- Email: [email protected]
Combinatorial equivalence of Chromatic Scheduling Polytopes
Electronic Notes in Discrete Mathematics 18 (2004) 177–180 www.elsevier.com/locate/endm
Combinatorial equivalence of Chromatic Scheduling Polytopes Javier Marenco1 Departamento de Computaci´ on, FCEN, Universidad de Buenos Aires, Pabell´ on I, Ciudad Universitaria, (1428) Buenos Aires, Argentina. and Instituto de Ciencias, Universidad Nacional de General Sarmiento, J. M. Guti´ errez y J. Verdi, Los Polvorines, (1613) Buenos Aires, Argentina.
Annegret Wagler2 Konrad-Zuse-Zentrum f¨ ur Informationstechik Berlin, Takustr. 7, 14195 Berlin, Germany.
Abstract Point-to-Multipoint systems are one kind of radio systems supplying wireless access to voice/data communication networks. Capacity constraints typically force the reuse of frequencies but, on the other hand, no interference must be caused thereby. This leads to the bandwidth allocation problem, a special case of so-called chromatic scheduling problems. Both problems are NP-Hard, and there exist no polynomial time algorithms with a guaranteed quality. In order to apply cutting plane methods to this problem, the associated chromatic scheduling polytopes must be studied. These polytopes have interesting theoretical properties. In this work, we present a characterization of the extreme points of chromatic scheduling polytopes, which enables to prove combinatorial equivalence properties. Keywords: polyhedral combinatorics, combinatorial equivalence
PMP-Systems. The purpose of a Point-to-Multipoint Radio Access System is to supply wireless access to voice/data communication networks. Base stations form the access points to the backbone network and customer terminals are linked to base stations by means of radio signals. 1
Email: [email protected] . Partially supported by Ubacyt Grant X036, Conicet Grant 644/98 and Anpcyt Grant 11-09112. 2 Email: [email protected] 1571-0653/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2004.06.028
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There are two main differences between PMP-Systems and cellular phone networks. Firstly, each customer is provided a fixed antenna and is assigned to a certain sector of a base station (see Figure 1a). Secondly, the customers do not have a unique communication demand, but each customer has an individual one, hence the task is to assign frequency intervals instead of single channels (see Figure 1c). A central issue is that a link connecting a customer terminal and a base station may be subject to interference from another link that uses the same frequency. In particular, links to customers of the same sector must not use the same frequency, since they are served by the same antenna. In addition, some links of customers in different sectors may also cause interferences (see Figure 1b). Bandwidth allocation. To maintain the links in PMP-Systems, some specific part of the radio frequency spectrum has to be used. This typically causes capacity problems and, therefore, it is necessary to reuse frequencies. The bandwidth allocation problem has to be solved in order to guarantee an interference-free communication. The goal is to assign a frequency interval within the available radio frequency spectrum to each customer (see Figure 1c), taking into account the individual communication demands, possible interference, and several technical and legal restrictions. This problem can be seen as a chromatic scheduling problem [3] or as a consecutive coloring problem [4] on a weighted graph (G, d) = (V, E, d), where V = {1, . . . , n} represents the set of customers, E the set of interfering links ij ∈ V × V , and d = (d1 , . . . , dn ) the demand vector. The interval [0, s], with s ∈ Z+ , represents the available frequency spectrum. Small instances of the bandwidth allocation problem could be solved by greedy-like heuristics [1], but in order to tackle problem sizes of real world applications, algorithms have to be designed that rely on a deeper insight of the problem structure. One kind of algorithms which turned out to be successful for many other applications uses cutting plane methods. To apply
J. Marenco, A. Wagler / Electronic Notes in Discrete Mathematics 18 (2004) 177–180
179
such methods to the bandwidth allocation problem, the polytope representing the convex hull of the incidence vectors of all feasible solutions of the problem has to be studied. Chromatic scheduling polytopes. We define the chromatic scheduling polytope P (G, d, s) to be the convex hull of all feasible solutions in terms of variables li , ri ∈ Z+ for each node i ∈ V (which represent the frequency interval assigned to customer i) and ordering variables xij ∈ {0, 1} for ij ∈ E with xij = 1 if ri < lj and xij = 0 otherwise. We also define the fixed-length chromatic scheduling polytope R(G, d, s) to be the convex hull of the feasible solutions having ri − li = di for every i ∈ V . For every integer solution y, there is a feasible solution which is symmetric to y with respect to the available spectrum [0, s]. The polytopes P (G, d, s) and R(G, d, s) reflect the symmetry of the frequency assignments, and this symmetry provides theoretical tools for the search of facet-inducing inequalities [7]. Chromatic scheduling polytopes typically have a huge number of facets, even for small graphs. For instance, when G is a complete graph on n nodes, the polytope R(G, d, s) is affinely equivalent to the linear ordering polytope n [8], which has over 480 million facets for n = 8 [2]. It has been empirPLO ically observed [5] that the number of facets of R(G, d, s) remains constant for s ≥ s0 , where s0 depends on the graph structure and the node weights. This suggests that there may exist some relationship between R(G, d, s) and R(G, d, s + 1) for s ≥ s0 . In this work we explore this issue. Extreme points. Since P (G, d, s) and R(G, d, s) are not 0/1-polytopes, an integer solution is not necessarily an extreme point. We give a characterization of the extreme points of these polytopes, based on the so-called adjacency graph associated with every feasible solution. This characterization provides a simple bijection between the extreme points of P (G, d, s) and P (G, d, s + 1) resp. R(G, d, s) and R(G, d, s + 1) for s ≥ τ (G, d) + 1, where τ (G, d) denotes the minimum frequency spectrum length s such that there exists a solution for every possible ordering among the intervals. Combinatorial equivalence. Based on the previous results on the extreme points of chromatic scheduling polytopes, we prove that P (G, d, s) and P (G, d, s + 1) are affinely isomorphic (and thus combinatorially equivalent) for s ≥ 2τ (G, d) [8]. This implies that the polytopes {P (G, d, s)}s≥2τ (G,d) are pairwise affinely isomorphic. The same result holds for R(G, d, s). In the particular case that G is the union of vertex-disjoint cliques, we can prove a stronger result asserting that R(G, d, s) and R(G, d, s + 1) are affinely isomorphic for s ≥ τ (G, d) + 1. This is the best possible bound, since this equivalence does not hold for s = τ (G, d). We conjecture this to be the case
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for arbitrary interference graphs.
References ¨tter, M. Gro ¨ tschel, A. Wagler, and R. Wessa ¨ly, Frequenzplanung [1] A. Bley, A. Eisenbla f¨ ur Punkt- zu Mehrpunkt-Funksysteme, Abschlußbericht eines Projekts der BOSCH Telecom GmbH und des Konrad-Zuse-Zentrums f¨ ur Informationstechnik Berlin (1999). [2] T. Christof and G. Reinelt, Decomposition and parallelization techniques for enumerating the facets of 0/1 polytopes. Preprint, Heidelberg University (1998). [3] D. de Werra and Y. Gay, Chromatic scheduling and frequency assignment. Discrete Appl. Math. 49 (1994) 165–174. [4] D. de Werra and A. Hertz, Consecutive colorings of graphs, ZOR 32 (1988) 1–8. [5] A. Gerhardt, Polyedrische Untersuchungen von Zwei-Maschinen-Scheduling-Problemen mit Antiparallelit¨ atsbedingungen. Master thesis, Technische Universit¨ at Berlin, 1999. [6] M. Kubale, Interval vertex-coloring of a graph with forbidden colors, Discrete Math. 74 (1989) 125–136. [7] J. Marenco and A. Wagler, Chromatic scheduling polytopes coming from the bandwidth allocation problem in point-to-multipoint radio access systems. Submitted to Annals of Operations Research. [8] J. Marenco, Chromatic scheduling polytopes coming from the bandwidth allocation problem in point-to-multipoint radio access systems. PhD Thesis (in preparation), Universidad de Buenos Aires, Argentina.
Combinatorial equivalence of Chromatic Scheduling Polytopes Javier Marenco1 Departamento de Computaci´ on, FCEN, Universidad de Buenos Aires, Pabell´ on I, Ciudad Universitaria, (1428) Buenos Aires, Argentina. and Instituto de Ciencias, Universidad Nacional de General Sarmiento, J. M. Guti´ errez y J. Verdi, Los Polvorines, (1613) Buenos Aires, Argentina.
Annegret Wagler2 Konrad-Zuse-Zentrum f¨ ur Informationstechik Berlin, Takustr. 7, 14195 Berlin, Germany.
Abstract Point-to-Multipoint systems are one kind of radio systems supplying wireless access to voice/data communication networks. Capacity constraints typically force the reuse of frequencies but, on the other hand, no interference must be caused thereby. This leads to the bandwidth allocation problem, a special case of so-called chromatic scheduling problems. Both problems are NP-Hard, and there exist no polynomial time algorithms with a guaranteed quality. In order to apply cutting plane methods to this problem, the associated chromatic scheduling polytopes must be studied. These polytopes have interesting theoretical properties. In this work, we present a characterization of the extreme points of chromatic scheduling polytopes, which enables to prove combinatorial equivalence properties. Keywords: polyhedral combinatorics, combinatorial equivalence
PMP-Systems. The purpose of a Point-to-Multipoint Radio Access System is to supply wireless access to voice/data communication networks. Base stations form the access points to the backbone network and customer terminals are linked to base stations by means of radio signals. 1
Email: [email protected] . Partially supported by Ubacyt Grant X036, Conicet Grant 644/98 and Anpcyt Grant 11-09112. 2 Email: [email protected] 1571-0653/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2004.06.028
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J. Marenco, A. Wagler / Electronic Notes in Discrete Mathematics 18 (2004) 177–180
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Fig. 1. Bandwidth allocation in Point-to-Multipoint radio access systems.
There are two main differences between PMP-Systems and cellular phone networks. Firstly, each customer is provided a fixed antenna and is assigned to a certain sector of a base station (see Figure 1a). Secondly, the customers do not have a unique communication demand, but each customer has an individual one, hence the task is to assign frequency intervals instead of single channels (see Figure 1c). A central issue is that a link connecting a customer terminal and a base station may be subject to interference from another link that uses the same frequency. In particular, links to customers of the same sector must not use the same frequency, since they are served by the same antenna. In addition, some links of customers in different sectors may also cause interferences (see Figure 1b). Bandwidth allocation. To maintain the links in PMP-Systems, some specific part of the radio frequency spectrum has to be used. This typically causes capacity problems and, therefore, it is necessary to reuse frequencies. The bandwidth allocation problem has to be solved in order to guarantee an interference-free communication. The goal is to assign a frequency interval within the available radio frequency spectrum to each customer (see Figure 1c), taking into account the individual communication demands, possible interference, and several technical and legal restrictions. This problem can be seen as a chromatic scheduling problem [3] or as a consecutive coloring problem [4] on a weighted graph (G, d) = (V, E, d), where V = {1, . . . , n} represents the set of customers, E the set of interfering links ij ∈ V × V , and d = (d1 , . . . , dn ) the demand vector. The interval [0, s], with s ∈ Z+ , represents the available frequency spectrum. Small instances of the bandwidth allocation problem could be solved by greedy-like heuristics [1], but in order to tackle problem sizes of real world applications, algorithms have to be designed that rely on a deeper insight of the problem structure. One kind of algorithms which turned out to be successful for many other applications uses cutting plane methods. To apply
J. Marenco, A. Wagler / Electronic Notes in Discrete Mathematics 18 (2004) 177–180
179
such methods to the bandwidth allocation problem, the polytope representing the convex hull of the incidence vectors of all feasible solutions of the problem has to be studied. Chromatic scheduling polytopes. We define the chromatic scheduling polytope P (G, d, s) to be the convex hull of all feasible solutions in terms of variables li , ri ∈ Z+ for each node i ∈ V (which represent the frequency interval assigned to customer i) and ordering variables xij ∈ {0, 1} for ij ∈ E with xij = 1 if ri < lj and xij = 0 otherwise. We also define the fixed-length chromatic scheduling polytope R(G, d, s) to be the convex hull of the feasible solutions having ri − li = di for every i ∈ V . For every integer solution y, there is a feasible solution which is symmetric to y with respect to the available spectrum [0, s]. The polytopes P (G, d, s) and R(G, d, s) reflect the symmetry of the frequency assignments, and this symmetry provides theoretical tools for the search of facet-inducing inequalities [7]. Chromatic scheduling polytopes typically have a huge number of facets, even for small graphs. For instance, when G is a complete graph on n nodes, the polytope R(G, d, s) is affinely equivalent to the linear ordering polytope n [8], which has over 480 million facets for n = 8 [2]. It has been empirPLO ically observed [5] that the number of facets of R(G, d, s) remains constant for s ≥ s0 , where s0 depends on the graph structure and the node weights. This suggests that there may exist some relationship between R(G, d, s) and R(G, d, s + 1) for s ≥ s0 . In this work we explore this issue. Extreme points. Since P (G, d, s) and R(G, d, s) are not 0/1-polytopes, an integer solution is not necessarily an extreme point. We give a characterization of the extreme points of these polytopes, based on the so-called adjacency graph associated with every feasible solution. This characterization provides a simple bijection between the extreme points of P (G, d, s) and P (G, d, s + 1) resp. R(G, d, s) and R(G, d, s + 1) for s ≥ τ (G, d) + 1, where τ (G, d) denotes the minimum frequency spectrum length s such that there exists a solution for every possible ordering among the intervals. Combinatorial equivalence. Based on the previous results on the extreme points of chromatic scheduling polytopes, we prove that P (G, d, s) and P (G, d, s + 1) are affinely isomorphic (and thus combinatorially equivalent) for s ≥ 2τ (G, d) [8]. This implies that the polytopes {P (G, d, s)}s≥2τ (G,d) are pairwise affinely isomorphic. The same result holds for R(G, d, s). In the particular case that G is the union of vertex-disjoint cliques, we can prove a stronger result asserting that R(G, d, s) and R(G, d, s + 1) are affinely isomorphic for s ≥ τ (G, d) + 1. This is the best possible bound, since this equivalence does not hold for s = τ (G, d). We conjecture this to be the case
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J. Marenco, A. Wagler / Electronic Notes in Discrete Mathematics 18 (2004) 177–180
for arbitrary interference graphs.
References ¨tter, M. Gro ¨ tschel, A. Wagler, and R. Wessa ¨ly, Frequenzplanung [1] A. Bley, A. Eisenbla f¨ ur Punkt- zu Mehrpunkt-Funksysteme, Abschlußbericht eines Projekts der BOSCH Telecom GmbH und des Konrad-Zuse-Zentrums f¨ ur Informationstechnik Berlin (1999). [2] T. Christof and G. Reinelt, Decomposition and parallelization techniques for enumerating the facets of 0/1 polytopes. Preprint, Heidelberg University (1998). [3] D. de Werra and Y. Gay, Chromatic scheduling and frequency assignment. Discrete Appl. Math. 49 (1994) 165–174. [4] D. de Werra and A. Hertz, Consecutive colorings of graphs, ZOR 32 (1988) 1–8. [5] A. Gerhardt, Polyedrische Untersuchungen von Zwei-Maschinen-Scheduling-Problemen mit Antiparallelit¨ atsbedingungen. Master thesis, Technische Universit¨ at Berlin, 1999. [6] M. Kubale, Interval vertex-coloring of a graph with forbidden colors, Discrete Math. 74 (1989) 125–136. [7] J. Marenco and A. Wagler, Chromatic scheduling polytopes coming from the bandwidth allocation problem in point-to-multipoint radio access systems. Submitted to Annals of Operations Research. [8] J. Marenco, Chromatic scheduling polytopes coming from the bandwidth allocation problem in point-to-multipoint radio access systems. PhD Thesis (in preparation), Universidad de Buenos Aires, Argentina.