Periodic Reporting for period 3 - LOPRE (Lossy Preprocessing)
Periodo di rendicontazione: 2022-05-01 al 2023-10-31
A critical component of the computational processing of data sets is the ‘preprocessing’ or ‘compression’ step. The goal of this step is the efficient computation of a succinct, sufficiently accurate representation of the given data. Preprocessing is ubiquitous and has widespread applications in machine learning, SAT solving, ILP solving and the processing of strings, images, audio and video arising from a wide range of sources. The kind of preprocessing performed on the data typically depends on whether or not it is required to be oblivious to the application. For instance, data compression algorithms for the ZIP file format are application oblivious in the sense that one can use these regardless of the kind of processing to be performed later on the data. On the other hand, there are data compression algorithms like JPEG which can be used only for specific purposes, for instance, compressing images. These are application sensitive in the sense that they are used when one has a good idea of the type of data being handled as well as what the data will be used for. Unfortunately, in spite of the extensive use of application sensitive preprocessing, there is no framework of rigorous mathematical analysis for these preprocessing algorithms. The overarching goal of this proposal is to develop
a mathematical framework for the rigorous analysis of appliction sensitive preprocessing algorithms.
a mathematical framework for the rigorous analysis of appliction sensitive preprocessing algorithms.
Polynomial Time Approximation: Developed PTAS’s and EPTASes for weighted version of vertex/edge deletion subset problems on H-minor graphs.Designed an optimal factor 2-approximating algorithm for Feedback Vertex Set in Tournaments.
FPT Approximation: Designed a Parameterized Approximation Scheme for Min k-Cut. Developed a framework for designing FPT-approximation algorithms for FPT problems and showed its usefulness by designing efficient FPT-approximation algorithms for Directed Feedback Vertex Set and Multicut. Designed an ``approximate version'' of representative families arising in Matroid theory for approximate counting in parameterized complexity. Used this to design faster counting algorithms for several problems including k-Path.
Computation Geometry: Obtained a fragment of CSP for which we designed a polynomial time algorithm and applied to solve a problem on Art Gallery problem. Designing a constant factor approximation algorithm for navigating through obstacles in plane.
Parameterized Algorithms: Designed first single exponential time algorithm for the Disjoint Paths problem on planar graphs.
Resolution of parameterized complexity of Hitting Topological Minors.
FPT Approximation: Designed a Parameterized Approximation Scheme for Min k-Cut. Developed a framework for designing FPT-approximation algorithms for FPT problems and showed its usefulness by designing efficient FPT-approximation algorithms for Directed Feedback Vertex Set and Multicut. Designed an ``approximate version'' of representative families arising in Matroid theory for approximate counting in parameterized complexity. Used this to design faster counting algorithms for several problems including k-Path.
Computation Geometry: Obtained a fragment of CSP for which we designed a polynomial time algorithm and applied to solve a problem on Art Gallery problem. Designing a constant factor approximation algorithm for navigating through obstacles in plane.
Parameterized Algorithms: Designed first single exponential time algorithm for the Disjoint Paths problem on planar graphs.
Resolution of parameterized complexity of Hitting Topological Minors.
Below are some of the significant papers on the project.
(1) Daniel Lokshtanov, Saket Saurabh, Vaishali Surianarayanan: A Parameterized Approximation Scheme for Min $k$-Cut. FOCS 2020: 798-809
(2) Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, Meirav Zehavi:
Approximation Schemes via Width/Weight Trade-offs on Minor-free Graphs. SODA 2020: 2299-2318
(3) Daniel Lokshtanov, Pranabendu Misra, Michal Pilipczuk, Saket Saurabh, Meirav Zehavi:
An exponential time parameterized algorithm for planar disjoint paths. STOC 2020: 1307-1316
(4) Daniel Lokshtanov, Pranabendu Misra, M. S. Ramanujan, Saket Saurabh, Meirav Zehavi:
FPT-approximation for FPT Problems. SODA 2021: 199-218
(5) Daniel Lokshtanov, Saket Saurabh, Meirav Zehavi:
Efficient Computation of Representative Weight Functions with Applications to Parameterized Counting (Extended Version). SODA 2021: 179-198
(6) Neeraj Kumar, Daniel Lokshtanov, Saket Saurabh, Subhash Suri:
A Constant Factor Approximation for Navigating Through Connected Obstacles in the Plane. SODA 2021: 822-839
All of the above are significant achievements and advances our research field significantly beyond the state of the art. A Parameterized Approximation Scheme for Min k-Cut was completely unplanned.
While developing a sparsification tool for our project, we realized that we could apply it to this problem. We belive that by the end of the project we will be able to achieve significant portion of stated goals and would have made significant contribution to the world of compression.
(1) Daniel Lokshtanov, Saket Saurabh, Vaishali Surianarayanan: A Parameterized Approximation Scheme for Min $k$-Cut. FOCS 2020: 798-809
(2) Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, Meirav Zehavi:
Approximation Schemes via Width/Weight Trade-offs on Minor-free Graphs. SODA 2020: 2299-2318
(3) Daniel Lokshtanov, Pranabendu Misra, Michal Pilipczuk, Saket Saurabh, Meirav Zehavi:
An exponential time parameterized algorithm for planar disjoint paths. STOC 2020: 1307-1316
(4) Daniel Lokshtanov, Pranabendu Misra, M. S. Ramanujan, Saket Saurabh, Meirav Zehavi:
FPT-approximation for FPT Problems. SODA 2021: 199-218
(5) Daniel Lokshtanov, Saket Saurabh, Meirav Zehavi:
Efficient Computation of Representative Weight Functions with Applications to Parameterized Counting (Extended Version). SODA 2021: 179-198
(6) Neeraj Kumar, Daniel Lokshtanov, Saket Saurabh, Subhash Suri:
A Constant Factor Approximation for Navigating Through Connected Obstacles in the Plane. SODA 2021: 822-839
All of the above are significant achievements and advances our research field significantly beyond the state of the art. A Parameterized Approximation Scheme for Min k-Cut was completely unplanned.
While developing a sparsification tool for our project, we realized that we could apply it to this problem. We belive that by the end of the project we will be able to achieve significant portion of stated goals and would have made significant contribution to the world of compression.