
A note on Cunningham's algorithm for matroid intersection
In the matroid intersection problem, we are given two matroids of rank r...
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Breaking the Quadratic Barrier for Matroid Intersection
The matroid intersection problem is a fundamental problem that has been ...
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Breaking O(nr) for Matroid Intersection
We present algorithms that break the Õ(nr)independencequery bound for ...
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Nearoptimal Approximate Discrete and Continuous Submodular Function Minimization
In this paper we provide improved running times and oracle complexities ...
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Decision times of infinite computations
The decision time of an infinite time algorithm is the supremum of its h...
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Sampling Based Approximate Skyline Calculation on Big Data
The existing algorithms for processing skyline queries cannot adapt to b...
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Minimizing Convex Functions with Integral Minimizers
Given a separation oracle 𝖲𝖮 for a convex function f that has an integra...
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Faster Matroid Intersection
In this paper we consider the classic matroid intersection problem: given two matroids _1=(V,_1) and _2=(V,_2) defined over a common ground set V, compute a set S∈_1∩_2 of largest possible cardinality, denoted by r. We consider this problem both in the setting where each _i is accessed through an independence oracle, i.e. a routine which returns whether or not a set S∈_i in time, and the setting where each _i is accessed through a rank oracle, i.e. a routine which returns the size of the largest independent subset of S in _i in time. In each setting we provide faster exact and approximate algorithms. Given an independence oracle, we provide an exact O(nrlog r ) time algorithm. This improves upon the running time of O(nr^1.5) due to Cunningham in 1986 and Õ(n^2+n^3) due to Lee, Sidford, and Wong in 2015. We also provide two algorithms which compute a (1ϵ)approximate solution to matroid intersection running in times Õ(n^1.5/^1.5) and Õ((n^2r^1ϵ^2+r^1.5ϵ^4.5) ), respectively. These results improve upon the O(nr/)time algorithm of Cunningham as noted recently by Chekuri and Quanrud. Given a rank oracle, we provide algorithms with even better dependence on n and r. We provide an O(n√(r)log n )time exact algorithm and an O(nϵ^1log n )time algorithm which obtains a (1)approximation to the matroid intersection problem. The former result improves over the Õ(nr +n^3)time algorithm by Lee, Sidford, and Wong. The rank oracle is of particular interest as the matroid intersection problem with this oracle is a special case of the submodular function minimization problem with an evaluation oracle.
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