Matroids Of Rank Research Articles

Maximal matroids in weak order posets

Let X be a family of subsets of a finite set E. A matroid on E is called an X-matroid if each set in X is a circuit. We develop techniques for determining when there exists a unique maximal X-matroid in the weak order poset of all X-matroids on E and formulate a conjecture which would characterise the rank function of this unique maximal matroid when it exists. The conjecture suggests a new type of matroid rank function which extends the concept of weakly saturated sequences from extremal graph theory. We verify the conjecture for various families X and show that, if true, the conjecture could have important applications in such areas as combinatorial rigidity and low rank matrix completion.

Multiagent MST Cover: Pleasing All Optimally via a Simple Voting Rule

Given a connected graph on whose edges we can build roads to connect the nodes, a number of agents hold possibly different perspectives on which edges should be selected by assigning different edge weights. Our task is to build a minimum number of roads so that every agent has a spanning tree in the built subgraph whose weight is the same as a minimum spanning tree in the original graph. We first show that this problem is NP-hard and does not admit better than ((1-o(1)) ln k)-approximation polynomial-time algorithms unless P = NP, where k is the number of agents. We then give a simple voting algorithm with an optimal approximation ratio. Moreover, our algorithm only needs to access the agents' rankings on the edges. Finally, we extend our problem to submodular objective functions and Matroid rank constraints.

Short proof of a theorem of Brylawski on the coefficients of the Tutte polynomial

In this short note we show that a system M=(E,r) with a ground set E of size m and (rank) function r:2E→Z≥0 satisfying r(S)≤min(r(E),|S|) for every set S⊆E, the Tutte polynomial TM(x,y)≔∑S⊆E(x−1)r(E)−r(S)(y−1)|S|−r(S),written as TM(x,y)=∑i,jtijxiyj, satisfies that for any integer h≥0, we have ∑i=0h∑j=0h−ih−ij(−1)jtij=(−1)m−rh−rh−m,where r=r(E), and we use the convention that when h<m, the binomial coefficient h−rh−m is interpreted as 0.This generalizes a theorem of Brylawski on matroid rank functions and h<m, and a theorem of Gordon for h≤m with the same assumptions on the rank function.The proof presented here is significantly shorter than the previous ones. We only use the fact that the Tutte polynomial TM(x,y) simplifies to (x−1)r(E)y|E| along the hyperbola (x−1)(y−1)=1.

Special Case of Rota's Basis Conjecture on Graphic Matroids

Gian-Carlo Rota conjectured that for any $n$ bases $B_1,B_2,\ldots,B_n$ in a matroid of rank $n$, there exist $n$ disjoint transversal bases of $B_1,B_2,\ldots,B_n$. The conjecture for graphic matroids corresponds to the problem of an edge-decomposition as follows; If an edge-colored connected multigraph $G$ has $n-1$ colors and the graph induced by the edges colored with $c$ is a spanning tree for each color $c$, then $G$ has $n-1$ mutually edge-disjoint rainbow spanning trees. In this paper, we prove that edge-colored graphs where the edges colored with $c$ induce a spanning star for each color $c$ can be decomposed into rainbow spanning trees.

Log-concave polynomials, I: Entropy and a deterministic approximation algorithm for counting bases of matroids

We give a deterministic polynomial-time 2O(r)-approximation algorithm for the number of bases of a given matroid of rank r and the number of common bases of any two matroids of rank r. To the best of our knowledge, this is the first nontrivial deterministic approximation algorithm that works for arbitrary matroids. Based on a lower bound of Azar, Broder, and Frieze, this is almost the best possible result assuming oracle access to independent sets of the matroid. There are two main ingredients in our result. For the first, we build upon recent results of Adiprasito, Huh, Katz, and Wang on combinatorial Hodge theory to show that the basis generating polynomial of any matroid is a (completely) log-concave polynomial. Formally, we prove that the multivariate generating polynomial of the bases of any matroid is (and all of its directional derivatives along the positive orthant are) log-concave as functions over the positive orthant. For the second ingredient, we develop a general framework for approximate counting in discrete problems, based on convex optimization. The connection goes through subadditivity of the entropy. For matroids, we prove that an approximate superadditivity of the entropy holds by relying on the log-concavity of the basis generating polynomial.

Serial exchanges in matroids

Suppose B1 and B2 are two bases in a matroid M. For subsets X⊆B1 and Y⊆B2, we say that X is serially exchangeable with Y if there exist orderings x1≺x2≺⋯≺xk and y1≺y2≺⋯≺yk of X and Y, respectively, such that for i=1,…,k, B1−x1−⋯−xi+y1+⋯+yi and B2−y1−⋯−yi+x1+⋯+xi are bases. Kotlar and Ziv [9] conjectured that for any X⊆B1, there exists Y⊆B2 for which X is serially exchangeable with Y. Let Mq be the set of matroids representable over GF(q). In this paper, we show that to prove the above conjecture for matroids in Mq, it suffices to prove it for such matroids of rank at most (k+2)q2k.In second part of the paper, we show that if M is binary, and |X|≤3, then there exists Y⊆B2 for which X is serially exchangeable with Y.

Finding Fair and Efficient Allocations for Matroid Rank Valuations

In this article, we present new results on the fair and efficient allocation of indivisible goods to agents whose preferences correspond to matroid rank functions . This is a versatile valuation class with several desirable properties (such as monotonicity and submodularity), which naturally lends itself to a number of real-world domains. We use these properties to our advantage; first, we show that when agent valuations are matroid rank functions, a socially optimal (i.e., utilitarian social welfare-maximizing) allocation that achieves envy-freeness up to one item (EF1) exists and is computationally tractable. We also prove that the Nash welfare-maximizing and the leximin allocations both exhibit this fairness/efficiency combination by showing that they can be achieved by minimizing any symmetric strictly convex function over utilitarian optimal outcomes. To the best of our knowledge, this is the first valuation function class not subsumed by additive valuations for which it has been established that an allocation maximizing Nash welfare is EF1. Moreover, for a subclass of these valuation functions based on maximum (unweighted) bipartite matching, we show that a leximin allocation can be computed in polynomial time. Additionally, we explore possible extensions of our results to fairness criteria other than EF1 as well as to generalizations of the above valuation classes.

Tropical ideals do not realise all Bergman fans

Every tropical ideal in the sense of Maclagan–Rincón has an associated tropical variety, a finite polyhedral complex equipped with positive integral weights on its maximal cells. This leads to the realisability question, ubiquitous in tropical geometry, of which weighted polyhedral complexes arise in this manner. Using work of Las Vergnas on the non-existence of tensor products of matroids, we prove that there is no tropical ideal whose variety is the Bergman fan of the direct sum of the Vámos matroid and the uniform matroid of rank two on three elements and in which all maximal cones have weight one.

Approximating nash social welfare under rado valuations

The Nash social welfare problem asks for an allocation of indivisible items to agents in order to maximize the geometric mean of agents' valuations. We give an overview of the constant-factor approximation algorithm for the problem when agents have Rado valuations [Garg et al. 2021]. Rado valuations are a common generalization of the assignment (OXS) valuations and weighted matroid rank functions. Our approach also gives the first constant-factor approximation algorithm for the asymmetric Nash social welfare problem under the same valuations, provided that the maximum ratio between the weights is bounded by a constant.

Market Pricing for Matroid Rank Valuations

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 15 December 2020Accepted: 25 August 2021Published online: 08 November 2021Keywordspricing scheme, Walrasian equilibrium, gross substitutes valuation, matroid rank functionAMS Subject Headings90C27, 91B52Publication DataISSN (print): 0895-4801ISSN (online): 1095-7146Publisher: Society for Industrial and Applied MathematicsCODEN: sjdmec

Quasi-matroidal classes of ordered simplicial complexes

We introduce the notion of a quasi-matroidal class of ordered simplicial complexes: an approximation to the idea of a matroid cryptomorphism in the landscape of ordered simplicial complexes. A quasi-matroidal class contains pure shifted simplicial complexes and ordered matroid independence complexes. The essential property is that if a fixed simplicial complex belongs to this class for every ordering of its vertex set, then it is a matroid independence complex. Some examples of such classes appear implicitly in the matroid theory literature. We introduce various such classes that highlight different apsects of matroid theory and its similarities with the theory of shifted simplicial complexes. For example, we lift the study of objects like the Tutte polynomial and broken circuit complexes to a quasi-matroidal class that allows us to define such objects for shifted complexes. Furthermore, some of the quasi-matroidal classes are amenable to inductive techniques that can't be applied directly in the context of matroid theory. As an example, we provide a suitable setting to reformulate and extend a conjecture of Stanley about h-vectors of matroids which is expected to be tractable with techniques that are out of reach for matroids alone. This new conjecture holds for pure shifted simplicial complexes and matroids of rank up to 4.

On the Construction of Substitutes

Gross substitutability is a central concept in economics and is connected to important notions in discrete convex analysis, number theory, and the analysis of greedy algorithms in computer science. Many different characterizations are known for this class, but providing a constructive description remains a major open problem. The construction problem asks how to construct all gross substitutes from a class of simpler functions using a set of operations. Because gross substitutes are a natural generalization of matroids to real-valued functions, matroid rank functions form a desirable such class of simpler functions. Shioura proved that a rich class of gross substitutes can be expressed as sums of matroid rank functions, but it is open whether all gross substitutes can be constructed this way. Our main result is a negative answer showing that some gross substitutes cannot be expressed as positive linear combinations of matroid rank functions. En route, we provide necessary and sufficient conditions for the sum to preserve substitutability, uncover a new operation preserving substitutability, and fully describe all substitutes with at most four items.

Powerful Sets: a Generalisation of Binary Matroids

A set $S\subseteq\{0,1\}^E$ of binary vectors, with positions indexed by $E$, is said to be a powerful code if, for all $X\subseteq E$, the number of vectors in $S$ that are zero in the positions indexed by $X$ is a power of 2. By treating binary vectors as characteristic vectors of subsets of $E$, we say that a set $S\subseteq2^E$ of subsets of $E$ is a powerful set if the set of characteristic vectors of sets in $S$ is a powerful code. Powerful sets (codes) include cocircuit spaces of binary matroids (equivalently, linear codes over $\mathbb{F}_2$), but much more besides. Our motivation is that, to each powerful set, there is an associated nonnegative-integer-valued rank function (by a construction of Farr), although it does not in general satisfy all the matroid rank axioms.In this paper we investigate the combinatorial properties of powerful sets. We prove fundamental results on special elements (loops, coloops, frames, near-frames, and stars), their associated types of single-element extensions, various ways of combining powerful sets to get new ones, and constructions of nonlinear powerful sets. We show that every powerful set is determined by its clutter of minimal nonzero members. Finally, we show that the number of powerful sets is doubly exponential, and hence that almost all powerful sets are nonlinear.

Energy Efficient Coverage Extension Relay Node Placement in LTE-A Networks

Deployed long term evolution-advanced (LTE-A) infrastructure may need coverage extension due to the exponential growth of mobile broadband data usages as well as poor network performance along the cell edges. A proper installation of relay nodes (RN) extends the network coverage in LTE-A networks. We propose an energy efficient and optimal RN placement (EEORNP) algorithm that maximizes the network coverage under the energy constraint, while maintaining the signal-to-interference ratio. The algorithm is based on an improved greedy algorithm, where an effective and optimal RN placement is guaranteed when the matroid rank function of the energy efficient coverage extension optimization is sub-modular and monotonic. The performance is investigated in terms of coverage percentage and number of RN needed to cover users. Results show that the proposed EEORNP outperforms both greedy and random placement algorithms.

Enumerating Matroids of Fixed Rank

It has been conjectured that asymptotically almost all matroids are sparse paving, i.e. that~$s(n) \sim m(n)$, where $m(n)$ denotes the number of matroids on a fixed groundset of size $n$, and $s(n)$ the number of sparse paving matroids. In an earlier paper, we showed that $\log s(n) \sim \log m(n)$. The bounds that we used for that result were dominated by matroids of rank $r\approx n/2$. In this paper we consider the relation between the number of sparse paving matroids $s(n,r)$ and the number of matroids $m(n,r)$ on a fixed groundset of size $n$ of fixed rank $r$. In particular, we show that $\log s(n,r) \sim \log m(n,r)$ whenever $r\ge 3$, by giving asymptotically matching upper and lower bounds.Our upper bound on $m(n,r)$ relies heavily on the theory of matroid erections as developed by Crapo and Knuth, which we use to encode any matroid as a stack of paving matroids. Our best result is obtained by relating to this stack of paving matroids an antichain that completely determines the matroid. We also obtain that the collection of essential flats and their ranks gives a concise description of matroids.

Optimal Mechanisms for Combinatorial Auctions and Combinatorial Public Projects via Convex Rounding

We design the first truthful-in-expectation, constant-factor approximation mechanisms for NP -hard cases of the welfare maximization problem in combinatorial auctions with nonidentical items and in combinatorial public projects. Our results apply to bidders with valuations that are nonnegative linear combinations of gross-substitute valuations, a class that encompasses many of the most well-studied subclasses of submodular functions, including coverage functions and weighted matroid rank functions. Our mechanisms have an expected polynomial runtime and achieve an approximation factor of 1 − 1/ e . This approximation factor is the best possible for both problems, even for known and explicitly given coverage valuations, assuming P ≠ NP . Recent impossibility results suggest that our results cannot be extended to a significantly larger valuation class. Both of our mechanisms are instantiations of a new framework for designing approximation mechanisms based on randomized rounding algorithms. The high-level idea of this framework is to optimize directly over the (random) output of the rounding algorithm , rather than the usual (and rarely truthful) approach of optimizing over the input to the rounding algorithm. This framework yields truthful-in-expectation mechanisms, which can be implemented efficiently when the corresponding objective function is concave. For bidders with valuations in the cone generated by gross-substitute valuations, we give novel randomized rounding algorithms that lead to both a concave objective function and a (1 − 1/ e )-approximation of the optimal welfare.

The complexity of minimizing the difference of two M[formula omitted]-convex set functions

In the context of discrete DC programming, Maehara and Murota (2015) posed the problem of determining the complexity of minimizing the difference of two M♮-convex set functions. In this paper, we show the NP-hardness of this minimization problem by proving a stronger result: maximizing the difference of two matroid rank functions is NP-hard.

Enumerating Neighborly Polytopes and Oriented Matroids

Neighborly polytopes are those that maximize the number of faces in each dimension among all polytopes with the same number of vertices. Despite their extremal properties, they form a surprisingly rich class of polytopes, which has been widely studied and is the subject of many open problems and conjectures. In this article, we study the enumeration of neighborly polytopes beyond the cases that have been computed so far. To this end, we enumerate neighborly oriented matroids—a combinatorial abstraction of neighborly polytopes—of small rank and corank. In particular, if we denote by OM(r, n) the set of all oriented matroids of rank r and n elements, we determine all uniform neighborly oriented matroids in OM(5, ≤12), OM(6, ≤9), OM(7, ≤11), and OM(9, ≤12) and all possible face lattices of neighborly oriented matroids in OM(6, 10) and OM(8, 11). Moreover, we classify all possible face lattices of uniform 2-neighborly oriented matroids in OM(7, 10) and OM(8, 11). Based on the enumeration, we construct many interesting examples and test open conjectures.

On Serial Symmetric Exchanges of Matroid Bases

AbstractWe study some properties of a serial (i.e., one‐by‐one) symmetric exchange of elements of two disjoint bases of a matroid. We show that any two elements of one base have a serial symmetric exchange with some two elements of the other base. As a result, we obtain that any two disjoint bases in a matroid of rank 4 have a full serial symmetric exchange

Matroid rank functions and discrete concavity

We discuss the relationship between matroid rank functions and a concept of discrete concavity called \({{\rm M}^{\natural}}\)-concavity. It is known that a matroid rank function and its weighted version called a weighted rank function are \({{\rm M}^{\natural}}\)-concave functions, while the (weighted) sum of matroid rank functions is not \({{\rm M}^{\natural}}\)-concave in general. We present a sufficient condition for a weighted sum of matroid rank functions to be an \({{\rm M}^{\natural}}\)-concave function, and show that every weighted rank function can be represented as a weighted sum of matroid rank functions satisfying this condition.