An Integer Programming Approach to the Bandwidth Packing Problem

Abstract

We consider the bandwidth packing problem arising from telecommunication networks. The problem is to determine the set of calls and an assignment of them to the paths in an arc-capacitated network to maximize profit. We propose an algorithm to solve the integer programming formulation of the problem. An efficient column generation technique to solve the linear programming relaxation is proposed, and a modified cover inequality is used to strengthen the IP formulation. The algorithm incorporates the column generation technique and the strong cutting plane approach into a branch-and-bound scheme. We test the proposed algorithm on some random problems. The results show that the algorithm can be used to solve the problems within reasonably small time limits.bandwidth packing, zero-one programming, polyhedral cuts, branch-and-cut

Selected Topics in Column Generation

“…Application(s) Agarwal et al (1989); Desaulniers et al (2001b); Desrochers et al (1992); Löbel (1997Löbel ( , 1998; Ribeiro and Soumis (1994) various vehicle routing problems Borndörfer et al (2003); Desaulniers et al (2001b); Desrochers and Soumis (1989) crew scheduling Desrosiers et al (1984) multiple traveling salesman problem with time windows Krumke et al (2002) real-time dispatching of automobile service units Lübbecke and Zimmermann (2003); Sol (1994) multiple pickup and delivery problem with time windows Anbil et al (1998); Crainic and Rousseau (1987) airline crew pairing Barnhart and Schneur (1996) air network design for express shipment service Erdmann et al (2001) airline schedule generation Barnhart et al (1998a); Desaulniers et al (1997); Ioachim et al (1999) fleet assignment and aircraft routing and scheduling Crama and Oerlemans (1994) job grouping for flexible manufacturing systems Eben-Chaime et al (1996) grouping and packaging of electronic circuits Park et al (1996) bandwidth packing in telecommunication networks Ribeiro et al (1989) traffic assignment in satellite communication systems Sankaran (1995) course registration at a business school Vanderbeck (1994) graph partitioning e.g., in VLSI, compiler design Vanderbeck (1994) single-machine multi-item lot-sizing Hurkens et al (1997); Valério de Carvalho (1999, 2002b; Vance (1998); Vance et al (1994); Vanderbeck (1999) bin packing and cutting stock problems Alvelos and Valério de Carvalho (2000); …”

“…The choice of the structure on which to impose decisions is a matter of algorithmic efficiency and performance. We remark that adding cutting planes in conjunction with column generation in a branch-and-bound search is usually called branch-and-price-and-cut, see e.g., Barnhart et al (1998a); Barnhart, Hane, and Vance (2000), and Park, Kang, and Park (1996).…”

Models and algorithms for network design problems

“…For technical reasons based on routing protocols, each commodity must be sent along a unique path from s(q) to t(q), see Barnhart et al (2000); Park et al (1996). Introducing the binary variable x q ij stating whether commodity q is routed through arc (i, j), the problem can be formulated as In practice, although the tra c size d q varies along time, it is not convenient to change the routing according tho these variations; x must be set once for a given time period.…”

A Column-Generation and Branch-and-Cut Approach to the Bandwidth-Packing Problem

“…Parker and Ryan do not use cutting planes. Park et al (1996) embed cutting planes based on lifted minimal covers (see Padberg 1975) within column generation to create the lp-relaxation, and incorporate these cuts into a branch-and-cut procedure. Cuts were not generated within the tree unless the branch-andprice routines failed to prove optimality within a reasonable period of time.…”