Abstract
We study the following linear classification problem in signal processing: Given a set Bof n black point and a set W of m white points in the plane (m = O(n)), compute a minimum number of lines L such that in the arrangement of L each face contain points with the same color (i.e., either all black points or all white points). We call this the Minimum Linear Classification (MLC) problem. We prove that MLC is NP-complete by a reduction from the Minimum Line Fitting (MLF) problem; moreover, a C-approximation to MLC implies a C-approximation to the MLF problem. Nevertheless, we obtain an O(log n)-factor algorithm for MLC and we also obtain an O(log Z)-factor algorithm for MLC where Z is the minimum number of disjoint axis-parallel black/white rectangles covering B and W.
Similar content being viewed by others
References
V. Chvátal (1979) ArticleTitleA greedy heuristic for the set-covering problem Mathematics of Operations Research 4 233–235 Occurrence Handle10.1287/moor.4.3.233
Cormen, T. Leiserson, C. and Rivest, R. (1990), Introduction to Algorithms, The MIT Press.
G.D. Forney SuffixJr. (1972) ArticleTitleMaximum-likelihood sequence estimation of digital sequences in the presence of intersymbol interference IEEE Transactions Information 18 IssueID30 363–378 Occurrence Handle10.1109/TIT.1972.1054829
R. Hassin N. Megiddo (1990) ArticleTitleApproximation algorithms for hitting objects with straight lines Discrete Applied Mathematics 30 29–42 Occurrence Handle10.1016/0166-218X(91)90011-K
H. Imai Ta. Asano (1986) ArticleTitleEfficient algorithms for geometric graph searching problems SIAM Journal on Computing 15 478–494 Occurrence Handle10.1137/0215033
D. Johnson (1974) ArticleTitleApproximation algorithms for combinatorial problems Journal of Computer and System Sciences 9 256–278 Occurrence Handle10.1016/S0022-0000(74)80044-9
Kim, Y. and Moon, J. (2000), Multi-dimensional signal space partitioning using a minimal set of hyperplanes for detecting ISI-corrupted symbols, IEEE Transactions on Communications, April 2000.
L. Lovász (1975) ArticleTitleOn the ratio of optimal integral and fractional covers Discrete Mathematics 13 383–390 Occurrence Handle10.1016/0012-365X(75)90058-8
Mata, C. and Mitchell, J. (1995), Approximation algorithms for geometric tour and network design problems, Proceedings of the 11th ACM Symposium on Computing Geometry, 360–369
N. Megiddo A. Tamir (1982) ArticleTitleOn the complexity of locating linear facilities in the plane Operations Research Letters 1 194–197 Occurrence Handle10.1016/0167-6377(82)90039-6
J. Moon L.R. Carley (1990) ArticleTitlePerformance comparison of detection methods in magnetic recording IEEE Transactions on Magnetics 26 IssueID6 3155–3172 Occurrence Handle10.1109/20.102903
J. Moon T. Jeon (1998) ArticleTitleSequence detection for binary ISI channels using signal space partitioning IEEE Transactions on Communications 46 IssueID7 891–901 Occurrence Handle10.1109/26.701316
Raz, R. and Safra, S. (1997), A sub-constant error-probability low-degree test, and a subconstant error-probability PCP characterization of NP, Proceedings of the 29th ACM Symposium on Theory of Computing (STOC97), 475–484.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lu, B., Du, H., Jia, X. et al. On a Minimum Linear Classification Problem. J Glob Optim 35, 103–109 (2006). https://doi.org/10.1007/s10898-005-8465-2
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10898-005-8465-2