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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
EverywhereSparse Spanners via Dense Subgraphs
, 2012
"... The significant progressg in constructing graph spanners that are sparse (small number of edges) or light (low total weight) has skipped spanners that are everywheresparse (small maximum degree). This disparity is in line with other network design problems, where the maximumdegree objective has b ..."
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Cited by 4 (1 self)
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The significant progressg in constructing graph spanners that are sparse (small number of edges) or light (low total weight) has skipped spanners that are everywheresparse (small maximum degree). This disparity is in line with other network design problems, where the maximumdegree objective has been a notorious technical challenge. Our main result is for the LOWEST DEGREE 2SPANNER (LD2S) problem, where the goal is to compute a 2spanner of an input graph so as to minimize the maximum degree. We design a polynomialtime algorithm achieving approximation factor Õ(∆3−2 √ 2) ≈ Õ( ∆ 0.172), where ∆ is the maximum degree of the input graph. The previous Õ( ∆ 1/4)–approximation was proved nearly two decades ago by Kortsarz and Peleg [SODA 1994, SICOMP 1998]. Our main conceptual contribution is to establish a formal connection between LD2S and a variant of the DENSEST kSUBGRAPH (DkS) problem. Specifically, we design for both problems strong relaxations based on the SheraliAdams linear programming (LP) hierarchy, and show that “faithful ” randomized rounding of the DkSvariant can be used to round LD2S solutions. Our notion of faithfulness intuitively means that all vertices and edges are chosen with probability proportional to their LP value, but the precise formulation is more subtle. Unfortunately, the best algorithms known for DkS use the LovászSchrijver LP hierarchy in a nonfaithful way [Bhaskara, Charikar, Chlamtac, Feige, and Vijayaraghavan, STOC 2010]. Our main technical contribution is to overcome this shortcoming, while still matching the gap that arises in random graphs by planting a subgraph with same logdensity.
Narrow proofs may be maximally long.
 In Proceedings of the 29th Annual IEEE Conference on Computational Complexity (CCC ’14),
, 2014
"... Abstract We prove that there are 3CNF formulas over n variables that can be refuted in resolution in width w but require resolution proofs of size n Ω(w) . This shows that the simple counting argument that any formula refutable in width w must have a proof in size n is essentially tight. Moreover, ..."
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Abstract We prove that there are 3CNF formulas over n variables that can be refuted in resolution in width w but require resolution proofs of size n Ω(w) . This shows that the simple counting argument that any formula refutable in width w must have a proof in size n is essentially tight. Moreover, our lower bound generalizes to polynomial calculus resolution (PCR) and SheraliAdams, implying that the corresponding size upper bounds in terms of degree and rank are tight as well. The lower bound does not extend all the way to Lasserre, however, since we show that there the formulas we study have proofs of constant rank and size polynomial in both n and w.
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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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
On integrality ratios for asymmetric TSP in the SheraliAdams hierarchy
, 2014
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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.
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.