Approximating Subdense Instances of Covering Problems

We study approximability of subdense instances of various covering problems on graphs, defined as instances in which the minimum or average degree is Omega(n/psi(n)) for some function psi(n)=omega(1) of the instance size. We design new approximation algorithms as well as new polynomial time approximation schemes (PTASs) for those problems and establish first approximation hardness results for them. Interestingly, in some cases we were able to prove optimality of the underlying approximation ratios, under usual complexity-theoretic assumptions. Our results for the Vertex Cover problem depend on an improved recursive sampling method which could be of independent interest

The parallel approximability of a subclass of quadratic programming

In this paper we deal with the parallel approximability of a special class of Quadratic Programming (QP), called Smooth Positive Quadratic Programming. This subclass of QP is obtained by imposing restrictions on the coefficients of the QP instance. The Smoothness condition restricts the magnitudes of the coefficients while the positiveness requires that all the coefficients be non-negative. Interestingly, even with these restrictions several combinatorial problems can be modeled by Smooth QP. We show NC Approximation Schemes for the instances of Smooth Positive QP. This is done by reducing the instance of QP to an instance of Positive Linear Programming, finding in NC an approximate fractional solution to the obtained program, and then rounding the fractional solution to an integer approximate solution for the original problem. Then we show how to extend the result for positive instances of bounded degree to Smooth Integer Programming problems. Finally, we formulate several important combinatorial problems as Positive Quadratic Programs (or Positive Integer Programs) in packing/covering form and show that the techniques presented can be used to obtain NC Approximation Schemes for "dense" instances of such problems.Peer ReviewedPostprint (published version

Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs

We study the family of intersection graphs of low density objects in low dimensional Euclidean space. This family is quite general, and includes planar graphs. We prove that such graphs have small separators. Next, we present efficient $(1+\varepsilon)$-approximation algorithms for these graphs, for Independent Set, Set Cover, and Dominating Set problems, among others. We also prove corresponding hardness of approximation for some of these optimization problems, providing a characterization of their intractability in terms of density

A Birthday Repetition Theorem and Complexity of Approximating Dense CSPs

A $(k \times l)$-birthday repetition $\mathcal{G}^{k \times l}$ of a two-prover game $\mathcal{G}$ is a game in which the two provers are sent random sets of questions from $\mathcal{G}$ of sizes $k$ and $l$ respectively. These two sets are sampled independently uniformly among all sets of questions of those particular sizes. We prove the following birthday repetition theorem: when $\mathcal{G}$ satisfies some mild conditions, $val(\mathcal{G}^{k \times l})$ decreases exponentially in $\Omega(kl/n)$ where $n$ is the total number of questions. Our result positively resolves an open question posted by Aaronson, Impagliazzo and Moshkovitz (CCC 2014). As an application of our birthday repetition theorem, we obtain new fine-grained hardness of approximation results for dense CSPs. Specifically, we establish a tight trade-off between running time and approximation ratio for dense CSPs by showing conditional lower bounds, integrality gaps and approximation algorithms. In particular, for any sufficiently large $i$ and for every $k \geq 2$, we show the following results: - We exhibit an $O(q^{1/i})$-approximation algorithm for dense Max $k$-CSPs with alphabet size $q$ via $O_k(i)$-level of Sherali-Adams relaxation. - Through our birthday repetition theorem, we obtain an integrality gap of $q^{1/i}$ for $\tilde\Omega_k(i)$-level Lasserre relaxation for fully-dense Max $k$-CSP. - Assuming that there is a constant $\epsilon > 0$ such that Max 3SAT cannot be approximated to within $(1-\epsilon)$ of the optimal in sub-exponential time, our birthday repetition theorem implies that any algorithm that approximates fully-dense Max $k$-CSP to within a $q^{1/i}$ factor takes $(nq)^{\tilde \Omega_k(i)}$ time, almost tightly matching the algorithmic result based on Sherali-Adams relaxation.Comment: 45 page

On the practically interesting instances of MAXCUT

The complexity of a computational problem is traditionally quantified based on the hardness of its worst case. This approach has many advantages and has led to a deep and beautiful theory. However, from the practical perspective, this leaves much to be desired. In application areas, practically interesting instances very often occupy just a tiny part of an algorithm's space of instances, and the vast majority of instances are simply irrelevant. Addressing these issues is a major challenge for theoretical computer science which may make theory more relevant to the practice of computer science. Following Bilu and Linial, we apply this perspective to MAXCUT, viewed as a clustering problem. Using a variety of techniques, we investigate practically interesting instances of this problem. Specifically, we show how to solve in polynomial time distinguished, metric, expanding and dense instances of MAXCUT under mild stability assumptions. In particular, $(1+\epsilon)$-stability (which is optimal) suffices for metric and dense MAXCUT. We also show how to solve in polynomial time $\Omega(\sqrt{n})$-stable instances of MAXCUT, substantially improving the best previously known result

Approximating Dense Max 2-CSPs

In this paper, we present a polynomial-time algorithm that approximates sufficiently high-value Max 2-CSPs on sufficiently dense graphs to within $O(N^{\varepsilon})$ approximation ratio for any constant $\varepsilon > 0$. Using this algorithm, we also achieve similar results for free games, projection games on sufficiently dense random graphs, and the Densest $k$-Subgraph problem with sufficiently dense optimal solution. Note, however, that algorithms with similar guarantees to the last algorithm were in fact discovered prior to our work by Feige et al. and Suzuki and Tokuyama. In addition, our idea for the above algorithms yields the following by-product: a quasi-polynomial time approximation scheme (QPTAS) for satisfiable dense Max 2-CSPs with better running time than the known algorithms