Avner Magen

Rank bounds and integrality gaps for cutting planes procedures

We present a new method for proving rank lower bounds for Cutting Planes (CP) and several procedures based on lifting due to Lovasz and Schrijver (LS), when viewed as proof systems for unsatisfiability. We apply this method to obtain the following new results: first, we prove near-optimal rank bounds for Cutting Planes and Lovasz-Schrijver proofs for several prominent unsatisfiable CNF examples, including random kCNF formulas and the Tseitin graph formulas. It follows from these lower bounds that a linear number of rounds of CP or LS procedures when applied to relaxations of integer linear programs is not sufficient for reducing the integrality gap. Secondly, we give unsatisfiable examples that have constant rank CP and LS proofs but that require linear rank resolution proofs. Thirdly, we give examples where the CP rank is O(log n) but the LS rank is linear. Finally, we address the question of size versus rank: we show that, for both proof systems, rank does not accurately reflect proof size. Specifically, there are examples with polynomial-size CP/LS proofs, but requiring linear rank.

Dimensionality Reductions That Preserve Volumes and Distance to Affine Spaces, and Their Algorithmic Applications

Let X be a subset of n points of the Euclidean space, and let 0 < ɛ < 1. A classical result of Johnson and Lindenstrauss [JL84] states that there is a projection of X onto a subspace of dimension O(ɛ-2 log n), with distortion ≤ 1 + ɛ. Here we show a natural extension of the above result, to a stronger preservation of the geometry of finite spaces. By a k-fold increase of the number of dimensions used compared to [JL84], a good preservation of volumes and of distances between points and affine spaces is achieved. Specifically, we show it is possible to embed a subset of size n of the Euclidean space into a O(ɛ-2 klogn)-dimensional Euclidean space, so that no set of size s ≤ k changes its volume by more than (1+ɛ)s-1. Moreover, distances of points from affine hulls of sets of at most k-1 points in the space do not change by more than a factor of 1 + ɛ. A consequence of the above with k = 3 is that angles can be preserved using asymptotically the same number of dimensions as the one used in [JL84]. Our method can be applied to many problems with high-dimensional nature such as Projective Clustering and Approximated Nearest Affine Neighbor Search. In particular, it shows a first poly-logarithmic query time approximation algorithm to the latter. We also show a structural application that for volume respecting embedding in the sense introduced by Feige [Fei00], the host space need not generally be of dimensionality greater than polylogarithmic in the size of the graph.

Integrality gaps of 2 - o(1) for Vertex Cover SDPs in the Lovész-Schrijver Hierarchy

Linear and semidefinite programming are highly successful approaches for obtaining good approximations for NP-hard optimization problems. For example, breakthrough approximation algorithms for Max Cut and Sparsest Cut use semidefinite programming. Perhaps the most prominent NP-hard problem whose exact approximation factor is still unresolved is Vertex Cover. PCP-based techniques of Dinur and Safra [7] show that it is not possible to achieve a factor better than 1.36; on the other hand no known algorithm does better than the factor of 2 achieved by the simple greedy algorithm. Furthermore, there is a widespread belief that SDP technicptes are the most promising methods available for improving upon this factor of 2. Following a line of study initiated by Arora et al. [3], our aim is to show that a large family of LP and SDP based algorithms fail to produce an approximation for Vertex Cover better than 2. Lovasz and Schrijver [21] introduced the systems LS and LS+for systematically tightening LP and SDP relaxations, respectively, over many rounds. These systems naturally capture large classes of LP and SDP relaxations; indeed, LS+ captures the celebrated SDP-based algorithms for Max Cur and Sparsest Cur mentioned above. We rule out polynomial-time 2 - Omega(lfloor) approximations for Vertex Cover using LS+. In particular, we prove an integrality gap of 2 - o(lfloor)for Vertex Cover SDPs obtained by tightening the standard LP relaxation with Omega(radiclog n/ log log n) rounds of LS+. While tight integrality gaps were known for Vertex Cover in the weaker LS system [23 ], previous results did not rule out a2 - Omega(1) approximation after even two rounds of LS+.

SDP Gaps from Pairwise Independence

We consider the problem of approximating fixed-predicate constraint satisfaction problems (MAX k-CSPq(P)), where the variables take values from (q) =f0; 1;:::; q 1g, and each constraint is on k variables and is defined by a fixed k-ary predicate P. Familiar problems like MAX 3-SAT and MAX-CUT belong to this category. Austrin and Mossel recently identified a general class of predicates P for which MAX k-CSPq(P) is hard to approximate. They study predicates P : (q) k !f0; 1g such that the set of assignments accepted by P contains the support of a balanced pairwise independent distribution over the domain of the inputs. We refer to such predicates as promising. Austrin and Mossel show that for any promising predicate P, the problem MAX k-CSPq(P) is Unique-Games-hard to approximate better than the trivial approximation obtained by a random assignment. We give an unconditional analogue of this result in a restricted model of computation. We consider the hierarchy of semidefinite relaxations of MAX k-CSPq(P) obtained by augmenting the canonical semidefinite relaxation with the Sherali-Adams hierarchy. We show that for any promising predicate P, the integrality gap remains the same as the approximation ratio achieved by a random assignment, even after W(n) levels of this hierarchy.

Toward a model for backtracking and dynamic programming

We consider a model (BT) for backtracking algorithms. Our model generalizes both the priority model of Borodin, Nielson and Rackoff, as well as a simple dynamic programming model due to Woeginger, and hence spans a wide spectrum of algorithms. After witnessing the strength of the model, we then show its limitations by providing lower bounds for algorithms in this model for several classical problems such as interval scheduling, knapsack and satisfiability.

Approximating the Weight of the Euclidean Minimum Spanning Tree in Sublinear Time

We consider the problem of computing the weight of a Euclidean minimum spanning tree for a set of n points in

Sublinear Geometric Algorithms

\mathbb R^d

Sublinear geometric algorithms

. We focus on the setting where the input point set is supported by certain basic (and commonly used) geometric data structures that can provide efficient access to the input in a structured way. We present an algorithm that estimates with high probability the weight of a Euclidean minimum spanning tree of a set of points to within

Dimensionality Reductions in ℓ 2 that Preserve Volumes and Distance to Affine Spaces

1 + \eps

How well can primal-dual and local-ratio algorithms perform?

using only