11 research outputs found
Sunflowers Meet Sparsity: A Linear-Vertex Kernel for Weighted Clique-Packing on Sparse Graphs
We study the kernelization complexity of the Weighted H-Packing problem on sparse graphs. For a fixed connected graph H, in the Weighted H-Packing problem the input is a graph G, a vertex-weight function w: V (G) → N, and positive integers k, t. The question is whether there exist k vertex-disjoint subgraphs H 1, ⋯, H k of G such that H i is isomorphic to H for each i ∈ [k] and the total weight of these k · |V (H)| vertices is at least t. It is known that the (unweighted) H-Packing problem admits a kernel with O(k |V (H)|-1) vertices on general graphs, and a linear kernel on planar graphs and graphs of bounded genus. In this work, we focus on case that H is a clique on h ≥ 3 vertices (which captures Triangle Packing) and present a linear-vertex kernel for Weighted Kh-Packing on graphs of bounded expansion, along with a kernel with O(k 1+ϵ) vertices on nowhere-dense graphs for all ϵ > 0. To obtain these results, we combine two powerful ingredients in a novel way: the Erdos-Rado Sunflower lemma and the theory of sparsity.</p
Sunflowers Meet Sparsity: A Linear-Vertex Kernel for Weighted Clique-Packing on Sparse Graphs
We study the kernelization complexity of the Weighted H-Packing problem on sparse graphs. For a fixed connected graph H, in the Weighted H-Packing problem the input is a graph G, a vertex-weight function w: V (G) → N, and positive integers k, t. The question is whether there exist k vertex-disjoint subgraphs H 1, ⋯, H k of G such that H i is isomorphic to H for each i ∈ [k] and the total weight of these k · |V (H)| vertices is at least t. It is known that the (unweighted) H-Packing problem admits a kernel with O(k |V (H)|-1) vertices on general graphs, and a linear kernel on planar graphs and graphs of bounded genus. In this work, we focus on case that H is a clique on h ≥ 3 vertices (which captures Triangle Packing) and present a linear-vertex kernel for Weighted Kh-Packing on graphs of bounded expansion, along with a kernel with O(k 1+ϵ) vertices on nowhere-dense graphs for all ϵ > 0. To obtain these results, we combine two powerful ingredients in a novel way: the Erdos-Rado Sunflower lemma and the theory of sparsity.</p
On Petri Nets with Hierarchical Special Arcs
We investigate the decidability of termination, reachability, coverability and deadlock-freeness of Petri nets endowed with a hierarchy of places, and with inhibitor arcs, reset arcs and transfer arcs that respect this hierarchy. We also investigate what happens when we have a mix of these special arcs, some of which respect the hierarchy, while others do not. We settle the decidability status of the above four problems for all combinations of hierarchy, inhibitor, reset and transfer arcs, except the termination problem for two combinations. For both these combinations, we show that deciding termination is as hard as deciding the positivity problem on linear recurrence sequences -- a long-standing open problem
Synchronizing Data Words for Register Automata
Register automata (RAs) are finite automata extended with a finite set of
registers to store and compare data from an infinite domain. We study the
concept of synchronizing data words in RAs: does there exist a data word that
sends all states of the RA to a single state?
For deterministic RAs with k registers (k-DRAs), we prove that inputting data
words with 2k+1 distinct data from the infinite data domain is sufficient to
synchronize. We show that the synchronization problem for DRAs is in general
PSPACE-complete, and it is NLOGSPACE-complete for 1-DRAs. For nondeterministic
RAs (NRAs), we show that Ackermann(n) distinct data (where n is the size of the
RA) might be necessary to synchronize. The synchronization problem for NRAs is
in general undecidable, however, we establish Ackermann-completeness of the
problem for 1-NRAs.
Another main result is the NEXPTIME-completeness of the length-bounded
synchronization problem for NRAs, where a bound on the length of the
synchronizing data word, written in binary, is given. A variant of this last
construction allows to prove that the length-bounded universality problem for
NRAs is co-NEXPTIME-complete
Cutting Barnette graphs perfectly is hard
A perfect matching cut is a perfect matching that is also a cutset, or equivalently, a perfect matching containing an even number of edges on every cycle. The corresponding algorithmic problem, Perfect Matching Cut, is known to be NP-complete in subcubic bipartite graphs [Le & Telle, TCS '22], but its complexity was open in planar graphs and cubic graphs. We settle both questions simultaneously by showing that Perfect Matching Cut is NP-complete in 3-connected cubic bipartite planar graphs or Barnette graphs. Prior to our work, among problems whose input is solely an undirected graph, only Distance-2 4-Coloring was known to be NP-complete in Barnette graphs. Notably, Hamiltonian Cycle would only join this private club if Barnette's conjecture were refuted