Loading...
Please wait, while we are loading the content...
Similar Documents
Expressing Combinatorial Optimization Problems by Systems of Polynomial Equations and the Nullstellensatz
| Content Provider | Semantic Scholar |
|---|---|
| Author | Loera, Jesús A. De Lee, Joonkyung Margulies, Susan Onn, Shmuel |
| Copyright Year | 2007 |
| Abstract | Systems of polynomial equations over the complex or real numbers can be used to model combinatorial problems. In this way, a combinatorial problem is feasible (e.g. a graph is 3-colorable, hamiltonian, etc.) if and only if a related system of polynomial equations has a solution. In the first part of this paper, we construct new polynomial encodings for the problems of finding in a graph its longest cycle, the largest planar subgraph, the edge-chromatic number, or the largest k-colorable subgraph. For an infeasible polynomial system, the (complex) Hilbert Nullstellensatz gives a certificate that the associated combinatorial problem is infeasible. Thus, unless P = NP, there must exist an infinite sequence of infeasible instances of each hard combinatorial problem for which the minimum degree of a Hilbert Nullstellensatz certificate of the associated polynomial system grows. We show that the minimum-degree of a Nullstellensatz certificate for the non-existence of a stable set of size greater than the stability number of the graph is the stability number of the graph. Moreover, such a certificate contains at least one term per stable set of G. In contrast, for non-3- colorability, we found only graphs with Nullstellensatz certificates of degree four. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://arxiv.org/pdf/0706.0578v1.pdf |
| Alternate Webpage(s) | https://ie.technion.ac.il/~onn/Preprints/graph_polynomials.pdf |
| Alternate Webpage(s) | http://ie.technion.ac.il/~onn/Preprints/graph_polynomials.pdf |
| Alternate Webpage(s) | http://www.math.psu.edu/margulies/papers/final_cpc.pdf |
| Alternate Webpage(s) | http://www.optimization-online.org/DB_FILE/2007/06/1688.pdf |
| Alternate Webpage(s) | http://arxiv.org/pdf/0706.0578v1.pdf |
| Alternate Webpage(s) | http://arxiv.org/pdf/0706.0578.pdf |
| Alternate Webpage(s) | http://www.math.ucdavis.edu/~deloera/MISC/MATHRESEARCH/nullsoda.pdf |
| Alternate Webpage(s) | https://www.usna.edu/Users/math/marguile/papers/final_cpc.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
