The DOI matches its Crossref record; 46 of the 46 references it cites were found in a registry or an index.
A Linear Programming Approach to Max-Sum Problem: A Review
Abstract
The max-sum labeling problem, defined as maximizing a sum of binary (i.e., pairwise) functions of discrete variables, is a general NP-hard optimization problem with many applications, such as computing the MAP configuration of a Markov random field. We review a not widely known approach to the problem, developed by Ukrainian researchers Schlesinger et al. in 1976, and show how it contributes to recent results, most importantly, those on the convex combination of trees and tree-reweighted max-product. In particular, we review Schlesinger et al.'s upper bound on the max-sum criterion, its minimization by equivalent transformations, its relation to the constraint satisfaction problem, the fact that this minimization is dual to a linear programming relaxation of the original problem, and the three kinds of consistency necessary for optimality of the upper bound. We revisit problems with Boolean variables and supermodular problems. We describe two algorithms for decreasing the upper bound. We present an example application for structural image analysis.
Overview
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Citations
Cites
(1988) · Cited by 16,582
Fast approximate energy minimization via graph cuts
Boykov, Y., Veksler, O., & Zabih, R. (2001) · IEEE Transactions on Pattern Analysis and Machine Intelligence · doi:10.1109/34.969114 · Cited by 6,988
Introduction to Lattices and Order
Davey, B. A., & Priestley, H. A. (2002) · Cambridge University Press eBooks · doi:10.1017/cbo9780511809088 · Cited by 4,901
Geometric Algorithms and Combinatorial Optimization
Grötschel, M., Lovász, L., & Schrijver, A. (1988) · Algorithms and Combinatorics · doi:10.1007/978-3-642-97881-4 · Cited by 3,431
Perceptrons: An Introduction to Computational Geometry
Minsky, M., & Papert, S. A. (1969) · Cited by 3,266
What energy functions can be minimized via graph cuts?
Kolmogorov, V., & Zabih, R. (2004) · IEEE Transactions on Pattern Analysis and Machine Intelligence · doi:10.1109/tpami.2004.1262177 · Cited by 3,127
Consistency in networks of relations
Mackworth, A. K. (1977) · Artificial Intelligence · doi:10.1016/0004-3702(77)90007-8 · Cited by 2,458
The ellipsoid method and its consequences in combinatorial optimization
Grötschel, M., Lovász, L., & Schrijver, A. (1981) · Combinatorica · doi:10.1007/bf02579273 · Cited by 2,011
Constructing Free-Energy Approximations and Generalized Belief Propagation Algorithms
Yedidia, J. S., Freeman, W. T., & Weiss, Y. (2005) · IEEE Transactions on Information Theory · doi:10.1109/tit.2005.850085 · Cited by 1,617
Scene Labeling by Relaxation Operations
Rosenfeld, A., Hummel, R. A., & Zucker, S. W. (1976) · IEEE Transactions on Systems, Man, and Cybernetics · doi:10.1109/tsmc.1976.4309519 · Cited by 1,399
Cited by
Komodakis, N., & Pesquet, J.-C. (2015) · IEEE Signal Processing Magazine · doi:10.1109/msp.2014.2377273 · Cited by 419
Feature Correspondence Via Graph Matching: Models and Global Optimization
Torresani, L., Kolmogorov, V., & Rother, C. (2008) · Lecture Notes in Computer Science · doi:10.1007/978-3-540-88688-4_44 · Cited by 388
Visual Semantic Information Pursuit: A Survey
Liu, D., Bober, M., & Kittler, J. (2021) · IEEE Transactions on Pattern Analysis and Machine Intelligence · doi:10.1109/tpami.2019.2950025 · Cited by 39
Metadata
- Authors
- Werner, T.
- Venue
- IEEE Transactions on Pattern Analysis and Machine Intelligence
- Published
- 2007-07
- Volume
- 29
- Issue
- 7
- Pages
- 1165-1179
- ISSN
- 0162-8828 (ISSN-L)
- OpenAlex
- W2135414191
- Cited by
- 342
- References
- 46
- FWCI
- 31.52
- Type
- journal article
- Language
- en
- Open access
- closed
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
How to cite
Cite CoChat for the checks on this page, and the paper for its own findings.
- This page
- CoChat. (2026). Citation check: A Linear Programming Approach to Max-Sum Problem: A Review. https://app.cochat.ai/papers/10.1109/tpami.2007.1036
- The paper
- Werner, T. (2007). A Linear Programming Approach to Max-Sum Problem: A Review. IEEE Transactions on Pattern Analysis and Machine Intelligence. https://doi.org/10.1109/tpami.2007.1036
Data sources
Bibliographic data: Semantic Scholar, OpenAlex, Crossref.
Contains information from the Semantic Scholar Academic Graph Datasets, made available under the ODC Attribution License (ODC-BY 1.0).
Semantic Scholar’s data is described in: Kinney, R., et al. (2023). The Semantic Scholar Open Data Platform. arXiv:2301.10140.
