tailieunhanh - On some interconnections between combinatorial optimization and extremal graph theory

The uniting feature of combinatorial optimization and extremal graph theory is that in both areas one should find extrema of a function defined in most cases on a finite set. While in combinatorial optimization the point is in developing efficient algorithms and heuristics for solving specified types of problems, the extremal graph theory deals with finding bounds for various graph invariants under some constraints and with constructing extremal graphs. | Yugoslav Journal of Operations Research 14 (2004), Number 2, 147-154 ON SOME INTERCONNECTIONS BETWEEN COMBINATORIAL OPTIMIZATION AND EXTREMAL GRAPH THEORY Dragoš CVETKOVIĆ Faculty of Electrical Engineering, University of Belgrade, Belgrade, Serbia and Montenegro, ecvetkod@ Pierre HANSEN GERAD and École des Hautes Études Commerciales, Montréal H3T 2A7, Canada Vera KOVAČEVIĆ-VUJČIĆ Faculty of Organizational Sciences, University of Belgrade, Belgrade, Serbia and Montenegro verakov@ Received: March 2004 / Accepted: August 2004 The uniting feature of combinatorial optimization and extremal graph theory is that in both areas one should find extrema of a function defined in most cases on a finite set. While in combinatorial optimization the point is in developing efficient algorithms and heuristics for solving specified types of problems, the extremal graph theory deals with finding bounds for various graph invariants under some constraints and with constructing extremal graphs. We analyze by examples some interconnections and interactions of the two theories and propose some conclusions. Keywords: Combinatorial optimization, extremal graph theory, variable neighborhood search, mathematical programming. 148 D. Cvetković, P. Hansen, V. Kovačević-Vujčić / On Some Interconnections 1. INTRODUCTION We list a few details from Mathematics Subject Classification 2000 which are relevant for our discussion. 05 Combinatorics 05C Graph Theory 05C35 Extremal Problems 05C90 Graph Algorithms 90 Operations Research, Mathematical Programming 90C Mathematical Programming 90C27 Combinatorial Optimization 90C35 Programming Involving Graphs and Networks 90C22 Semidefinite Programming 68 Computer Science 68R Discrete Mathematics in Relation to Computer Science 68R10 Graph Theory 68W Algorithms 68W05 Non-numerical Algorithms We shal discuss some relations between combinatoral optimization (90C27) and extremal problems in graph theory or .