TY - GEN
T1 - On Streaming Algorithms for Geometric Independent Set and Clique
AU - Bhore, Sujoy
AU - Klute, Fabian
AU - Oostveen, Jelle J.
N1 - Publisher Copyright:
© 2022, The Author(s), under exclusive license to Springer Nature Switzerland AG.
PY - 2022/1/1
Y1 - 2022/1/1
N2 - We study the maximum geometric independent set and clique problems in the streaming model. Given a collection of geometric objects arriving in an insertion only stream, the aim is to find a subset such that all objects in the subset are pairwise disjoint or intersect respectively. We show that no constant factor approximation algorithm exists to find a maximum set of independent segments or 2-intervals without using a linear number of bits. Interestingly, our proof only requires a set of segments whose intersection graph is also an interval graph. This reveals an interesting discrepancy between segments and intervals as there does exist a 2-approximation for finding an independent set of intervals that uses only O(α(I) log | I| ) bits of memory for a set of intervals I with α(I) being the size of the largest independent set of I. On the flipside we show that for the geometric clique problem there is no constant-factor approximation algorithm using less than a linear number of bits even for unit intervals. On the positive side we show that the maximum geometric independent set in a set of axis-aligned unit-height rectangles can be 4-approximated using only O(α(R) log | R| ) bits.
AB - We study the maximum geometric independent set and clique problems in the streaming model. Given a collection of geometric objects arriving in an insertion only stream, the aim is to find a subset such that all objects in the subset are pairwise disjoint or intersect respectively. We show that no constant factor approximation algorithm exists to find a maximum set of independent segments or 2-intervals without using a linear number of bits. Interestingly, our proof only requires a set of segments whose intersection graph is also an interval graph. This reveals an interesting discrepancy between segments and intervals as there does exist a 2-approximation for finding an independent set of intervals that uses only O(α(I) log | I| ) bits of memory for a set of intervals I with α(I) being the size of the largest independent set of I. On the flipside we show that for the geometric clique problem there is no constant-factor approximation algorithm using less than a linear number of bits even for unit intervals. On the positive side we show that the maximum geometric independent set in a set of axis-aligned unit-height rectangles can be 4-approximated using only O(α(R) log | R| ) bits.
KW - Communication lower bounds
KW - Geometric independent set
KW - Geometric intersection graphs
KW - Streaming algorithms
UR - https://www.scopus.com/pages/publications/85142713250
U2 - 10.1007/978-3-031-18367-6_11
DO - 10.1007/978-3-031-18367-6_11
M3 - Conference contribution
AN - SCOPUS:85142713250
SN - 9783031183669
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 211
EP - 224
BT - Approximation and Online Algorithms - 20th International Workshop, WAOA 2022, Proceedings
A2 - Chalermsook, Parinya
A2 - Laekhanukit, Bundit
PB - Springer Science and Business Media Deutschland GmbH
T2 - 20th International Workshop on Approximation and Online Algorithms, WAOA 2022
Y2 - 8 September 2022 through 9 September 2022
ER -