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Polynomial integrality gaps for strong SDP relaxations of Densest ksubgraph
"... The Densest ksubgraph problem (i.e. find a size k subgraph with maximum number of edges), is one of the notorious problems in approximation algorithms. There is a significant gap between known upper and lower bounds for Densest ksubgraph: the current best algorithm gives an ≈ O(n 1/4) approximatio ..."
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The Densest ksubgraph problem (i.e. find a size k subgraph with maximum number of edges), is one of the notorious problems in approximation algorithms. There is a significant gap between known upper and lower bounds for Densest ksubgraph: the current best algorithm gives an ≈ O(n 1/4) approximation, while even showing a small constant factor hardness requires significantly stronger assumptions than P ̸ = NP. In addition to interest in designing better algorithms, a number of recent results have exploited the conjectured hardness of Densest ksubgraph and its variants. Thus, understanding the approximability of Densest ksubgraph is an important challenge. In this work, we give evidence for the hardness of approximating Densest ksubgraph within polynomial factors. Specifically, we expose the limitations of strong semidefinite programs from SDP hierarchies in solving Densest ksubgraph. Our results include: • A lower bound of Ω ( n 1/4 / log 3 n) on the integrality gap for Ω(log n / log log n) rounds of the SheraliAdams relaxation for Densest ksubgraph. This also holds for the relaxation obtained from SheraliAdams with an added SDP constraint. Our gap instances are in
On integrality ratios for asymmetric tsp in the sheraliadams hierarchy
 In Automata, Languages, and Programming
, 2013
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How to Sell Hyperedges: The Hypermatching Assignment Problem
, 2013
"... We are given a set of clients with budget constraints and a set of indivisible items. Each client is willing to buy one or more bundles of (at most) k items each (bundles can be seen as hyperedges in a khypergraph). If client i gets a bundle e, she pays bi,e and yields a net profit wi,e. The Hyperm ..."
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Cited by 3 (2 self)
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We are given a set of clients with budget constraints and a set of indivisible items. Each client is willing to buy one or more bundles of (at most) k items each (bundles can be seen as hyperedges in a khypergraph). If client i gets a bundle e, she pays bi,e and yields a net profit wi,e. The Hypermatching Assignment Problem (HAP) is to assign a set of pairwise disjoint bundles to clients so as to maximize the total profit while respecting the budgets. This problem has various applications in production planning and budgetconstrained auctions and generalizes wellstudied problems in combinatorial optimization: for example the weighted (unweighted) khypergraph matching problem is the special case of HAP with one client having unbounded budget and general (unit) profits; the Generalized Assignment Problem (GAP) is the special case of HAP with k = 1. Let ε> 0 denote an arbitrarily small constant. In this paper we obtain the following main results: • We give a randomized (k + 1 + ) approximation algorithm for HAP, which is based on rounding the 1round Lasserre strengthening of a novel LP. This is one of a few approximation results based on Lasserre hierarchies and our approach might be of independent interest. We remark that for weighted khypergraph matching no LP nor SDP relaxation is known to have integrality gap better than k − 1 + 1/k for general k [Chan and Lau, SODA’10]. • For the relevant special case that one wants to maximize the total revenue (i.e., bi,e = wi,e), we present a local search based (k + O( k))/2 approximation algorithm for k = O(1). This almost matches the best known (k + 1 + )/2 approximation ratio by Berman [SWAT’00] for
The Lasserre hierarchy in Approximation algorithms  Lecture Notes for the MAPSP 2013 Tutorial
, 2013
"... The Lasserre hierarchy is a systematic procedure to strengthen a relaxation for an optimization problem by adding additional variables and SDP constraints. In the last years this hierarchy moved into the focus of researchers in approximation algorithms as the obtain relaxations have provably nice pr ..."
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The Lasserre hierarchy is a systematic procedure to strengthen a relaxation for an optimization problem by adding additional variables and SDP constraints. In the last years this hierarchy moved into the focus of researchers in approximation algorithms as the obtain relaxations have provably nice properties. In particular on the tth level, the relaxation can be solved in time n O(t) and every constraint that one could derive from looking just at t variables is automatically satisfied. Additionally, it provides a vector embedding of events so that probabilities are expressable as inner products. The goal of these lecture notes is to give short but rigorous proofs of all key properties of the Lasserre hierarchy. In the second part we will demonstrate how the Lasserre SDP can be applied to (mostly NPhard) optimization problems such as KNAPSACK, MATCHING, MAXCUT (in general and in dense graphs), 3COLORING and
Graph Pricing Problem on Bounded Treewidth, Bounded Genus and kPartite Graphs
, 2013
"... Consider the following problem. A seller has infinite copies of n products represented by nodes in a graph. There are m consumers, each has a budget and wants to buy two products. Consumers are represented by weighted edges. Given the prices of products, each consumer will buy both products she wa ..."
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Consider the following problem. A seller has infinite copies of n products represented by nodes in a graph. There are m consumers, each has a budget and wants to buy two products. Consumers are represented by weighted edges. Given the prices of products, each consumer will buy both products she wants, at the given price, if she can afford to. Our objective is to help the seller price the products to maximize her profit. This problem is called graph vertex pricing (GVP) problem and has resisted several recent attempts despite its current simple solution. This motivates the study of this problem on special classes of graphs. In this paper, we study this problem on a large class of graphs such as graphs with bounded treewidth, bounded genus and kpartite graphs. We show that there exists an FPTAS for GVP on graphs with bounded treewidth. This result is also extended to an FPTAS for the more general singleminded pricing problem. On bounded genus graphs we present a PTAS and show that GVP is NPhard even on planar graphs.
A Lagrangian relaxation view of linear and semidefinite hierarchies
, 2012
"... We consider the generalpolynomial optimization problem P: f ∗ = min{f(x) : x ∈ K} where K is a compact basic semialgebraic set. We first show that the standard Lagrangian relaxation yields a lower bound as close as desired to the global optimum f ∗ , provided that it is applied to a problem ˜ P equ ..."
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We consider the generalpolynomial optimization problem P: f ∗ = min{f(x) : x ∈ K} where K is a compact basic semialgebraic set. We first show that the standard Lagrangian relaxation yields a lower bound as close as desired to the global optimum f ∗ , provided that it is applied to a problem ˜ P equivalent to P, in which sufficiently many redundant constraints (products of the initial ones) are added to the initial description of P. Next we show that the standard hierarchy of LPrelaxations of P (in the spirit of SheraliAdams ’ RLT) can be interpreted as a brute force simplification of the above Lagrangian relaxation in which a nonnegative polynomial (with coefficients to be determined) is replaced with a constant polynomial equal to zero. Inspired by this interpretation, we provide a systematic improvement of the LPhierarchy by doing a much less brutal simplification which results into a parametrized hierarchy of semidefinite programs (and not linear programs any more). For each semidefinite program in the parametrized hierarchy, the semidefinite constraint has a fixed size O(n k), independently of the rank in the hierarchy, in contrast with the standard hierarchy of semidefinite relaxations. The parameter k is to be decided by the user. When applied to a non trivial class of convex problems, the first relaxation of the parametrized hierarchy is exact, in contrast with the LPhierarchy where convergence cannot be finite. When applied to 0/1 programs it is at least as good as the first one in the hierarchy of semidefinite relaxations. However obstructions to exactness still exist and are briefly analyzed. Finally, the standard semidefinite hierarchy can also be viewed as a simplification of an extended Lagrangianrelaxation, but different in spirit as sums of squares (and not scalars) multipliers are allowed.
Statistical Limits of Convex Relaxations
"... Many high dimensional sparse learning problems are formulated as nonconvex optimization. A popular approach to solve these nonconvex optimization problems is through convex relaxations such as linear and semidefinite programming. In this paper, we study the statistical limits of convex relaxations. ..."
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Many high dimensional sparse learning problems are formulated as nonconvex optimization. A popular approach to solve these nonconvex optimization problems is through convex relaxations such as linear and semidefinite programming. In this paper, we study the statistical limits of convex relaxations. Particularly, we consider two problems: Mean estimation for sparse principal submatrix and edge probability estimation for stochastic block model. We exploit the sumofsquares relaxation hierarchy to sharply characterize the limits of a broad class of convex relaxations. Our result shows statistical optimality needs to be compromised for achieving computational tractability using convex relaxations. Compared with existing results on computational lower bounds for statistical problems, which consider general polynomialtime algorithms and rely on computational hardness hypotheses on problems like planted clique detection, our theory focuses on a broad class of convex relaxations and does not rely on unproven hypotheses. 1
Linear Programming Hierarchies Suffice for Directed Steiner Tree
"... Abstract. We demonstrate that ` rounds of the SheraliAdams hierarchy and 2 ` rounds of the LovászSchrijver hierarchy suffice to reduce the integrality gap of a natural LP relaxation for Directed Steiner Tree in `layered graphs from Ω( k) to O( ` · log k) where k is the number of terminals. Th ..."
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Abstract. We demonstrate that ` rounds of the SheraliAdams hierarchy and 2 ` rounds of the LovászSchrijver hierarchy suffice to reduce the integrality gap of a natural LP relaxation for Directed Steiner Tree in `layered graphs from Ω( k) to O( ` · log k) where k is the number of terminals. This is an improvement over Rothvoss ’ result that 2 ` rounds of the considerably stronger Lasserre SDP hierarchy reduce the integrality gap of a similar formulation to O( ` · log k). We also observe that Directed Steiner Tree instances with 3 layers of edges have only an O(log k) integrality gap in the standard LP relaxation, complementing the known fact that the gap can be as large as Ω( k) in graphs with 4 layers. 1