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39
Faster SDP hierarchy solvers for local rounding algorithms
, 2012
"... Convex relaxations based on different hierarchies of linear/semidefinite programs have been used recently to devise approximation algorithms for various optimization problems. The approximation guarantee of these algorithms improves with the number of rounds r in the hierarchy, though the complexi ..."
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Cited by 4 (0 self)
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the complexity of solving (or even writing down the solution for) the r’th level program grows as nΩ(r) where n is the input size. In this work, we observe that many of these algorithms are based on local rounding procedures that only use a small part of the SDP solution (of size nO(1)2O(r) instead of nΩ(r)). We
SDP gaps and UGChardness for MaxCutGain
, 2008
"... Given a graph with maximum cut of (fractional) size c, the Goemans–Williamson semidefinite programming (SDP)based algorithm is guaranteed to find a cut of size at least.878 · c. However this guarantee becomes trivial when c is near 1/2, since making random cuts guarantees a cut of size 1/2 (i.e., ..."
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Cited by 25 (4 self)
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(ε/log(1/ε)) integrality gap for the MaxCut SDP based on Euclidean space with the Gaussian probability distribution. This shows that the SDProunding algorithm of CharikarWirth is essentially best possible. 2. We show how this SDP gap can be translated into a Long Code test with the same parameters
A Combinatorial, PrimalDual approach to Semidefinite Programs
"... Semidefinite programs (SDP) have been used in many recent approximation algorithms. We develop a general primaldual approach to solve SDPs using a generalization of the wellknown multiplicative weights update rule to symmetric matrices. For a number of problems, such as Sparsest Cut and Balanced ..."
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Cited by 94 (10 self)
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Semidefinite programs (SDP) have been used in many recent approximation algorithms. We develop a general primaldual approach to solve SDPs using a generalization of the wellknown multiplicative weights update rule to symmetric matrices. For a number of problems, such as Sparsest Cut
Rounding Semidefinite Programming Hierarchies via Global Correlation
, 2011
"... We show a new way to round vector solutions of semidefinite programming (SDP) hierarchies into integral solutions, based on a connection between these hierarchies and the spectrum of the input graph. We demonstrate the utility of our method by providing a new SDPhierarchy based algorithm for constr ..."
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Cited by 35 (4 self)
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We show a new way to round vector solutions of semidefinite programming (SDP) hierarchies into integral solutions, based on a connection between these hierarchies and the spectrum of the input graph. We demonstrate the utility of our method by providing a new SDPhierarchy based algorithm
On the Optimality of a Class of LPbased Algorithms ∗
"... In this paper we will be concerned with a class of packing and covering problems which includes Vertex Cover and Independent Set. Typically, one can write an LP relaxation and then round the solution. For instance, for Vertex Cover one can obtain a 2approximation via this approach. On the other han ..."
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Cited by 3 (0 self)
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satisfaction problems (CSPs). Unfortunately, we do not know how to extend his framework so that it applies for problems such as Vertex Cover where the constraints are strict. In this paper, we explain why the simple LPbased rounding algorithm for the Vertex Cover problem is optimal assuming the UGC
Solving the MaxCut Problem using Semidefinite Optimization in a Cutting Plane Algorithm
, 2008
"... A central graph theory problem that occurs in experimental physics, circuit layout, and computational linear algebra is the maxcut problem. The maxcut problem is to find a bipartition of the vertex set of a graph with the objective to maximize the number of edges between the two partitions. The p ..."
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iteration of our cutting plane algorithm has the following features: (a) an SDP relaxation of the maxcut problem, whose objective value provides an upper bound on the maxcut value, (b) the GoemansWilliamson heuristic to round the solution to the SDP relaxation into a feasible cut vector, that provides a
Finding good nearly balanced cuts in power law graphs
, 2004
"... In power law graphs, cut quality varies inversely with cut balance. Using some million node social graphs as a test bed, we empirically investigate this property and its implications for graph partitioning. We use six algorithms, including Metis and MQI (state of the art methods for finding bisectio ..."
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Cited by 20 (2 self)
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bisections and quotient cuts) and four relaxation/rounding methods. We find that an SDP relaxation avoids the Spectral method’s tendency to break off tiny pieces of the graph. We also find that a flowbased rounding method works better than hyperplane rounding. 1
Integrality gaps for strong SDP relaxations of unique games
"... Abstract — With the work of Khot and Vishnoi [18] as a starting point, we obtain integrality gaps for certain strong SDP relaxations of Unique Games. Specifically, we exhibit a Unique Games gap instance for the basic semidefinite program strengthened by all valid linear inequalities on the inner pro ..."
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Cited by 38 (7 self)
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products of up to exp(Ω(log log n) 1/4) vectors. For a stronger relaxation obtained from the basic semidefinite program by R rounds of Sherali–Adams liftandproject, we prove a Unique Games integrality gap for R = Ω(log log n) 1/4. By composing these SDP gaps with UGChardness reductions, the above results
Unique games with entangled provers are easy
 In Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
, 2008
"... We consider oneround games between a classical verifier and two provers who share entanglement. We show that when the constraints enforced by the verifier are ‘unique ’ constraints (i.e., permutations), the value of the game can be well approximated by a semidefinite program. Essentially the only a ..."
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Cited by 33 (9 self)
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rounding technique’, showing how to take a solution to an SDP and transform it to a strategy for entangled provers. Using our approximation by a semidefinite program we also show a parallel repetition theorem for unique entangled games. 1
Approximating Kmeanstype clustering via semidefinite programming
, 2005
"... One of the fundamental clustering problems is to assign n points into k clusters based on the minimal sumofsquares (MSSC), which is known to be NPhard. In this paper, by using matrix arguments, we first model MSSC as a socalled 01 semidefinite programming (SDP). We show that our 01 SDP model p ..."
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Cited by 13 (2 self)
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feasible solution of the original MSSC model from the approximate solution of the relaxed SDP problem. By using principal component analysis, we develop a rounding procedure to construct a feasible partitioning from a solution of the relaxed problem. In our rounding procedure, we need to solve a k
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