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Quasipolynomial-Time Deterministic Kernelization and (Gammoid) Representation

Authors Rohit Gurjar , Daniel Lokshtanov , Pranabendu Misra , Fahad Panolan , Saket Saurabh , Meirav Zehavi

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Matroids and greedoids
  • Theory of computation → Pseudorandomness and derandomization
  • Mathematics of computing → Graph algorithms
Keywords
  • Network Flows
  • Gammoids
  • Matchings
  • Transversal Matroids
  • Matroid Representation
  • Derandomization

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Abstract

In this paper, we suggest to extend the notion of a kernel to permit the kernelization algorithm to be executed in quasi-polynomial time rather than polynomial time. So far, we are only aware of one work that addressed this negatively, showing that some lower bounds on kernel sizes proved for kernelization also hold when quasi-polynomial time complexity is allowed. When we, anyway, deal with an NP-hard problem, sacrificing polynomial time in preprocessing for quasi-polynomial time may often not be a big deal, but, of course, the question is - does it give us more power? The only known work, mentioned above, seems to suggest that the answer is "no". In this paper, we show that this is not the case - in particular, we show that this notion is extremely powerful for derandomization. Some of the most basic kernelization algorithms in the field are based on inherently randomized tools whose derandomization is a huge problem that has remained (and may still remain) open for many decades. Still, some breakthrough advances for derandomization in quasi-polynomial time have been made. Can we harness these advancements to design quasi-polynomial deterministic kernelization algorithms for basic problems in the field? To this end, we revisit the question of deterministic polynomial-time computation of a linear representation of transversal matroids and gammoids, which is a longstanding open problem. We present a deterministic computation of a representation matrix of a transversal matroid in time quasipolynomial in the rank of the matroid, where each entry of the matrix can be represented in quasipolynomial (in the rank of the matroid) bits. As a corollary, we obtain a linear representation of a gammoid in deterministic quasipolynomial time and quasipolynomial bits in the size of the underlying ground set of the gammoid. In turn, as applications of our results, we present deterministic quasi-polynomial time kernels of polynomial size for several central problems in the field.

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Rohit Gurjar, Daniel Lokshtanov, Pranabendu Misra, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Quasipolynomial-Time Deterministic Kernelization and (Gammoid) Representation. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 54:1-54:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.MFCS.2025.54

Author Details

Rohit Gurjar
  • Indian Institute of Technology Bombay, India
Daniel Lokshtanov
  • University of California, Santa Barbara, CA, USA
Pranabendu Misra
  • Chennai Mathematical Institute, India
Fahad Panolan
  • School of Computing, University of Leeds, UK
Saket Saurabh
  • Institute of Mathematical Sciences, HBNI, Chennai, India
Meirav Zehavi
  • Ben-Gurion University of the Negev, Beersheba, Israel

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