Packing of Mixed Hyperarborescences with Flexible Roots via Matroid Intersection

Given a mixed hypergraph $\mathcal{F}=(V,\mathcal{A}\cup \mathcal{E})$, a non-negative integer $k$ and functions $f,g:V\rightarrow \mathbb{Z}_{\geq 0}$, a packing of $k$ spanning mixed hyperarborescences of $\mathcal{F}$ is called $(k,f,g)$-flexible if every $v \in V$ is the root of at least $f(v)$ and at most $g(v)$ of the mixed hyperarborescences. We give a characterization of the mixed hypergraphs admitting such packings. This generalizes results of Frank and, more recently, Gao and Yang. Our approach is based on matroid intersection, generalizing a construction of Edmonds. We also obtain an algorithm for finding a minimum weight solution to the problem mentioned above.

Matroid bases with cardinality constraints on the intersection

Given two matroids $$\mathcal {M}_{1} = (E, \mathcal {B}_{1})$$ M 1 = ( E , B 1 ) and $$\mathcal {M}_{2} = (E, \mathcal {B}_{2})$$ M 2 = ( E , B 2 ) on a common ground set E with base sets $$\mathcal {B}_1$$ B 1 and $$\mathcal {B}_2$$ B 2 , some integer $$k \in \mathbb {N}$$ k ∈ N , and two cost functions $$c_{1}, c_{2} :E \rightarrow \mathbb {R}$$ c 1 , c 2 : E → R , we consider the optimization problem to find a basis $$X \in \mathcal {B}_{1}$$ X ∈ B 1 and a basis $$Y \in \mathcal {B}_{2}$$ Y ∈ B 2 minimizing the cost $$\sum _{e\in X} c_1(e)+\sum _{e\in Y} c_2(e)$$ ∑ e ∈ X c 1 ( e ) + ∑ e ∈ Y c 2 ( e ) subject to either a lower bound constraint $$|X \cap Y| \le k$$ | X ∩ Y | ≤ k , an upper bound constraint $$|X \cap Y| \ge k$$ | X ∩ Y | ≥ k , or an equality constraint $$|X \cap Y| = k$$ | X ∩ Y | = k on the size of the intersection of the two bases X and Y. The problem with lower bound constraint turns out to be a generalization of the Recoverable Robust Matroid problem under interval uncertainty representation for which the question for a strongly polynomial-time algorithm was left as an open question in Hradovich et al. (J Comb Optim 34(2):554–573, 2017). We show that the two problems with lower and upper bound constraints on the size of the intersection can be reduced to weighted matroid intersection, and thus be solved with a strongly polynomial-time primal-dual algorithm. We also present a strongly polynomial, primal-dual algorithm that computes a minimum cost solution for every feasible size of the intersection k in one run with asymptotic running time equal to one run of Frank’s matroid intersection algorithm. Additionally, we discuss generalizations of the problems from matroids to polymatroids, and from two to three or more matroids. We obtain a strongly polynomial time algorithm for the recoverable robust polymatroid base problem with interval uncertainties.

Budget Feasible Mechanisms on Matroids

Motivated by many practical applications, in this paper we study budget feasible mechanisms with the goal of procuring an independent set of a matroid. More specifically, we are given a matroid $${\mathcal {M}}=(E,{\mathcal {I}})$$ M = ( E , I ) . Each element of the ground set E is controlled by a selfish agent and the cost of the element is private information of the agent itself. A budget limited buyer has additive valuations over the elements of E. The goal is to design an incentive compatible budget feasible mechanism which procures an independent set of the matroid of largest possible value. We also consider the more general case of the pair $${\mathcal {M}}=(E,{\mathcal {I}})$$ M = ( E , I ) satisfying only the hereditary property. This includes matroids as well as matroid intersection. We show that, given a polynomial time deterministic algorithm that returns an $$\alpha $$ α -approximation to the problem of finding a maximum-value independent set in $${\mathcal {M}}$$ M , there exists an individually rational, truthful and budget feasible mechanism which is $$(3\alpha +1)$$ ( 3 α + 1 ) -approximated and runs in polynomial time, thus yielding also a 4-approximation for the special case of matroids.