Integer Rounding for Polymatroid and Branching Optimization Problems

Baum, Sanford; Trotter, Leslie E.

doi:10.1137/0602044

Cited by 75 publications

(43 citation statements)

References 14 publications

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“…In particular, the Integer Round-Up Property (IRUP) seems to hold for our problem. As defined in [4], an integer programming problem has the IRUP if its optimal value is given by the smallest integer greater than or equal to the optimal value of its LP relaxation.…”

Section: Introductionmentioning

confidence: 99%

Bottleneck Analysis for Routing and Call Scheduling in Multi-Hop Wireless Networks

Gomes

Pérennès

Rivano

2008

2008 IEEE Globecom Workshops

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Abstract-In this paper, we address the routing and call scheduling problem in which one has to find a minimum-length schedule of selected links in a TDMA (Time Division Multiple Access) based wireless network. As we deal with multi-hop networks, these selected links represent a routing solution (paths) providing enough capacity to achieve the routers requirements of bandwidth. We present a cross-layer formulation of the problem that computes joint routing and scheduling.We use a branch-and-price algorithm to solve optimally the problem. A column generation algorithm is used to cope with the exponential set of rounds. The branch-and-bound algorithm provides mono-routing. We run experiments on networks from the literature, with different number of gateways. Experimental results as well as theoretical insights let us conjecture that the bottleneck region analysis is enough to find the optimal solution. The Integer Round-Up Property (IRUP) seems to hold for our problem. I. INTRODUCTIONIn wireless networks, the communication channels are shared among the terminals. Thus, one of the major problems faced is the reduction of capacity due to interferences caused by simultaneous transmissions [1]. In this work, we call a round a collection of links that can be simultaneously activated in the network. We address the problem called Round Weighting Problem (RWP) [2] that consider joint routing and scheduling. We present a cross-layer formulation of the problem. We have to find a minimum-length schedule of selected links in a TDMA (Time Division Multiple Access) based wireless network. As we deal with multi-hop networks, these selected links represent a routing solution (paths) providing enough capacity to achieve the routers requirements of bandwidth. Scheduling methods such TDMA can guarantee achieving higher capacities by allowing time slots to be shared by simultaneous transmissions.A communication graph G = (V, E) represents the network topology, where the nodes are the routers and the edges are the links. Interferences between links are given as a conflict graph G c . We consider the RWP as a flow routing problem. Therefore, the flows in the edges of the problem solution represent the allocated bandwidth.We work with a special case of RW P where data are exchanged only between the routers in V r and the gateways in V g such as in Wireless Mesh networks (WMNs) or sensor networks. The input of the RW P corresponds to the communication graph G(V r ∪ V g , E), the conflict graph G c representing the edge interferences, and the network bandwidth b v

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Section: Introductionmentioning

confidence: 99%

Bottleneck Analysis for Routing and Call Scheduling in Multi-Hop Wireless Networks

Gomes

Pérennès

Rivano

2008

2008 IEEE Globecom Workshops

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“…An integer minimization problem is said to have the MIRUP if for any instance the optimal value is bounded from above by the linear programming (LP) lower bound rounded up plus 1. This definition generalizes the integer round-up property (IRUP) introduced by Baum and Trotter [1]. An integer minimization problem is said to have the IRUP if for any instance the optimal value equals the rounded-up optimal value of its LP relaxation.…”

Section: Introductionmentioning

confidence: 97%

“…gives the LP or continuous relaxation of (1). For an instance E = (m, , b, L) of the CSP let z d = z d (E) and z c = z c (E) denote the optimal values to (1) and (3), respectively.…”

Section: Notationmentioning

confidence: 99%

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New cases of the cutting stock problem having MIRUP

Altman

Avrachenkov

Filar

1998

Mathematical Methods of OR

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The modified integer round-up property (MIRUP) for a linear integer minimization problem means that the optimal value of this problem is not greater than the optimal value of the corresponding LP relaxation rounded up plus one. In earlier papers the MIRUP was shown to hold for the so-called divisible case and some other subproblems of the one-dimensional cutting stock problem. In this paper we extend the results by using special transformations, proving the existence of non-elementary patterns in the solution, and using more detailed greedy techniques.

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“…According to [1] and [16], the instance E possesses the integer round-up property (IRUP) if ∆(E) < 1, and E possesses the modified integer round-up property (MIRUP) if ∆(E) < 2. Let M * and M denote the sets of all instances of the 1CSP which have IRUP and MIRUP, respectively.…”

Section: Continuous Relaxationmentioning

confidence: 99%