A Hierarchy of Polynomial Kernels

In parameterized algorithmics, the process of kernelization is defined as a polynomial time algorithm that transforms the instance of a given problem to an equivalent instance of a size that is limited by a function of the parameter. As, afterwards, this smaller instance can then be solved to find an answer to the original question, kernelization is often presented as a form of preprocessing. A natural generalization of kernelization is the process that allows for a number of smaller instances to be produced to provide an answer to the original problem, possibly also using negation. This generalization is called Turing kernelization. Immediately, questions of equivalence occur or, when is one form possible and not the other. These have been long standing open problems in parameterized complexity. In the present paper, we answer many of these. In particular, we show that Turing kernelizations differ not only from regular kernelization, but also from intermediate forms as truth-table kernelizations. We achieve absolute results by diagonalizations and also results on natural problems depending on widely accepted complexity theoretic assumptions. In particular, we improve on known lower bounds for the kernel size of compositional problems using these assumptions

Feedback Vertex Set Inspired Kernel for Chordal Vertex Deletion

Given a graph $G$ and a parameter $k$, the Chordal Vertex Deletion (CVD) problem asks whether there exists a subset $U\subseteq V(G)$ of size at most $k$ that hits all induced cycles of size at least 4. The existence of a polynomial kernel for CVD was a well-known open problem in the field of Parameterized Complexity. Recently, Jansen and Pilipczuk resolved this question affirmatively by designing a polynomial kernel for CVD of size $O(k^{161}\log^{58}k)$, and asked whether one can design a kernel of size $O(k^{10})$. While we do not completely resolve this question, we design a significantly smaller kernel of size $O(k^{12}\log^{10}k)$, inspired by the $O(k^2)$-size kernel for Feedback Vertex Set. Furthermore, we introduce the notion of the independence degree of a vertex, which is our main conceptual contribution

A polynomial Turing-kernel for weighted independent set in bull-free graphs

The maximum stable set problem is NP-hard, even when restricted to triangle-free graphs. In particular, one cannot expect a polynomial time algorithm deciding if a bull-free graph has a stable set of size k, when k is part of the instance. Our main result in this paper is to show the existence of an FPT algorithm when we parameterize the problem by the solution size k. A polynomial kernel is unlikely to exist for this problem. We show however that our problem has a polynomial size Turingkernel. More precisely, the hard cases are instances of size O(k5). As a byproduct, if we forbid odd holes in addition to the bull, we show the existence of a polynomial time algorithm for the stable set problem. We also prove that the chromatic number of a bull-free graph is bounded by a function of its clique number and the maximum chromatic number of its triangle-free induced subgraphs. All our results rely on a decomposition theorem for bull-free graphs due to Chudnovsky which is modified here, allowing us to provide extreme decompositions, adapted to our computational purpose

Turing kernelization for finding long paths and cycles in restricted graph classes

\u3cp\u3eThe k-PATH problem asks whether a given undirected graph has a (simple) path of length k. We prove that k-PATH has polynomial-size Turing kernels when restricted to planar graphs, graphs of bounded degree, claw-free graphs, or to K \u3csub\u3e3,t\u3c/sub\u3e-minor-free graphs. This means that there is an algorithm that, given a k-PATH instance (G,k) belonging to one of these graph classes, computes its answer in polynomial time when given access to an oracle that solves k-PATH instances of size polynomial in k in a single step. Our techniques also apply to k-CYCLE, which asks for a cycle of length at least k. \u3c/p\u3

Turing kernelization for finding long paths and cycles in restricted graph classes

The k-PATH problem asks whether a given undirected graph has a (simple) path of length k. We prove that k-PATH has polynomial-size Turing kernels when restricted to planar graphs, graphs of bounded degree, claw-free graphs, or to K 3,t-minor-free graphs. This means that there is an algorithm that, given a k-PATH instance (G,k) belonging to one of these graph classes, computes its answer in polynomial time when given access to an oracle that solves k-PATH instances of size polynomial in k in a single step. Our techniques also apply to k-CYCLE, which asks for a cycle of length at least k

Turing Kernelization for Finding Long Paths and Cycles in Restricted Graph Classes

Abstract. We analyze the potential for provably effective preprocessing for the problems of finding paths and cycles with at least k edges. Several years ago, the question was raised whether the existing superpolynomial kernelization lower bounds for k-Path and k-Cycle can be circumvented by relaxing the requirement that the preprocessing algorithm outputs a single instance. To this date, very few examples are known where the relaxation to Turing kernelization is fruitful. We provide a novel example by giving polynomial-size Turing kernels for k-Path and k-Cycle on planar graphs, graphs of maximum degree t, claw-free graphs, and K3,t-minor-free graphs, for each constant t ≥ 3. Concretely, we present algorithms for k-Path (k-Cycle) on these restricted graph families that run in polynomial time when they are allowed to query an external oracle for the answers to k-Path (k-Cycle) instances of size and parameter bounded polynomially in k. Our kernelization schemes are based on a new methodology called Decompose-Query-Reduce.