Quasi M-convex and L-convex functions—quasiconvexity in discrete optimization

Murota, Kazuo; Shioura, Akiyoshi

doi:10.1016/s0166-218x(02)00468-7

Cited by 41 publications

(58 citation statements)

References 11 publications

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“…We remark that the relation between valuated delta-matroids in the sense of [6] and those in our sense is similar to that between M-concave functions and M ♮ -concave functions [20]. The next contribution of this paper is new algorithms for the weighted matching forest problem: we design a simpler algorithm and a faster algorithm than Giles' algorithm [15].…”

Section: Theorem 2 For Any Weighted Mixed Graph (G W) It Holds Thamentioning

confidence: 67%

Optimal Matching Forests and Valuated Delta-Matroids

Takazawa

2011

Integer Programming and Combinatoral Optimization

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The matching forest problem in mixed graphs is a common generalization of the matching problem in undirected graphs and the branching problem in directed graphs. Giles presented an O(n 2 m)-time algorithm for finding a maximum-weight matching forest, where n is the number of vertices and m is that of edges, and a linear system describing the matching forest polytope. Later, Schrijver proved total dual integrality of the linear system.In the present paper, we reveal another nice property of matching forests: the degree sequences of the matching forests in any mixed graph form a delta-matroid and the weighted matching forests induce a valuated delta-matroid. We remark that the delta-matroid is not necessarily even, and the valuated delta-matroid induced by weighted matching forests slightly generalizes the well-known notion of Dress and Wenzel's valuated delta-matroids. By focusing on the delta-matroid structure and reviewing Giles' algorithm, we design a simpler O(n 2 m)-time algorithm for the weighted matching forest problem. We also present a faster O(n 3 )-time algorithm by using Gabow's method for the weighted matching problem.

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Section: Theorem 2 For Any Weighted Mixed Graph (G W) It Holds Thamentioning

confidence: 67%

Optimal Matching Forests and Valuated Delta-Matroids

Takazawa

2011

Integer Programming and Combinatoral Optimization

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“…3] for discussion about discontinuity adaptive potentials). We should note here that, as we have shown in Section III-E, formally, a nonconvex clique potential is allowed in the algorithm, as long as every 2π-periodic sampling is convex (about the issue of convex functions on discrete domains see, e.g., [61]). It is, however, a trivial reasoning to conclude that this kind of nonconvex potentials are not discontinuity preserving.…”

Section: Nonconvex Potentialsmentioning

confidence: 99%

Phase Unwrapping via Graph Cuts

Bioucas-Dias

Valadão

2007

IEEE Trans. on Image Process.

474

220

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Abstract-Phase unwrapping is the inference of absolute phase from modulo-2π phase. This paper introduces a new energy minimization framework for phase unwrapping. The considered objective functions are first-order Markov random fields. We provide an exact energy minimization algorithm, whenever the corresponding clique potentials are convex, namely for the phase unwrapping classical L p norm, with p ≥ 1. Its complexity is KT (n, 3n), where K is the length of the absolute phase domain measured in 2π units and T (n, m) is the complexity of a max-flow computation in a graph with n nodes and m edges. For nonconvex clique potentials, often used owing to their discontinuity preserving ability, we face an NP-hard problem for which we devise an approximate solution. Both algorithms solve integer optimization problems, by computing a sequence of binary optimizations, each one solved by graph cut techniques. Accordingly, we name the two algorithms PUMA, for phase unwrapping max-flow/min-cut. A set of experimental results illustrates the effectiveness of the proposed approach and its competitiveness in comparison with state-of-the-art phase unwrapping algorithms.

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“…An even more general class of functions, the so-called L-convex and L -convex functions, was considered in [36]. References [33,34], in addition to introducing new results, give an excellent account of the connections between the dual of the convex cost network flow problems and of L-convex and L -convex based problems.…”

Section: Integer Optimizationmentioning

confidence: 99%

CAPE: combinatorial absolute phase estimation

Valadão

Bioucas-Dias

2009

J. Opt. Soc. Am. A

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An absolute phase estimation algorithm for interferometric applications is introduced. The approach is Bayesian. Besides coping with the 2-periodic sinusoidal nonlinearity in the observations, the proposed methodology assumes a first-order Markov random field prior and a maximum a posteriori probability (MAP) viewpoint. For computing the MAP solution, we provide a combinatorial suboptimal algorithm that involves a multiprecision sequence. In the coarser precision, it unwraps the phase by using, essentially, the previously introduced PUMA algorithm [IEEE Trans. Image Proc. 16, 698 (2007)], which blindly detects discontinuities and yields a piecewise smooth unwrapped phase. In the subsequent increasing precision iterations, the proposed algorithm denoises each piecewise smooth region, thanks to the previously detected location of the discontinuities. For each precision, we map the problem into a sequence of binary optimizations, which we tackle by computing min-cuts on appropriate graphs. This unified rationale for both phase unwrapping and denoising inherits the fast performance of the graph min-cuts algorithms. In a set of experimental results, we illustrate the effectiveness of the proposed approach.

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Quasi M-convex and L-convex functions—quasiconvexity in discrete optimization

Cited by 41 publications

References 11 publications

Optimal Matching Forests and Valuated Delta-Matroids

Optimal Matching Forests and Valuated Delta-Matroids

Phase Unwrapping via Graph Cuts

CAPE: combinatorial absolute phase estimation

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