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Optimal inapproximability results for MAXCUT and other 2variable CSPs?
, 2005
"... In this paper we show a reduction from the Unique Games problem to the problem of approximating MAXCUT to within a factor of ffGW + ffl, for all ffl> 0; here ffGW ss.878567 denotes the approximation ratio achieved by the GoemansWilliamson algorithm [25]. This implies that if the Unique Games ..."
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Cited by 238 (32 self)
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In this paper we show a reduction from the Unique Games problem to the problem of approximating MAXCUT to within a factor of ffGW + ffl, for all ffl> 0; here ffGW ss.878567 denotes the approximation ratio achieved by the GoemansWilliamson algorithm [25]. This implies that if the Unique Games
Fixing Variables in Semidefinite Relaxations
 SIAM J. MATRIX ANAL. APPL
, 1996
"... The standard technique of reduced cost fixing from linear programming is not trivially extensible to semidefinite relaxations as the corresponding Lagrange multipliers are usually not available. We propose a general technique for computing reasonable Lagrange multipliers to constraints which are not ..."
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Cited by 27 (2 self)
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The standard technique of reduced cost fixing from linear programming is not trivially extensible to semidefinite relaxations as the corresponding Lagrange multipliers are usually not available. We propose a general technique for computing reasonable Lagrange multipliers to constraints which are not part of the problem description. Its specialization to the semidefinite f\Gamma1; 1g relaxation of quadratic 01 programming yields an efficient routine for fixing variables. The routine offers the possibility to exploit problem structure. We extend the traditional bijective map between f0; 1g and f\Gamma1; 1g formulations to the constraints such that the dual variables remain the same and structural properties are preserved. In consequence the fixing routine can efficiently be applied to optimal solutions of the semidefinite f0; 1g relaxation of constrained quadratic 01 programming, as well. We provide numerical results showing the efficacy of the approach.
Semidefinite Programming
, 1999
"... Due to its many applications in control theory, robust optimization, combinatorial optimization and eigenvalue optimization, semidefinite programming had been in wide spread use even before the development of efficient algorithms brought it into the realm of tractability. Today it is one of the basi ..."
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Cited by 9 (0 self)
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Due to its many applications in control theory, robust optimization, combinatorial optimization and eigenvalue optimization, semidefinite programming had been in wide spread use even before the development of efficient algorithms brought it into the realm of tractability. Today it is one of the basic modeling and optimization tools along with linear and quadratic programming. Our survey is an introduction to semidefinite programming, its duality and complexity theory, its applications and algorithms.
Design of Phase Codes for Radar Performance Optimization with a Similarity Constraint
 IEEE Transactions on Signal Processing
, 2009
"... Abstract—This paper deals with the design of coded waveforms which optimize radar performances in the presence of colored Gaussian disturbance. We focus on the class of phase coded pulse trains and determine the radar code which approximately maximizes the detection performance under a similarity c ..."
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Cited by 7 (4 self)
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Abstract—This paper deals with the design of coded waveforms which optimize radar performances in the presence of colored Gaussian disturbance. We focus on the class of phase coded pulse trains and determine the radar code which approximately maximizes the detection performance under a similarity constraint with a prefixed radar code. This is tantamount to forcing a similarity between the ambiguity functions of the devised waveform and of the pulse train encoded with the prefixed sequence. We consider the cases of both continuous and finite phase alphabet, and formulate the code design in terms of a nonconvex, NPhard quadratic optimization problem. In order to approximate the optimal solutions, we propose techniques (with polynomial computational complexity) based on the method of semidefinite program (SDP) relaxation and randomization. Moreover, we also derive approximation bounds yielding a “measure of goodness ” of the devised algorithms. At the analysis stage, we assess the performance of the new encoding techniques both in terms of detection performance and ambiguity function, under different choices for the similarity parameter. We also show that the new algorithms achieve an accurate approximation of the optimal solution with a modest number of randomizations. Index Terms—Radar signal processing, nonconvex quadratic optimization, semidefinite program relaxation, randomization. I.
Cutting Plane Algorithms for Semidefinite Relaxations
 TOPICS IN SEMIDEFINITE AND INTERIORPOINT METHODS. FIELDS INSTITUTE COMMUNICATIONS SERIES VOL. 18, AMS
, 1997
"... We investigate the potential and limits of interior point based cutting ..."
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Cited by 3 (0 self)
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We investigate the potential and limits of interior point based cutting
SDPbased approach for channel assignment in multiradio wireless networks
, 2007
"... We consider the following channel assignment problem in multiradio multichannel wireless networks: Given a wireless network where k orthogonal channels are available and each node has multiple wireless interfaces, assign a channel to each link so that the total number of conflicts is minimized. We ..."
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Cited by 1 (0 self)
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We consider the following channel assignment problem in multiradio multichannel wireless networks: Given a wireless network where k orthogonal channels are available and each node has multiple wireless interfaces, assign a channel to each link so that the total number of conflicts is minimized. We present an integer semidefinite programming formulation for the problem and show that it is equivalent to an optimal channel assignment. By relaxing integrality constraints, we can find a lowerbound on the optimal channel assignment. We develop several channel assignment algorithms based on the solution to the SDP relaxation. Our results from numerical evaluations and packetlevel simulations show that our SDPbased rounding algorithms outperform other simple heuristics (up to 200 % improvement in throughput). In particular, our schemes achieve even larger performance improvement when nodes
Soft edge coloring
 In APPROX
, 2007
"... We consider the following channel assignment problem arising in wireless networks. We are given a graph G = (V, E), and the number of wireless cards Cv for all v, which limit the number of colors that edges incident to v can use. We also have the total number of channels CG available in the network. ..."
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Cited by 1 (1 self)
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We consider the following channel assignment problem arising in wireless networks. We are given a graph G = (V, E), and the number of wireless cards Cv for all v, which limit the number of colors that edges incident to v can use. We also have the total number of channels CG available in the network. For a pair of edges incident to a vertex, they are said to be conflicting if the colors assigned to them are the same. Our goal is to color edges (assign channels) so that the number of conflicts is minimized. We first consider the homogeneous network where Cv = k and CG ≥ Cv for all nodes v. The problem is NPhard by a reduction from EDGE COLORING and we present two combinatorial algorithms for this case. The first algorithm is a distributed greedy method, which gives a solution with at most (1 − 1 k)E  more conflicts than the optimal solution. We also present an algorithm yielding at most V  more conflicts than the optimal solution. The algorithm generalizes Vizing’s algorithm in the sense that it gives the same result as Vizing’s algorithm when k = ∆ + 1. Moreover, we show that this approximation result is best possible unless P = NP. For the case where Cv = 1 or k, we show that the problem is NPhard even when Cv = 1 or 2, and CG = 2, and present two algorithms. The first algorithm is completely combinatorial and produces a solution with at most (2 − 1 1)OP T + (1 −)E  conflicts. We also develop an SDPbased algorithm, producing a solution with at most 1.122OP T + 0.122E  conflicts for k = 2, ln k and (2 − Θ ())OP T + (1 − Θ ())E  conflicts in general.