arXiv subjects

Per Austrin

Publications and source records attributed to Per Austrin.

24 records · Page 2Linked to original sources

arXiv2011

Inapproximability of Treewidth, One-Shot Pebbling, and Related Layout Problems

We study the approximability of a number of graph problems: treewidth and pathwidth of graphs, one-shot black (and black-white) pebbling costs of directed acyclic graphs, and a variety of different graph layout problems such as minimum cut linear arrangement and interval graph completion. We show that, assuming the recently introduced Small Set Expansion Conjecture, all of these problems are hard to approximate within any constant factor.

cs.CC↗

arXiv2011

Inapproximability of NP-Complete Variants of Nash Equilibrium

In recent work of Hazan and Krauthgamer (SICOMP 2011), it was shown that finding an $\eps$-approximate Nash equilibrium with near-optimal value in a two-player game is as hard as finding a hidden clique of size $O(\log n)$ in the random graph $G(n,1/2)$. This raises the question of whether a similar intractability holds for approximate Nash equilibrium without such constraints. We give evidence that the constraint of near-optimal value makes the problem distinctly harder: a simple algorithm finds an optimal 1/2-approximate equilibrium, while finding strictly better than 1/2-approximate equilibria is as hard as the Hidden Clique problem. This is in contrast to the unconstrained problem where more sophisticated algorithms, achieving better approximations, are known. Unlike general Nash equilibrium, which is in PPAD, optimal (maximum value) Nash equilibrium is NP-hard. We proceed to show that optimal Nash equilibrium is just one of several known NP-hard problems related to Nash equilibrium, all of which have approximate variants which are as hard as finding a planted clique. In particular, we show this for approximate variants of the following problems: finding a Nash equilibrium with value greater than $η$ (for any $η>0$, even when the best Nash equilibrium has value $1-η$), finding a second Nash equilibrium, and finding a Nash equilibrium with small support. Finally, we consider the complexity of approximate pure Bayes Nash equilibria in two-player games. Here we show that for general Bayesian games the problem is NP-hard. For the special case where the distribution over types is uniform, we give a quasi-polynomial time algorithm matched by a hardness result based on the Hidden Clique problem.

cs.CC↗

arXiv2010

Noise Correlation Bounds for Uniform Low Degree Functions

We study correlation bounds under pairwise independent distributions for functions with no large Fourier coefficients. Functions in which all Fourier coefficients are bounded by $δ$ are called $δ$-{\em uniform}. The search for such bounds is motivated by their potential applicability to hardness of approximation, derandomization, and additive combinatorics. In our main result we show that $\E[f_1(X_1^1,...,X_1^n) ... f_k(X_k^1,...,X_k^n)]$ is close to 0 under the following assumptions: 1. The vectors $\{(X_1^j,...,X_k^j) : 1 \leq j \leq n\}$ are i.i.d, and for each $j$ the vector $(X_1^j,...,X_k^j)$ has a pairwise independent distribution. 2. The functions $f_i$ are uniform. 3. The functions $f_i$ are of low degree. We compare our result with recent results by the second author for low influence functions and to recent results in additive combinatorics using the Gowers norm. Our proofs extend some techniques from the theory of hypercontractivity to a multilinear setup.

math.PR↗

arXiv2010

A Simple Deterministic Reduction for the Gap Minimum Distance of Code Problem

We present a simple deterministic gap-preserving reduction from SAT to the Minimum Distance of Code Problem over $\F_2$. We also show how to extend the reduction to work over any finite field. Previously a randomized reduction was known due to Dumer, Micciancio, and Sudan, which was recently derandomized by Cheng and Wan. These reductions rely on highly non-trivial coding theoretic constructions whereas our reduction is elementary. As an additional feature, our reduction gives a constant factor hardness even for asymptotically good codes, i.e., having constant rate and relative distance. Previously it was not known how to achieve deterministic reductions for such codes.

cs.CC↗

arXiv2010

Improved Inapproximability For Submodular Maximization

We show that it is Unique Games-hard to approximate the maximum of a submodular function to within a factor 0.695, and that it is Unique Games-hard to approximate the maximum of a symmetric submodular function to within a factor 0.739. These results slightly improve previous results by Feige, Mirrokni and Vondrák (FOCS 2007) who showed that these problems are NP-hard to approximate to within $3/4 + ε \approx 0.750$ and $5/6 + ε \approx 0.833$, respectively.

cs.CC↗

arXiv2008

Approximation Resistant Predicates From Pairwise Independence

We study the approximability of predicates on $k$ variables from a domain $[q]$, and give a new sufficient condition for such predicates to be approximation resistant under the Unique Games Conjecture. Specifically, we show that a predicate $P$ is approximation resistant if there exists a balanced pairwise independent distribution over $[q]^k$ whose support is contained in the set of satisfying assignments to $P$.

cs.CC↗