116 research outputs found
Directed Multicut is W[1]-hard, Even for Four Terminal Pairs
We prove that Multicut in directed graphs, parameterized by the size of the
cutset, is W[1]-hard and hence unlikely to be fixed-parameter tractable even if
restricted to instances with only four terminal pairs. This negative result
almost completely resolves one of the central open problems in the area of
parameterized complexity of graph separation problems, posted originally by
Marx and Razgon [SIAM J. Comput. 43(2):355-388 (2014)], leaving only the case
of three terminal pairs open.
Our gadget methodology allows us also to prove W[1]-hardness of the Steiner
Orientation problem parameterized by the number of terminal pairs, resolving an
open problem of Cygan, Kortsarz, and Nutov [SIAM J. Discrete Math.
27(3):1503-1513 (2013)].Comment: v2: Added almost tight ETH lower bound
A Polynomial Kernel For Multicut In Trees
The MULTICUT IN TREES problem consists in deciding, given a tree, a set of
requests (i.e. paths in the tree) and an integer k, whether there exists a set
of k edges cutting all the requests. This problem was shown to be FPT by Guo
and Niedermeyer. They also provided an exponential kernel. They asked whether
this problem has a polynomial kernel. This question was also raised by Fellows.
We show that MULTICUT IN TREES has a polynomial kernel
Fixed-parameter tractability of Directed Multicut with three terminal pairs parameterized by the size of the cutset: twin-width meets flow-augmentation
We show fixed-parameter tractability of the Directed Multicut problem with
three terminal pairs (with a randomized algorithm). This problem, given a
directed graph , pairs of vertices (called terminals) ,
, and , and an integer , asks to find a set of at most
non-terminal vertices in that intersect all -paths, all
-paths, and all -paths. The parameterized complexity of this
case has been open since Chitnis, Cygan, Hajiaghayi, and Marx proved
fixed-parameter tractability of the 2-terminal-pairs case at SODA 2012, and
Pilipczuk and Wahlstr\"{o}m proved the W[1]-hardness of the 4-terminal-pairs
case at SODA 2016.
On the technical side, we use two recent developments in parameterized
algorithms. Using the technique of directed flow-augmentation [Kim, Kratsch,
Pilipczuk, Wahlstr\"{o}m, STOC 2022] we cast the problem as a CSP problem with
few variables and constraints over a large ordered domain.We observe that this
problem can be in turn encoded as an FO model-checking task over a structure
consisting of a few 0-1 matrices. We look at this problem through the lenses of
twin-width, a recently introduced structural parameter [Bonnet, Kim,
Thomass\'{e}, Watrigant, FOCS 2020]: By a recent characterization [Bonnet,
Giocanti, Ossona de Mendes, Simon, Thomass\'{e}, Toru\'{n}czyk, STOC 2022] the
said FO model-checking task can be done in FPT time if the said matrices have
bounded grid rank. To complete the proof, we show an irrelevant vertex rule: If
any of the matrices in the said encoding has a large grid minor, a vertex
corresponding to the ``middle'' box in the grid minor can be proclaimed
irrelevant -- not contained in the sought solution -- and thus reduced
Fixed-Parameter Tractability of Directed Multicut with Three Terminal Pairs Parameterized by the Size of the Cutset: Twin-width Meets Flow-Augmentation
We show fixed-parameter tractability of the Directed Multicut problem withthree terminal pairs (with a randomized algorithm). This problem, given adirected graph , pairs of vertices (called terminals) ,, and , and an integer , asks to find a set of at most non-terminal vertices in that intersect all -paths, all-paths, and all -paths. The parameterized complexity of thiscase has been open since Chitnis, Cygan, Hajiaghayi, and Marx provedfixed-parameter tractability of the 2-terminal-pairs case at SODA 2012, andPilipczuk and Wahlstr\"{o}m proved the W[1]-hardness of the 4-terminal-pairscase at SODA 2016. On the technical side, we use two recent developments in parameterizedalgorithms. Using the technique of directed flow-augmentation [Kim, Kratsch,Pilipczuk, Wahlstr\"{o}m, STOC 2022] we cast the problem as a CSP problem withfew variables and constraints over a large ordered domain.We observe that thisproblem can be in turn encoded as an FO model-checking task over a structureconsisting of a few 0-1 matrices. We look at this problem through the lenses oftwin-width, a recently introduced structural parameter [Bonnet, Kim,Thomass\'{e}, Watrigant, FOCS 2020]: By a recent characterization [Bonnet,Giocanti, Ossona de Mendes, Simon, Thomass\'{e}, Toru\'{n}czyk, STOC 2022] thesaid FO model-checking task can be done in FPT time if the said matrices havebounded grid rank. To complete the proof, we show an irrelevant vertex rule: Ifany of the matrices in the said encoding has a large grid minor, a vertexcorresponding to the ``middle'' box in the grid minor can be proclaimedirrelevant -- not contained in the sought solution -- and thus reduced.<br
MILP Formulations for Unsupervised and Interactive Image Segmentation and Denoising
Image segmentation and denoising are two key components of modern computer vision systems. The Potts model plays an important role for denoising of piecewise defined
functions, and Markov Random Field (MRF) using Potts terms are popular in image segmentation. We propose Mixed Integer Linear Programming (MILP) formulations for both models, and utilize standard MILP solvers to efficiently solve them.
Firstly, we investigate the discrete first derivative (piecewise constant) Potts model with the ` 1 norm data term. We propose a novel MILP formulation by introducing binary edge variables to model the Potts prior. We look into the facet-defining inequalities for the associated integer polytope. We apply the model for generating superpixels on noisy images.
Secondly, we propose a MILP formulation for the discrete piecewise affine Potts model. To obtain consistent partitions, the inclusion of multicut constraints is necessary, which is added iteratively using the cutting plane method. We apply the model for simultaneously segmenting and denoising depth images.
Thirdly, MILP formulations of MRF models with global connectivity constraints were investigated previously, but only simplified versions of the problem were solved.
We investigate this problem via a branch-and-cut method and propose a user-interactive way for segmentation.
Our proposed MILPs are in general NP-hard, but they can be used to generate globally optimal solutions and ground-truth results. We also propose three fast heuristic algorithms that provide good solutions in very short time. The MILPs can be applied as a post-processing method on top of any algorithms, not only providing a guarantee on the quality, but also seek for better solutions within the branch-and-cut framework of the solver.
We demonstrate the power and usefulness of our methods by extensive experiments against other state-of-the-art methods on synthetic images, standard image datasets, as well as medical images with trained probability maps
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