minimum) solution. We try to establish the relationship between the maximization of information divergences, the optimal-likelihood theory, and classified sharing is instituted. We propose integration of unknown rough sets to matroids in this paper. Particularly, we devise methodically the upper and lower tightening bounds on rough matroids which may expand up to the generic combinatorial matroid structure. The relationships are established by the upper and lower tightening bounds approximations of generalized combinatorial rough sets based on different interdependent relation sets, respectively. As we define the generalized lower/upper bounds for rough matroid, we define a new structure for lower/upper greedoid leading to generalization of the greedoid. Additionally, based on the new established relation, the generalized rough set also provides a theory of poset matroid.
The capacitated minimum spanning tree problem
Tdx, 2013
Along the development of this thesis I've had the pleasure of working with some researchers from whom I learnt a lot. I am specially grateful to Elena Fernández and Mari Albareda for their unconditional support and patience, specially in the most difficult moments. I also want to thank Mauricio Resende for his contributions to do part of this work. Most of this work was developed in the Departament d'Estadística i Investigació Operativa of the Universitat Politècnica de Catalunya. I want to thank the department and all its members for all the facilities and help I found here. Specially to Sonia Navarro and Laura Marí for their help during the thesis deposit. I also want to thank the CONACYT for their financial support. Finally, I want to thank my parents for the gift of life; my brothers and sister for all the time and experiences that we have shared; and all my friends for their support and friendship.
Reoptimization of the minimum spanning tree
Wiley Interdisciplinary Reviews: Computational Statistics, 2011
We implement a fast reoptimization algorithm for MIN SPANNING TREE under vertex insertions, initially proposed and analyzed in the work of Boria and Paschos [Boria N, Paschos VTh. Fast reoptimization for the minimum spanning tree problem. J Discrete Algor 2010, 8:296-310] and study its experimental approximation behavior in randomly generated graphs. The reoptimization setting can briefly be formulated as follows: given an instance of the problem for which we already know some optimal solution, and given some 'small' perturbations on this instance, is it possible to compute a new (optimal or at least near-optimal) solution for the modified instance without computation from scratch? We focus in this article on the most popular modification: vertex-insertion.
Fast reoptimization for the minimum spanning tree problem
Journal of Discrete Algorithms, 2010
Minimum spanning tree is a classical polynomial problem very well known in operational research and in theoretical computer science. In this paper, we settle the reoptimization versions of this problem, which can be formulated as follows: given an instance of the problem for which we already know some optimal solution, and given some "small" perturbations on this initial instance, is it possible to compute a new (optimal or at least near-optimal) solution for the modified instance without ex nihilo computation? We focus on two kinds of modifications: node-insertions and node-deletions. For the former type of modifications, where k new nodes are inserted together with their incident edges, we first propose a fast strategy with complexity O(kn) which provides a max{2, 3 − (2/(k − 1))}-approximation ratio, in complete metric graphs. We then devise a more elaborated strategy that computes optimal solutions in any graph with complexity O(kn log n). When k nodes are deleted, we devise a strategy which in O(n) achieves approximation ratio bounded above by 2⌈|L max |/2⌉ in complete metric graphs, where L max is the longest deleted path and |L max | is the number of its edges. For any of the approximation strategies, we also provide lower bounds on their approximation ratios.
A note on packing spanning trees in graphs and bases in matroids
Cornell University - arXiv, 2012
We consider the class of graphs for which the edge connectivity is equal to the maximum number of edge-disjoint spanning trees, and the natural generalization to matroids, where the cogirth is equal to the number of disjoint bases. We provide descriptions of such graphs and matroids, showing that such a graph (or matroid) has a unique decomposition. In the case of graphs, our results are relevant for certain communication protocols.
A new exchange property for matroids and its application to max-min-problems
Zeitschrift für Operations Research, 1984
We provide a new characterization for matroids which is useful for sensitivity analysis and for solving certain max min matroid optimization problems. Zusammenfassung: Es wird eine neue Charakterisierung von Matroiden angegeben, welche ftir eine Sensitivitiitsanalyse und fiir eine L6sung yon Max-Min-Optimierungsproblemen ffir Matroide niitzlich ist.
Matroid Secretary for Regular and Decomposable Matroids
SIAM Journal on Computing, 2014
In the matroid secretary problem we are given a stream of elements and asked to choose a set of elements that maximizes the total value of the set, subject to being an independent set of a matroid given in advance. The difficulty comes from the assumption that decisions are irrevocable: if we choose to accept an element when it is presented by the stream then we can never get rid of it, and if we choose not to accept it then we cannot later add it. Babaioff, Immorlica, and Kleinberg [SODA 2007] introduced this problem, gave O(1)-competitive algorithms for certain classes of matroids, and conjectured that every matroid admits an O(1)-competitive algorithm. However, most matroids that are known to admit an O(1)-competitive algorithm can be easily represented using graphs (e.g. graphic, cographic, and transversal matroids). In particular, there is very little known about F -representable matroids (the class of matroids that can be represented as elements of a vector space over a field F ), which are one of the foundational types of matroids. Moreover, most of the known techniques are as dependent on graph theory as they are on matroid theory. We go beyond graphs by giving O(1)-competitive algorithms for regular matroids (the class of matroids that are representable over any field), and use techniques that are fundamentally matroid-theoretic rather than graph-theoretic.
Lecture Notes in Computer Science, 2014
This article introduces the lazy matroid problem, which captures the goal of saving time or money in certain task selection scenarios. We are given a budget B and a matroid M with weights on its elements. The problem consists in finding an independent set F of minimum weight. In addition, F is feasible if its augmentation with any new element x implies that either F + x exceeds B or F + x is dependent. Our first result is a polynomial time approximation scheme for this NPhard problem which generalizes a recently studied version of the lazy bureaucrat problem. We next study the approximability of a more general setting called lazy staff matroid. In this generalization, every element of M has a multidimensional weight. We show that approximating this generalization is much harder than for the lazy matroid problem since it includes the independent dominating set problem.