Abstract
The notion of Turing kernelization investigates whether a polynomial-time algorithm can solve an NP-hard problem, when it is aided by an oracle that can be queried for the answers to bounded-size subproblems. One of the main open problems in this direction is whether k-Path admits a polynomial Turing kernel: can a polynomial-time algorithm determine whether an undirected graph has a simple path of length k, using an oracle that answers queries of size k^{mathscr {O} (1)}? We show this can be done when the input graph avoids a fixed graph H as a topological minor, thereby significantly generalizing an earlier result for bounded-degree and K_{3,t}-minor-free graphs. Moreover, we show that k-Path even admits a polynomial Turing kernel when the input graph is not H-topological-minor-free itself, but contains a known vertex modulator of size bounded polynomially in the parameter, whose deletion makes it so. To obtain our results, we build on the graph minors decomposition to show that any H-topological-minor-free graph that does not contain a k-path, has a separation that can safely be reduced after communication with the oracle.
Highlights
Suppose that Alice is a polynomial-time agent faced with an input to an NP-hard problem that she wishes to solve exactly
Theorem 45 One can solve in polynomial time a given k-Path instance (G, k), given access to a set M ⊆ V (G) such that G − M admits a tree decomposition of width less than w and adhesion size h = O(1), and an oracle that solves the Auxiliary Linkage problem for instances (G, k, S, (Ri )ri=1) with G being a subgraph of G, max(r, k ) ≤ k, |S| ≤ |M| + O(1), and |V (G )| being bounded polynomially in k, w, and |M|
We significantly extended the graph classes on which k-Path has a polynomial Turing kernel
Summary
Suppose that Alice is a polynomial-time agent faced with an input to an NP-hard problem that she wishes to solve exactly. In this decomposition the adhesions, i.e. intersections of two adjacent bags, have constant size (at most 2) This makes it possible to find a separation (A, B) of the graph of constant order |A ∩ B| and with A moderately large: polynomially bounded in k (so that G[A] can be sent to the oracle) but large enough to contain an irrelevant vertex (one whose removal will not change the answer). These guarantee a tree decomposition of constant adhesion (depending only on the fixed graph H ) with bags that are possibly large but nearly-embeddable or have bounded degree (for topological minors) As before, if these bags are triconnected, known theorems [7,9,30] can be used to bound their size polynomially.
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