Abstract
We study the Art Gallery Problem under k-hop visibility in polyominoes. In this visibility model, two unit squares of a polyomino can see each other if and only if the shortest path between the respective vertices in the dual graph of the polyomino has length at most k. In this paper, we show that the VC dimension of this problem is 3 in simple polyominoes, and 4 in polyominoes with holes. Furthermore, we provide a reduction from Planar Monotone 3Sat, thereby showing that the problem is NP-complete even in thin polyominoes (i.e., polyominoes that do not a contain a 2×2 block of cells). Complementarily, we present a linear-time 4-approximation algorithm for simple 2-thin polyominoes (which do not contain a 3×3 block of cells) for all k∈N.
| Original language | English |
|---|---|
| Title of host publication | LATIN 2024 |
| Subtitle of host publication | Theoretical Informatics - 16th Latin American Symposium, 2024, Proceedings |
| Editors | José A. Soto, Andreas Wiese |
| Publisher | Springer Science and Business Media Deutschland GmbH |
| Pages | 288-302 |
| Number of pages | 15 |
| ISBN (Print) | 9783031555978 |
| DOIs | |
| State | Published - 2024 |
| Event | 16th Latin American Symposium on Theoretical Informatics, LATIN 2042 - Puerto Varas, Chile Duration: 18 Mar 2024 → 22 Mar 2024 |
Publication series
| Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
|---|---|
| Volume | 14578 LNCS |
| ISSN (Print) | 0302-9743 |
| ISSN (Electronic) | 1611-3349 |
Conference
| Conference | 16th Latin American Symposium on Theoretical Informatics, LATIN 2042 |
|---|---|
| Country/Territory | Chile |
| City | Puerto Varas |
| Period | 18/03/24 → 22/03/24 |
Bibliographical note
Publisher Copyright:© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.
Keywords
- approximation
- Art Gallery problem
- k-hop dominating set
- k-hop visibility
- polyominoes
- VC dimension