Showing papers by "Johan Håstad published in 2011"

Beating the Random Ordering is Hard: Every ordering CSP is approximation resistant.

Abstract: We prove that, assuming the Unique Games conjecture (UGC), every problem in the class of ordering constraint satisfaction problems (OCSPs) where each constraint has constant arity is approximation resistant In other words, we show that if $\rho$ is the expected fraction of constraints satisfied by a random ordering, then obtaining a $\rho'$ approximation for any $\rho'>\rho$ is UG-hard For the simplest OCSP, the Maximum Acyclic Subgraph (MAS) problem, this implies that obtaining a $\rho$-approximation for any constant $\rho>1/2$ is UG-hard Specifically, for every constant $\varepsilon>0$ the following holds: given a directed graph $G$ that has an acyclic subgraph consisting of a fraction $(1-\varepsilon)$ of its edges, it is UG-hard to find one with more than $(1/2+\varepsilon)$ of its edges Note that it is trivial to find an acyclic subgraph with $1/2$ the edges by taking either the forward or backward edges in an arbitrary ordering of the vertices of $G$ The MAS problem has been well studied, and beating the random ordering for MAS has been a basic open problem An OCSP of arity $k$ is specified by a subset $\Pi\subseteq S_k$ of permutations on $\{1,2,\dots,k\}$ An instance of such an OCSP is a set $V$ and a collection of constraints, each of which is an ordered $k$-tuple of $V$ The objective is to find a global linear ordering of $V$ while maximizing the number of constraints ordered as in $\Pi$ A random ordering of $V$ is expected to satisfy a $\rho=\frac{|\Pi|}{k!}$ fraction We show that, for any fixed $k$, it is hard to obtain a $\rho'$-approximation for $\Pi$-OCSP for any $\rho'>\rho$ The result is in fact stronger: we show that for every $\Lambda\subseteq\Pi\subseteq S_k$, and an arbitrarily small $\varepsilon$, it is hard to distinguish instances where a $(1-\varepsilon)$ fraction of the constraints can be ordered according to $\Lambda$ from instances where at most a $(\rho+\varepsilon)$ fraction can be ordered as in $\Pi$ A special case of our result is that the Betweenness problem is hard to approximate beyond a factor $1/3$ The results naturally generalize to OCSPs which assign a payoff to the different permutations Finally, our results imply (unconditionally) that a simple semidefinite relaxation for MAS does not suffice to obtain a better approximation

Beating the Random Ordering Is Hard: Every Ordering CSP Is Approximation Resistant

Abstract: We prove that, assuming the Unique Games conjecture (UGC), every problem in the class of ordering constraint satisfaction problems (OCSPs) where each constraint has constant arity is approximation resistant. In other words, we show that if $\rho$ is the expected fraction of constraints satisfied by a random ordering, then obtaining a $\rho'$ approximation for any $\rho'>\rho$ is UG-hard. For the simplest OCSP, the Maximum Acyclic Subgraph (MAS) problem, this implies that obtaining a $\rho$-approximation for any constant $\rho>1/2$ is UG-hard. Specifically, for every constant $\varepsilon>0$ the following holds: given a directed graph $G$ that has an acyclic subgraph consisting of a fraction $(1-\varepsilon)$ of its edges, it is UG-hard to find one with more than $(1/2+\varepsilon)$ of its edges. Note that it is trivial to find an acyclic subgraph with $1/2$ the edges by taking either the forward or backward edges in an arbitrary ordering of the vertices of $G$. The MAS problem has been well studied, and beating the random ordering for MAS has been a basic open problem. An OCSP of arity $k$ is specified by a subset $\Pi\subseteq S_k$ of permutations on $\{1,2,\dots,k\}$. An instance of such an OCSP is a set $V$ and a collection of constraints, each of which is an ordered $k$-tuple of $V$. The objective is to find a global linear ordering of $V$ while maximizing the number of constraints ordered as in $\Pi$. A random ordering of $V$ is expected to satisfy a $\rho=\frac{|\Pi|}{k!}$ fraction. We show that, for any fixed $k$, it is hard to obtain a $\rho'$-approximation for $\Pi$-OCSP for any $\rho'>\rho$. The result is in fact stronger: we show that for every $\Lambda\subseteq\Pi\subseteq S_k$, and an arbitrarily small $\varepsilon$, it is hard to distinguish instances where a $(1-\varepsilon)$ fraction of the constraints can be ordered according to $\Lambda$ from instances where at most a $(\rho+\varepsilon)$ fraction can be ordered as in $\Pi$. A special case of our result is that the Betweenness problem is hard to approximate beyond a factor $1/3$. The results naturally generalize to OCSPs which assign a payoff to the different permutations. Finally, our results imply (unconditionally) that a simple semidefinite relaxation for MAS does not suffice to obtain a better approximation.

Randomly Supported Independence and Resistance

Abstract: We prove that for any positive integers $q$ and $k$ there is a constant $c_{q,k}$ such that a uniformly random set of $c_{q,k}n^k\log n$ vectors in $[q]^n$ with high probability supports a balanced $k$-wise independent distribution. In the case of $k\leq2$ a more elaborate argument gives the stronger bound, $c_{q,k}n^k$. Using a recent result by Austrin and Mossel, this shows that a predicate on $t$ bits, chosen at random among predicates accepting $c_{q,2}t^2$ input vectors, is, assuming the unique games conjecture, likely to be approximation resistant. These results are close to tight: we show that there are other constants, $c_{q,k}'$, such that a randomly selected set of cardinality $c_{q,k}'n^k$ points is unlikely to support a balanced $k$-wise independent distribution and, for some $c>0$, a random predicate accepting $ct^2/\log t$ input vectors is nontrivially approximable with high probability. In a different application of the result of Austrin and Mossel we prove that, again assuming the unique games conjecture, any predicate on $t$ Boolean inputs accepting at least $(32/33)\cdot2^t$ inputs is approximation resistant. The results extend from balanced distributions to arbitrary product distributions.

Making the long code shorter, with applications to the Unique Games Conjecture

Abstract: The long code is a central tool in hardness of approximation, especially in questions related to the unique games conjecture. We construct a new code that is exponentially more efficient, but can still be used in many of these applications. Using the new code we obtain exponential improvements over several known results, including the following: 1. For any eps > 0, we show the existence of an n vertex graph G where every set of o(n) vertices has expansion 1 - eps, but G's adjacency matrix has more than exp(log^delta n) eigenvalues larger than 1 - eps, where delta depends only on eps. This answers an open question of Arora, Barak and Steurer (FOCS 2010) who asked whether one can improve over the noise graph on the Boolean hypercube that has poly(log n) such eigenvalues. 2. A gadget that reduces unique games instances with linear constraints modulo K into instances with alphabet k with a blowup of K^polylog(K), improving over the previously known gadget with blowup of 2^K. 3. An n variable integrality gap for Unique Games that that survives exp(poly(log log n)) rounds of the SDP + Sherali Adams hierarchy, improving on the previously known bound of poly(log log n). We show a connection between the local testability of linear codes and small set expansion in certain related Cayley graphs, and use this connection to derandomize the noise graph on the Boolean hypercube.

Satisfying degree-d equations over GF[2]n

Abstract: We study the problem where we are given a system of polynomial equations defined by multivariate polynomials over GF[2] of fixed constant degree d > 1 and the aim is to satisfy the maximal number of equations. A random assignment approximates this problem within a factor 2-d and we prove that for any e > 0, it is NP-hard to obtain a ratio 2-d + e. When considering instances that are perfectly satisfiable we give a probabilistic polynomial time algorithm that, with high probability, satisfies a fraction 21-d - 21-2d and we prove that it is NP-hard to do better by an arbitrarily small constant. The hardness results are proved in the form of inapproximability results of Max-CSPs where the predicate in question has the desired form and we give some immediate results on approximation resistance of some predicates.

Making the long code shorter, with applications to the Unique Games Conjecture

Abstract: The long code is a central tool in hardness of approximation, especially in questions related to the unique games conjecture. We construct a new code that is exponentially more efficient, but can still be used in many of these applications. Using the new code we obtain exponential improvements over several known results, including the following: 1. For any e > 0, we show the existence of an n vertex graph G where every set of o(n) vertices has expansion 1 − e, but G’s adjacency matrix has more than exp(log n) eigenvalues larger than 1 − e, where δ depends only on e. This answers an open question of Arora, Barak and Steurer (FOCS 2010) who asked whether one can improve over the noise graph on the Boolean hypercube that has poly(log n) such eigenvalues. 2. A gadget that reduces unique games instances with linear constraints modulo K into instances with alphabet k with a blowup of Kpolylog(K), improving over the previously known gadget with blowup of 2Ω(K). 3. An n variable integrality gap for Unique Games that survives exp(poly(log log n)) rounds of the SDP + Sherali Adams hierarchy, improving on the previously known bound of poly(log log n). We show a connection between the local testability of linear codes and small set expansion in certain related Cayley graphs, and use this connection to derandomize the noise graph on the Boolean hypercube. ∗Microsoft Research New England, Cambridge MA. †Microsoft Research-Silicon Valley. ‡Royal Insitute of Technology, Stockholm, Sweden. §IAS, Princeton. Work done in part while visiting Microsoft Research, Silicon Valley. ¶Georgia Institute of Technology, Atlanta, GA. ‖Microsoft Research New England, Cambridge MA.