Chromatic scheduling polytopes coming from the bandwidth allocation problem in point-to-multipoint radio access systems

Abstract

Point-to-Multipoint systems are a kind of radio systems supplying wireless access to voice/data communication networks. Such systems have to be run using a certain frequency spectrum, which typically causes capacity problems. Hence it is, on the one hand, necessary to reuse frequencies but, on the other hand, no interference must be caused thereby. This leads to a combinatorial optimization problem, the bandwidth allocation problem, a special case of so-called chromatic scheduling problems. Both problems are NP-hard and it is known that, for these problems, there exist no polynomial time algorithms with a fixed approximation ratio. Algorithms based on cutting planes have shown to be successful for many other combinatorial optimization problems. In order to apply such methods, knowledge on the associated polytopes is required. The present paper contributes to this issue, exploring basic properties of chromatic scheduling polytopes and several classes of facet-defining inequalities. Keywords Bandwidth allocation . Polyhedral combinatoricsPoint-to-Multipoint Radio Access Systems (PMP-Systems) are a kind of radio systems supplying wireless access to voice/data communication networks. Base stations form the access points to the backbone network; customer terminals are linked to the base stations by means of radio signals. Some specific part of the radio frequency spectrum has to be used only to J. L.

Facet-inducing inequalities for chromatic scheduling polytopes based on covering cliques

“…Let (G, d, s, g) be an instance of the bandwidth allocation problem in PMPsystems. We define the chromatic scheduling polytope P(G, d, s, g) ⊆ R 2n+m to be the convex hull of all feasible solutions (l, r, x) ∈ Z 2n+m satisfying constraints (1)- (6).…”

“…Chromatic scheduling polytopes are empty if the frequency span s is too small and pass through several stages as s increases: from a nonempty but low-dimensional stage to full-dimensionality and, finally, to a combinatorially steady state (where increasing s further does not change the structure of the faces of the polytope anymore), see [6,7] for more details.…”

“…It is worth noting that there exist instances such that s min (G, d, g) < s full (G, d, g), especially when the customer demands d are not uniform. Conversely, the full-dimensionality threshold can be bounded by s full (G, d, g) ≤ s min (G, d, g)+2d max +g =: α(G, d, g), where d max = max i∈V d i [6,7].…”

“…Chromatic scheduling polytopes admit very interesting symmetry properties [6,8]. In particular, every feasible solution y = (l, r, x) ∈ P(G, d, s, g) has an associated symmetrical solution y = (s 1 − r, s 1 − l, 1 − x) ∈ P (G, d, s, g).…”

“…The theoretical strength of the model constraints (1)- (5) was analyzed in [6,8], where it was shown that the interval bounds (2) and the antiparallelity constraints (3) and (4) do not define facets in general. A procedure to strengthen these families was presented, proving that the resulting inequalities are facet-defining for…”