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21
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.
Mining compression sequential patterns
"... Compression based pattern mining has been successfully applied to many data mining tasks. We propose an approach based on the minimum description length principle to extract sequential patterns that compress a database of sequences well. We show that mining compressing patterns is NPHard and belong ..."
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Compression based pattern mining has been successfully applied to many data mining tasks. We propose an approach based on the minimum description length principle to extract sequential patterns that compress a database of sequences well. We show that mining compressing patterns is NPHard and belongs to the class of inapproximable problems. We propose two heuristic algorithms to mining compressing patterns. The first uses a twophase approach similar to Krimp for itemset data. To overcome performance with the required candidate generation we propose GoKrimp, an effective greedy algorithm that directly mines compressing patterns. We conduct an empirical study on six reallife datasets to compare the proposed algorithms by run time, compressibility, and classification accuracy using the patterns found as features for SVM classifiers.
A note on a maximum ksubset intersection problem
 Inf. Process. Lett
, 2012
"... Consider the following problem which we call Maximum kSubset Intersection (MSI): Given a collection C = {S1,..., Sm} of m subsets over a finite set of elements E = {e1,..., en}, and a positive integer k, the objective is to select exactly k subsets Sj1,..., Sjk whose intersection size Sj1 ∩... ∩ ..."
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Consider the following problem which we call Maximum kSubset Intersection (MSI): Given a collection C = {S1,..., Sm} of m subsets over a finite set of elements E = {e1,..., en}, and a positive integer k, the objective is to select exactly k subsets Sj1,..., Sjk whose intersection size Sj1 ∩... ∩ Sjk  is maximum. In [2], Clifford and Popa studied a related problem and left as an open problem the status of the MSI problem. In this paper we show that this problem is hard to approximate.
Sparsest cut on bounded treewidth graphs: Algorithms and hardness results
 In 45th Annual ACM Symposium on Symposium on Theory of Computing
, 2013
"... We give a 2approximation algorithm for NonUniform Sparsest Cut that runs in time nO(k), where k is the treewidth of the graph. This improves on the previous 22 kapproximation in time poly(n)2O(k) due to Chlamtác ̌ et al. [CKR10]. To complement this algorithm, we show the following hardness resul ..."
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We give a 2approximation algorithm for NonUniform Sparsest Cut that runs in time nO(k), where k is the treewidth of the graph. This improves on the previous 22 kapproximation in time poly(n)2O(k) due to Chlamtác ̌ et al. [CKR10]. To complement this algorithm, we show the following hardness results: If the NonUniform Sparsest Cut problem has a ρapproximation for seriesparallel graphs (where ρ ≥ 1), then the MaxCut problem has an algorithm with approximation factor arbitrarily close to 1/ρ. Hence, even for such restricted graphs (which have treewidth 2), the Sparsest Cut problem is NPhard to approximate better than 17/16 − ε for ε> 0; assuming the Unique Games Conjecture the hardness becomes 1/αGW − ε. For graphs with large (but constant) treewidth, we show a hardness result of 2 − ε assuming the Unique Games Conjecture. Our algorithm rounds a linear program based on (a subset of) the SheraliAdams lift of the standard Sparsest Cut LP. We show that even for treewidth2 graphs, the LP has an integrality gap close to 2 even after polynomially many rounds of SheraliAdams. Hence our approach cannot be improved even on such restricted graphs without using a stronger relaxation. 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.
The parameterized complexity of kBiclique
 In Proc. 26th SODA
, 2014
"... Given a graph G and a parameter k, the kBiclique problem asks whether G contains a complete bipartite subgraph Kk,k. This is one of the most easily stated problems on graphs whose parameterized complexity has been long unknown. We prove that kBiclique is W[1]hard by giving an fptreduction from k ..."
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Given a graph G and a parameter k, the kBiclique problem asks whether G contains a complete bipartite subgraph Kk,k. This is one of the most easily stated problems on graphs whose parameterized complexity has been long unknown. We prove that kBiclique is W[1]hard by giving an fptreduction from kClique to kBiclique, thus solving this longstanding open problem. Our reduction uses a class of bipartite graphs with a certain threshold property, which might be of some independent interest. More precisely, for positive integers n, s and t, we consider a bipartite graph G = (A ∪ ̇ B,E) such that A can be partitioned into A = V1 ∪ ̇ V2 ∪̇, · · · , ∪ ̇ Vn and for every s distinct indices i1, · · · , is, there exist vi1 ∈ Vi1, · · · , vis ∈ Vis such that vi1, · · · , vis have at least t+ 1 common neighbors in B; on the other hand, every s+1 distinct vertices in A have at most t common neighbors in B. We prove that given such threshold bipartite graphs, we can construct an fptreduction from kClique to kBiclique. Using the Paleytype graphs and Weil’s character sum theorem, we show that for t = (s+1)! and n large enough, such threshold bipartite graphs can be computed in polynomial time. One corollary of our reduction is that there is no f(k) ·no(k) time algorithm to decide whether a graph contains a subgraph isomorphic to Kk!,k! unless the Exponential Time Hypothesis (ETH) fails. We also provide a probabilistic construction with better parameters t = Θ(s2), which indicates that kBiclique has no f(k) · no( k)time algorithm unless 3SAT with m clauses can be solved in 2o(m)time with high probability. Besides the lower bound for exact computation of kBiclique, our result also implies a dichotomy classification of the parameterized complexity of cardinality constraint satisfaction problems and the inapproximability of the maximum kintersection problem.
Hardness of Approximation for Crossing Number
 Discrete Comput. Geom
"... We show that, if P6=NP, there is a constant c0> 1 such that there is no c0approximation algorithm for the crossing number, even when restricted to 3regular graphs. 1 ..."
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We show that, if P6=NP, there is a constant c0> 1 such that there is no c0approximation algorithm for the crossing number, even when restricted to 3regular graphs. 1
A Tight Approximation Bound for the Stable Marriage Problem with Restricted Ties
"... The problem of finding a maximum cardinality stable matching in the presence of ties and unacceptable partners, called MAX SMTI, is a wellstudied NPhard problem. The MAX SMTI is NPhard even for highly restricted instances where (i) ties appear only in women’s preference lists and (ii) each tie ap ..."
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The problem of finding a maximum cardinality stable matching in the presence of ties and unacceptable partners, called MAX SMTI, is a wellstudied NPhard problem. The MAX SMTI is NPhard even for highly restricted instances where (i) ties appear only in women’s preference lists and (ii) each tie appears at the end of each woman’s preference list. The current best lower bounds on the approximation ratio for this variant are 1.1052 unless P=NP and 1.25 under the unique games conjecture, while the current best upper bound is 1.4616. In this paper, we improve the upper bound to 1.25, which matches the lower bound under the unique games conjecture. Note that this is the first special case of the MAX SMTI where the tight approximation bound is obtained. The improved ratio is achieved via a new analysis technique, which avoids the complicated casebycase analysis used in earlier studies. As a byproduct of our analysis, we show that the integrality gap of natural IP and LP formulations for this variant is 1.25. We also show that the unrestricted MAX SMTI cannot be approximated with less than 1.5 unless the approximation ratio of a certain special case of the minimum maximal matching problem can be improved.
Graph Partitioning and Semidefinite Programming Hierarchies
, 2012
"... contained in this document are those of the author and should not be interpreted as representing ..."
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contained in this document are those of the author and should not be interpreted as representing