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Lasserre hierarchy, higher eigenvalues, and approximation schemes for graph partitioning and quadratic integer programming with PSD objectives
 In Proceedings of 52nd Annual Symposium on Foundations of Computer Science (FOCS
, 2011
"... We present an approximation scheme for optimizing certain Quadratic Integer Programming problems with positive semidefinite objective functions and global linear constraints. This framework includes well known graph problems such as Minimum graph bisection, Edge expansion, Uniform sparsest cut, and ..."
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We present an approximation scheme for optimizing certain Quadratic Integer Programming problems with positive semidefinite objective functions and global linear constraints. This framework includes well known graph problems such as Minimum graph bisection, Edge expansion, Uniform sparsest cut, and Small Set expansion, as well as the Unique Games problem. These problems are notorious for the existence of huge gaps between the known algorithmic results and NPhardness results. Our algorithm is based on rounding semidefinite programs from the Lasserre hierarchy, and the analysis uses bounds for lowrank approximations of a matrix in Frobenius norm using columns of the matrix. For all the above graph problems, we give an algorithm running in time nO(r/ε 2) with approximation ratio 1+εmin{1,λr} , where λr is the r’th smallest eigenvalue of the normalized graph Laplacian L. In the case of graph bisection and small set expansion, the number of vertices in the cut is within lowerorder terms of the stipulated bound. Our results imply (1 + O(ε)) factor approximation in time nO(r
Minmax graph partitioning and small set expansion
, 2011
"... We study graph partitioning problems from a minmax perspective, in which an input graph on n vertices should be partitioned into k parts, and the objective is to minimize the maximum number of edges leaving a single part. The two main versions we consider are: (i) the k parts need to be of equal s ..."
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We study graph partitioning problems from a minmax perspective, in which an input graph on n vertices should be partitioned into k parts, and the objective is to minimize the maximum number of edges leaving a single part. The two main versions we consider are: (i) the k parts need to be of equal size, and (ii) the parts must separate a set of k given terminals. We consider a common generalization of these two problems, and design for it an O ( √ log n log k)approximation algorithm. This improves over an O(log 2 n) approximation for the second version due to Svitkina and Tardos [22], and roughly O(k log n) approximation for the first version that follows from other previous work. We also give an improved O(1)approximation algorithm for graphs that exclude any fixed minor. Our algorithm uses a new procedure for solving the SmallSet Expansion problem. In this problem, we are given a graph G and the goal is to find a nonempty set S ⊆ V of size S  ≤ ρn with minimum edgeexpansion. We give an O ( √ log n log (1/ρ)) bicriteria approximation algorithm for the general case of SmallSet Expansion, and O(1) approximation algorithm for graphs that exclude any fixed minor.
SumofSquares Proofs and the Quest toward Optimal Algorithms
"... Abstract. In order to obtain the bestknown guarantees, algorithms are traditionally tailored to the particular problem we want to solve. Two recent developments, the Unique Games Conjecture (UGC) and the SumofSquares (SOS) method, surprisingly suggest that this tailoring is not necessary and that ..."
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Abstract. In order to obtain the bestknown guarantees, algorithms are traditionally tailored to the particular problem we want to solve. Two recent developments, the Unique Games Conjecture (UGC) and the SumofSquares (SOS) method, surprisingly suggest that this tailoring is not necessary and that a single efficient algorithm could achieve best possible guarantees for a wide range of different problems. The Unique Games Conjecture (UGC) is a tantalizing conjecture in computational complexity, which, if true, will shed light on the complexity of a great many problems. In particular this conjecture predicts that a single concrete algorithm provides optimal guarantees among all efficient algorithms for a large class of computational problems. The SumofSquares (SOS) method is a general approach for solving systems of polynomial constraints. This approach is studied in several scientific disciplines, including real algebraic geometry, proof complexity, control theory, and mathematical programming, and has found applications in fields as diverse as quantum information theory, formal verification, game theory and many others. We survey some connections that were recently uncovered between the Unique Games Conjecture and the SumofSquares method. In particular, we discuss new tools to rigorously bound the running time of the SOS method for obtaining approximate solutions to hard optimization problems, and how these tools give the potential for the sumofsquares method to provide new guarantees for many problems of interest, and possibly to even refute the UGC.
Finding Small Sparse Cuts by Random Walk
"... Abstract. We study the problem of finding a small sparse cut in an undirected graph. Given an undirected graph G = (V,E) and a parameter k ≤ E, the small sparsest cut problem is to find a set S ⊆ V with minimum conductance among all sets with volume at most k. Using ideas developed in local graph ..."
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Abstract. We study the problem of finding a small sparse cut in an undirected graph. Given an undirected graph G = (V,E) and a parameter k ≤ E, the small sparsest cut problem is to find a set S ⊆ V with minimum conductance among all sets with volume at most k. Using ideas developed in local graph partitioning algorithms, we obtain the following bicriteria approximation algorithms for the small sparsest cut problem: – If there is a set U ⊆ V with conductance φ and vol(U) ≤ k, then there isapolynomial timealgorithm tofindaset S with conductance O ( √ φ/ǫ) and vol(S) ≤ k 1+ǫ for any ǫ> 1/k. – If there is a set U ⊆ V with conductance φ and vol(U) ≤ k, then there isapolynomial timealgorithm tofindaset S with conductance O ( √ φlogk/ǫ) and vol(S) ≤ (1+ǫ)k for any ǫ> 2logk/k. These algorithms can be implemented locally using truncated random walk, with running time almost linear to k. 1
Constant Factor Lasserre Integrality Gaps for Graph Partitioning Problems
"... Partitioning the vertices of a graph into two roughly equal parts while minimizing the number of edges crossing the cut is a fundamental problem (called Balanced Separator) that arises in many settings. For this problem, and variants such as the Uniform Sparsest Cut problem where the goal is to mini ..."
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Partitioning the vertices of a graph into two roughly equal parts while minimizing the number of edges crossing the cut is a fundamental problem (called Balanced Separator) that arises in many settings. For this problem, and variants such as the Uniform Sparsest Cut problem where the goal is to minimize the fraction of pairs on opposite sides of the cut that are connected by an edge, there are large gaps between the known approximation algorithms and nonapproximability results. While no constant factor approximation algorithms are known, even APXhardness is not known either. In this work we prove that for balanced separator and uniform sparsest cut, semidefinite programs from the Lasserre hierarchy (which are the most powerful relaxations studied in the literature) have an integrality gap bounded away from 1, even for Ω(n) levels of the hierarchy. This complements recent algorithmic results in [GS11] which used the Lasserre hierarchy to give an approximation scheme for these problems (with runtime depending on the spectrum of the graph). Along the way, we make an observation that simplifies the task of lifting “polynomial constraints ” (such as the global balance constraint in balanced separator) to higher levels of the Lasserre hierarchy. We also obtain a similar result for Max Cut and prove that even linear number of levels of the Lasserre hierarchy have an integrality gap exceeding 18/17 − o(1), though in this case there are known NPhardness results with this gap.
NonUniform Graph Partitioning
"... We consider the problem of NONUNIFORM GRAPH PARTITIONING, where the input is an edgeweighted undirected graph G = (V, E) and k capacities n1,..., nk, and the goal is to find a partition {S1, S2,..., Sk} of V satisfying Sj  ≤ nj for all 1 ≤ j ≤ k, that minimizes the total weight of edges crossi ..."
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We consider the problem of NONUNIFORM GRAPH PARTITIONING, where the input is an edgeweighted undirected graph G = (V, E) and k capacities n1,..., nk, and the goal is to find a partition {S1, S2,..., Sk} of V satisfying Sj  ≤ nj for all 1 ≤ j ≤ k, that minimizes the total weight of edges crossing between different parts. This natural graph partitioning problem arises in practical scenarios, and generalizes wellstudied balanced partitioning problems such as MINIMUM BISECTION, MINIMUM BALANCED CUT, and MINIMUM kPARTITIONING. Unlike these problems, NONUNIFORM GRAPH PARTITIONING seems to be resistant to many of the known partitioning techniques, such as spreading metrics, recursive partitioning, and Räcke’s tree decomposition, because k can be a function of n and the capacities could be of different magnitudes. We present a bicriteria approximation algorithm for NONUNIFORM GRAPH PARTITIONING that approximates the objective within O(log n) factor while deviating from the required capacities by at most a constant factor. Our approach is to apply stoppingtime based concentration results to a simple randomized rounding of a configuration LP. These concentration bounds are needed as the commonly used techniques of bounded differences and bounded conditioned variances do not suffice. 1
MINMAX GRAPH PARTITIONING AND SMALL SET EXPANSION∗
"... Abstract. We study graph partitioning problems from a minmax perspective, in which an input graph on n vertices should be partitioned into k parts, and the objective is to minimize the maximum number of edges leaving a single part. The two main versions we consider are where the k parts need to be ..."
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Abstract. We study graph partitioning problems from a minmax perspective, in which an input graph on n vertices should be partitioned into k parts, and the objective is to minimize the maximum number of edges leaving a single part. The two main versions we consider are where the k parts need to be of equal size, and where they must separate a set of k given terminals. We consider a common generalization of these two problems, and design for it an O( logn log k) approximation algorithm. This improves over an O(log2 n) approximation for the second version due to Svitkina and Tardos [Minmax multiway cut, in APPROXRANDOM, 2004, Springer, Berlin, 2004], and roughly O(k logn) approximation for the first version that follows from other previous work. We also give an O(1) approximation algorithm for graphs that exclude any fixed minor. Our algorithm uses a new procedure for solving the smallset expansion problem. In this problem, we are given a graph G and the goal is to find a nonempty set S ⊆ V of size S  ≤ ρn with minimum edge expansion. We give an O( logn log (1/ρ)) bicriteria approximation algorithm for smallset expansion in general graphs, and an improved factor of O(1) for graphs that exclude any fixed minor.
Truth vs. Proof in Computational Complexity
, 2012
"... Theoretical Computer Science is blessed (or cursed?) with many open problems. For some of these questions, such as the P vs NP problem, it seems like it could be decades or more before they reach resolution. So, if we have no proof either way, what do we assume about the answer? We could remain agno ..."
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Theoretical Computer Science is blessed (or cursed?) with many open problems. For some of these questions, such as the P vs NP problem, it seems like it could be decades or more before they reach resolution. So, if we have no proof either way, what do we assume about the answer? We could remain agnostic, saying that we simply don’t know, but there can be such a thing as too much skepticism in science. For example, Scott Aaronson once claimed [Aar10] that in other sciences P 6 = NP would by now have been declared a law of nature. I tend to agree. After all, we are trying to uncover the truth about the nature of computation and this quest won’t go any faster if we insist on discarding all evidence that is not in the form of mathematical proofs from first principles. But what other methods can we use to get evidence for questions in computational complexity? After all, it seems completely hopeless to experimentally verify even a nonasymptotic statement such as “There is no circuit of size 2100 that can solve 3SAT on 10, 000 variables”. There is in some sense only one tool us scientists can use to predict the answer to open questions, and this is Occam’s Razor. That is, if we want