Approximation algorithm for maximum edge coloring

Wangsen Feng, Li'ang Zhang, Hanpin Wang

Abstract: We propose a polynomial time approximation algorithm for a novel maximum edge coloring problem which arises from wireless mesh networks [Ashish Raniwala, Tzi-cker Chiueh, Architecture and algorithms for an IEEE 802.11-based multi-channel wireless mesh network, in: INFOCOM 2005, pp. 2223–2234; Ashish Raniwala, Kartik Gopalan, Tzi-cker Chiueh, Centralized channel assignment and routing algorithms for multi-channel wireless mesh networks, Mobile Comput. Commun. Rev. 8 (2) (2004) 50–65]. The problem is to color all the edges in a graph with maximum number of colors under the following q<math><mi is="true">q</mi></math>-Constraint: for every vertex in the graph, all the edges incident to it are colored with no more than q<math><mi is="true">q</mi></math> (q∈Z,q≥2<math><mi is="true">q</mi><mo is="true">∈</mo><mi mathvariant="double-struck" is="true">Z</mi><mo is="true">,</mo><mspace class="nbsp" is="true"></mspace><mi is="true">q</mi><mo is="true">≥</mo><mn is="true">2</mn></math>) colors. We show that the algorithm is a 2-approximation for the case q=2<math><mi is="true">q</mi><mo is="true">=</mo><mn is="true">2</mn></math> and a (1+4q−23q2−5q+2)<math><mrow is="true"><mo is="true">(</mo><mn is="true">1</mn><mo is="true">+</mo><mfrac is="true"><mrow is="true"><mn is="true">4</mn><mi is="true">q</mi><mo is="true">−</mo><mn is="true">2</mn></mrow><mrow is="true"><mn is="true">3</mn><msup is="true"><mrow is="true"><mi is="true">q</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup><mo is="true">−</mo><mn is="true">5</mn><mi is="true">q</mi><mo is="true">+</mo><mn is="true">2</mn></mrow></mfrac><mo is="true">)</mo></mrow></math>-approximation for the case q>2<math><mi is="true">q</mi><mo is="true">&gt;</mo><mn is="true">2</mn></math> respectively. The case q=2<math><mi is="true">q</mi><mo is="true">=</mo><mn is="true">2</mn></math> is of great importance in practice. For complete graphs and trees, polynomial time accurate algorithms are found for them when q=2<math><mi is="true">q</mi><mo is="true">=</mo><mn is="true">2</mn></math>. The approximation algorithm gives a feasible solution to channel assignment in multi-channel wireless mesh networks.