Degree bounded matroids and submodular flows

We consider two related problems, the Minimum Bounded Degree Matroid Basis problem and the Minimum Bounded Degree Submodular Flow problem. The first problem is a generalization of the Minimum Bounded Degree Spanning Tree problem: We are given a matroid and a hypergraph on its ground set with lower and upper bounds f(e)≤g(e) for each hyperedge e. The task is to find a minimum cost basis which contains at least f(e) and at most g(e) elements from each hyperedge e. In the second problem we have a submodular flow problem, a lower bound f(v) and an upper bound g(v) for each node v, and the task is to find a minimum cost 0-1 submodular flow with the additional constraint that the sum of the incoming and outgoing flow at each node v is between f(v) and g(v). Both of these problems are NP-hard (even the feasibility problems are NP-complete), but we show that they can be approximated in the following sense. Let opt be the value of the optimal solution. For the first problem we give an algorithm that finds a basis B of cost no more than opt such that f(e)-2Δ+1≤|B∩e|≤g(e)+2Δ-1 for every hyperedge e, where Δ is the maximum degree of the hypergraph. If there are only upper bounds (or only lower bounds), then the violation can be decreased to Δ-1. For the second problem we can find a 0-1 submodular flow of cost at most opt where the sum of the incoming and outgoing flow at each node v is between f(v)-1 and g(v)+1. These results can be applied to obtain approximation algorithms for several combinatorial optimization problems with degree constraints, including the Minimum Crossing Spanning Tree problem, the Minimum Bounded Degree Spanning Tree Union problem, the Minimum Bounded Degree Directed Cut Cover problem, and the Minimum Bounded Degree Graph Orientation problem. © 2012 János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg

Improved Algorithm for Degree Bounded Survivable Network Design Problem

We consider the Degree-Bounded Survivable Network Design Problem: the objective is to find a minimum cost subgraph satisfying the given connectivity requirements as well as the degree bounds on the vertices. If we denote the upper bound on the degree of a vertex v by b(v), then we present an algorithm that finds a solution whose cost is at most twice the cost of the optimal solution while the degree of a degree constrained vertex v is at most 2b(v) + 2. This improves upon the results of Lau and Singh and that of Lau, Naor, Salavatipour and Singh

Matroidal Degree-Bounded Minimum Spanning Trees

We consider the minimum spanning tree (MST) problem under the restriction that for every vertex v, the edges of the tree that are adjacent to v satisfy a given family of constraints. A famous example thereof is the classical degree-constrained MST problem, where for every vertex v, a simple upper bound on the degree is imposed. Iterative rounding/relaxation algorithms became the tool of choice for degree-bounded network design problems. A cornerstone for this development was the work of Singh and Lau, who showed for the degree-bounded MST problem how to find a spanning tree violating each degree bound by at most one unit and with cost at most the cost of an optimal solution that respects the degree bounds. However, current iterative rounding approaches face several limits when dealing with more general degree constraints. In particular, when several constraints are imposed on the edges adjacent to a vertex v, as for example when a partition of the edges adjacent to v is given and only a fixed number of elements can be chosen out of each set of the partition, current approaches might violate each of the constraints by a constant, instead of violating all constraints together by at most a constant number of edges. Furthermore, it is also not clear how previous iterative rounding approaches can be used for degree constraints where some edges are in a super-constant number of constraints. We extend iterative rounding/relaxation approaches both on a conceptual level as well as aspects involving their analysis to address these limitations. This leads to an efficient algorithm for the degree-constrained MST problem where for every vertex v, the edges adjacent to v have to be independent in a given matroid. The algorithm returns a spanning tree T of cost at most OPT, such that for every vertex v, it suffices to remove at most 8 edges from T to satisfy the matroidal degree constraint at v

On a generalization of iterated and randomized rounding

We give a general method for rounding linear programs that combines the commonly used iterated rounding and randomized rounding techniques. In particular, we show that whenever iterated rounding can be applied to a problem with some slack, there is a randomized procedure that returns an integral solution that satisfies the guarantees of iterated rounding and also has concentration properties. We use this to give new results for several classic problems where iterated rounding has been useful

Fair Algorithms for Clustering

As algorithms play a large role in our decision making, the possibility of algorithmic bias has led researchers to explore the realm of fair algorithms. In this thesis, we explore the design of a fair algorithm for clustering a problem in unsupervised machine learning algorithm. Our algorithm aims to balance the representation of an arbitrary number of protected groups in each cluster. We extend prior work by allowing the points to belong to multiple protected groups and for users to compromise between stricter fairness and the clustering objective. We provide experimental validation of our work on the k-median, k-means and k-center objectives

On Generalizations of Network Design Problems with Degree Bounds

Iterative rounding and relaxation have arguably become the method of choice in dealing with unconstrained and constrained network design problems. In this paper we extend the scope of the iterative relaxation method in two directions: (1) by handling more complex degree constraints in the minimum spanning tree problem (namely, laminar crossing spanning tree), and (2) by incorporating `degree bounds' in other combinatorial optimization problems such as matroid intersection and lattice polyhedra. We give new or improved approximation algorithms, hardness results, and integrality gaps for these problems.Comment: v2, 24 pages, 4 figure

On a generalization of iterated and randomized rounding

We give a general method for rounding linear programs that combines the commonly used iterated rounding and randomized rounding techniques. In particular, we show that whenever iterated rounding can be applied to a problem with some slack, there is a randomized procedure that returns an integral solution that satisfies the guarantees of iterated rounding and also has concentration properties. We use this to give new results for several classic problems such as rounding column-sparse LPs, makespan minimization on unrelated machines, degree-bounded spanning trees and multi-budgeted matchings