Two algorithms for weighted matroid intersection
TL;DR: This paper presents an algorithm of complexity O(nr(r+c+logn)) for the weighted matroid intersection problem, and presents a second algorithm that, given a feasible solution of cardinalityk, finds an optimal one of the same cardinality.
Abstract: Consider a finite setE, a weight functionw:EźR, and two matroidsM1 andM2 defined onE The weighted matroid intersection problem consists of finding a setI⊆E, independent in both matroids, that maximizes Σ{w(e):e inI} We present an algorithm of complexity O(nr(r+c+logn)) for this problem, wheren=|E|,r=min(rank(M1), rank (M2)),c=max (c1,c2) and, fori=1,2,ci is the complexity of finding the circuit ofIź{e} inMi (or show that none exists) wheree is inE andI⊆E is independent inM1 andM2 A related problem is to find a maximum weight set, independent in both matroids, and of given cardinalityk (if one exists) Our algorithm also solves this problem In addition, we present a second algorithm that, given a feasible solution of cardinalityk, finds an optimal one of the same cardinality A sensitivity analysis on the weights is easy to perform using this approach Our two algorithms are related to existing algorithms In fact, our framework provides new simple proofs of their validity Other contributions of this paper are the existence of nonnegative reduced weights (Theorem 6), allowing the improved complexity bound, and the introduction of artificial elements, allowing an improved start and flexibility in the implementation of the algorithms
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17 Oct 2015
TL;DR: In this article, the authors improved the running time for finding a point in a convex set given a separation oracle to O(n3 logO(1) nR=a#x03B5).
Abstract: In this paper we improve upon the running time for finding a point in a convex set given a separation oracle. In particular, given a separation oracle for a convex set K a#x2282; Rn that is contained in a box of radius R we show how to either compute a point in K or prove that K does not contain a ball of radius a#x03B5; using an expected O(n log(nR=a#x03B5;)) evaluations of the oracle and additional time O(n3 logO(1)(nR=a#x03B5;)). This matches the oracle complexity and improves upon the O(na#x03C9;+1 log(nR=a#x03B5;)) additional time of the previous fastest algorithm achieved over 25 years ago by Vaidya [91] for the current value of the matrix multiplication constant a#x03C9; < 2:373 [98], [36] when R=a#x03B5; = O(poly(n)). Using a mix of standard reductions and new techniques we show how our algorithm can be used to improve the running time for solving classic problems in continuous and combinatorial optimization. In particular we provide the following running time improvements: a#x03B5; Sub modular Function Minimization: n is the size of the ground set, M is the maximum absolute value of function values and EO is the time for function evaluation. Our weakly and strongly polynomial time algorithms have a running time of O(n2 lognM EO+n3 logO(1) nM) and O(n3 log2 n EO+n4 logO(1) n), improving upon the previous best of O((n4 · EO+n5) logM) and O(n5 · EO + n6) respectively. a#x03B5; Sub modular Flow: n = |V|, m = |E|, C is the maximum edge cost in absolute value and U is maximum edge capacity in absolute value. We obtain a faster weakly polynomial running time of O(n2 log nCU · EO + n3 logO(1) nCU), improving upon the previous best of O(mn5 log nU · EO) and O(n4h min {log C, log U}) from 15 years ago by a factor of O(n4). We also achieve faster strongly polynomial time algorithms as a consequence of our result on sub modular minimization. a#x03B5; Matroid Intersection: n is the size of the ground set, r is the maximum size of independent sets, M is the maximum absolute value of element weight, Trank and Tind are the time for each rank and independence oracle query. We obtain a running time of O((nr log2 nTrank+n3 logO(1) n) lognM) and O((n2 log nTind+n3 logO(1) n) lognM), achieving the first quadratic bound on the query complexity for the independence and rank oracles. In the unweighted case, this is the first improvement since 1986 for independence oracle. a#x03B5; Semi definite Programming: n is the number of constraints, m is the number of dimensions and S is the total number of non-zeros in the constraint matrices. We obtain a running time of O(n(n2 +ma#x03C9; +S)), improving upon the previous best of O(n(na#x03C9; +ma#x03C9; +S)) for the regime S is small.
277 citations
TL;DR: A random pseudo-polynomial algorithm for the problem of finding a base of specified value in a weighted represented matroid, subject to parity conditions is presented and a specialized version of the algorithm suitable for finding a bases in the intersection of two matroids is described.
72 citations
TL;DR: A polynomial time approximation algorithm for the Max-ECP with performance ratio O(n(loglogn)^2log^3n), improving the best previously known bound of O( n).
64 citations
Cites background from "Two algorithms for weighted matroid..."
...for the maximum edge clique partitioning problem with performance ratio O ( n(log log n)2 log3 n ) , improving the previously best known bound of O(n)....
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TL;DR: This work presents an efficient polynomial time approximation scheme (EPTAS) based on Lagrangian relaxation and matroid intersection for the constrained minimum spanning tree problem.
Abstract: Given an undirected graph G=(V,E) with |V|=n and |E|=m, nonnegative integers ce and de for each edge $e \in E$, and a bound D, the constrained minimum spanning tree problem (CST) is to find a spanning tree T=(V,ET) such that $\sum_{e \in E_T} d_e \leq D$ and $\sum_{e \in E_T} c_e$ is minimized. We present an efficient polynomial time approximation scheme (EPTAS) for this problem. Specifically, for every $\epsilon>0$ we present a $(1+\epsilon)$-approximation algorithm with time complexity $O((\frac{1}{\epsilon})^{O(\frac{1}{\epsilon})}n^4)$. Our method is based on Lagrangian relaxation and matroid intersection.
55 citations
Cites background from "Two algorithms for weighted matroid..."
...minimum cost subset S such that S is a base of both M and M0) [ 3 ]....
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...as follows (see [ 3 ] and [4]): For every e 2 T we have a node in V , and for every e0 2 E n T...
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TL;DR: The valuated matroid intersection theorem is applied to the analysis of the degree of the determinant of a mixed polynomial matrix to obtain a novel duality identity together with an efficient algorithm.
Abstract: The mixed polynomial matrix, introduced as a convenient mathematical tool for the description of physical/engineering dynamical systems, is a polynomial matrix of which the coefficients are classified into fixed constants and independent parameters. The valuated matroid, invented by Dress and Wenzel [ Appl. Math. Lett., 3 (1990), pp. 33--35], is a combinatorial abstraction of the degree of minors (subdeterminants) of a polynomial matrix. We discuss a number of implications of the recent developments in the theory of valuated matroids in the context of polynomial matrix theory. In particular, we apply the valuated matroid intersection theorem to the analysis of the degree of the determinant of a mixed polynomial matrix to obtain a novel duality identity together with an efficient algorithm.
41 citations
References
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24 Oct 1984
TL;DR: The structure, Fibonacci heaps (abbreviated F-heaps), extends the binomial queues proposed by Vuillemin and studied further by Brown to obtain improved running times for several network optimization algorithms.
Abstract: In this paper we develop a new data structure for implementing heaps (priority queues). Our structure, Fibonacci heaps (abbreviated F-heaps), extends the binomial queues proposed by Vuillemin and studied further by Brown. F-heaps support arbitrary deletion from an n-item heap in 0(log n) amortized time and all other standard heap operations in 0(1) amortized time. Using F-heaps we are able to obtain improved running times for several network optimization algorithms.
1,757 citations
01 Jan 1970
TL;DR: The viewpoint of the subject of matroids, and related areas of lattice theory, has always been abstraction of algebraic dependence or, equivalently, abstraction of the incidence relations in geometric representations of algebra.
Abstract: The viewpoint of the subject of matroids, and related areas of lattice theory, has always been, in one way or another, abstraction of algebraic dependence or, equivalently, abstraction of the incidence relations in geometric representations of algebra. Often one of the main derived facts is that all bases have the same cardinality. (See Van der Waerden, Section 33.)
774 citations
155 citations
TL;DR: In this paper, the ordinary assignment problem on a bipartite graph with weighted arcs is extended to the case where both of the two sets of vertices of the graph are given matroidal structures.
Abstract: The ordinary assignment problem on a bipartite graph with weighted arcs is extended to the case where both of the two sets of vertices of the graph are given matroidal structures, and a practical solution algorithm is presented to this extended problem.
70 citations