Uéverton S. Souza

Parameterized Complexity of Flood-Filling Games on Trees

This work presents new results on flood-filling games, Flood-It and Free-Flood-It, in which the player aims to make the board monochromatic with a minimum number of flooding moves. As for many colored graph problems, Flood-filling games have relevant interpretations in bioinformatics. The standard versions of Flood-It and Free-Flood-It are played on n ×m grids. In this paper we analyze the complexity of these games when played on trees. We prove that Flood-It remains NP-hard on trees whose leaves are at distance at most d = 2 from the pivot, and that Flood-It is in FPT when parameterized by the number of colors c in such trees (for any constant d). We also show that Flood-It on trees and Restricted Shortest Common Supersequence (RSCS) are analogous problems, in the sense that they can be translated from one to another, keeping complexity features; this implies that Flood-It on trees inherits several complexity results already proved for RSCS, such as some interesting FPT and W[1]-hard cases. We introduce a new variant of Flood-It, called Multi-Flood-It, where each move of the game is played on various pivots. We also present a general framework for reducibility from Flood-It to Free-Flood-It, by defining a special graph operator ψ such that Flood-It played on a graph class \(\mathcal{F}\) is reducible to Free-Flood-It played on the image of \(\mathcal{F}\) under ψ. An interesting particular case occurs when \(\mathcal{F}\) is closed under ψ. Some NP-hard cases for Free-Flood-It on trees can be derived using this approach. We conclude by showing some results on parameterized complexity for Free-Flood-It played on pc-trees (phylogenetic colored trees). We prove that some results valid for Flood-It on pc-trees can be inherited by Free-Flood-It on pc-trees, using another type of reducibility framework.

Maximum induced matchings close to maximum matchings

Extending results of Kobler and Rotics (2003), Cameron and Walker (2005) gave a complete structural description of the graphs G where the matching number ? ( G ) equals the induced matching number ? 2 ( G ) . We present a short proof of their result and use it to study graphs G with ? ( G ) - ? 2 ( G ) ? k . We show that the recognition of these graphs can be done in polynomial time for fixed k, and is fixed parameter tractable when parameterized by k for graphs of bounded maximum degree.

Tractability and hardness of flood-filling games on trees

This work presents new results on flood-filling games, Flood-It and Free-Flood-It, in which the player aims to make the board monochromatic with a minimum number of flooding moves. A flooding move consists of changing the color of the monochromatic component containing a vertex p (the pivot of the move). These games are originally played on grids; however, when played on trees, we have interesting applications and significant effects on problem complexity. In this paper, a complete mapping of the complexity of flood-filling games on trees is made, charting the consequences of single and aggregate parameterizations by: number of colors, number of moves, maximum distance from the pivot, number of occurrences of a color, number of leaves, and difference between number of moves and number of colors.We show that Flood-It on trees and Restricted Shortest Common Supersequence (RSCS) are analogous problems, in the sense that they can be translated from one to another, preserving complexity issues; this implies interesting FPT and W1]-hard cases to Flood-It. Restricting attention to phylogenetic colored trees (where each color occurs at most once in any root-leaf path, in order to model phylogenetic sequences), we also show some impressive NP-hard and FPT results for both games. In addition, we prove that Flood-It and Free-Flood-It remain NP-hard when played on 3-colored trees, which closes an open question posed by Fleischer and Woeginger. Finally, we present a general framework for reducibility from Flood-It to Free-Flood-It; some NP-hard cases for Free-Flood-It can be derived using this approach.

Revisiting the complexity of and/or graph solution

An and/or graph is an acyclic, edge-weighted directed graph containing a single source vertex such that every vertex v has a label f(v)∈{and,or}. A solution subgraph H of an and/or-graph must contain the source and obey the following rule: if an and-vertex (resp. or-vertex) is included in H then all (resp. one) of its out-edges must also be included in H. X-y graphs are defined as a generalization of and/or graphs: every vertex vi of an x-y graph has a label xi-yi, in such a way that if a vertex vi is included in a solution subgraph H of an x-y graph then xi of its yi out-edges must also be included in H. In this work, we analyze complexity aspects (both from the classical and the parameterized point of view) of finding solution subgraphs of minimum weight for and/or and x-y graphs.

Deletion Graph Problems Based on Deadlock Resolution

A deadlock occurs in a distributed computation if a group of processes wait indefinitely for resources from each other. In this paper we study actions to be taken after deadlock detection, especially the action of searching a small deadlock-resolution set. More precisely, given a “snapshot” graph G representing a deadlocked state of a distributed computation governed by a certain deadlock model \(\mathbb {M}\), we investigate the complexity of vertex/arc deletion problems that aim at finding minimum vertex/arc subsets whose removal turns G into a deadlock-free graph (according to model \(\mathbb {M}\)). Our contributions include polynomial algorithms and hardness proofs, for general graphs and for special graph classes. Among other results, we show that the arc deletion problem in the OR model can be solved in polynomial time, and the vertex deletion problem in the OR model remains NP-Complete even for graphs with maximum degree four, but it is solvable in \(O (m \sqrt{n})\) time for graphs with \(\varDelta \le 3\).

Complexity properties of complementary prisms

The complementary prism

On the (Parameterized) Complexity of Recognizing Well-Covered

G\bar{G}

-graphs

GG¯ of a graph G arises from the disjoint union of the graph G and its complement