5 research outputs found
Hamiltonian Cycle Parameterized by Treedepth in Single Exponential Time and Polynomial Space
For many algorithmic problems on graphs of treewidth , a standard dynamic
programming approach gives an algorithm with time and space complexity
. It turns out that when one
considers the more restrictive parameter treedepth, it is often the case that a
variation of this technique can be used to reduce the space complexity to
polynomial, while retaining time complexity of the form
, where is the treedepth. This
transfer of methodology is, however, far from automatic. For instance, for
problems with connectivity constraints, standard dynamic programming techniques
give algorithms with time and space complexity on graphs of treewidth , but it is not clear how to
convert them into time-efficient polynomial space algorithms for graphs of low
treedepth.
Cygan et al. (FOCS'11) introduced the Cut&Count technique and showed that a
certain class of problems with connectivity constraints can be solved in time
and space complexity . Recently,
Hegerfeld and Kratsch (STACS'20) showed that, for some of those problems, the
Cut&Count technique can be also applied in the setting of treedepth, and it
gives algorithms with running time
and polynomial space usage. However, a number of important problems eluded such
a treatment, with the most prominent examples being Hamiltonian Cycle and
Longest Path.
In this paper we clarify the situation by showing that Hamiltonian Cycle,
Hamiltonian Path, Long Cycle, Long Path, and Min Cycle Cover all admit
-time and polynomial space algorithms on graphs of
treedepth . The algorithms are randomized Monte Carlo with only false
negatives.Comment: Presented at WG2020. 20 pages, 2 figure
Turing kernelization for finding long paths in graph classes excluding a topological minor
\u3cp\u3eThe 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\u3csup\u3eO\u3c/sup\u3e \u3csup\u3e(\u3c/sup\u3e \u3csup\u3e1\u3c/sup\u3e \u3csup\u3e)\u3c/sup\u3e? 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\u3csub\u3e3\u3c/sub\u3e \u3csub\u3e,\u3c/sub\u3e \u3csub\u3et\u3c/sub\u3e-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.\u3c/p\u3
Turing kernelization for finding long paths in graph classes excluding a topological minor
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 boundedsize 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 kO(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 K3,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
Turing kernelization for finding long paths in graph classes excluding a topological minor
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 poly(k)? 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