We study online social networks in which relationships can be either positive (indicating relations such as friendship) or negative (indicating relations such as opposition or antagonism). Such a mix of positive and negative links arise in a variety of online settings; we study datasets from Epinions, Slashdot and Wikipedia. We find that the signs of links in the underlying social networks can be predicted with high accuracy, using models that generalize across this diverse range of sites. These models provide insight into some of the fundamental principles that drive the formation of signed links in networks, shedding light on theories of balance and status from social psychology; they also suggest social computing applications by which the attitude of one user toward another can be estimated from evidence provided by their relationships with other members of the surrounding social network.

Predicting Positive and Negative Links in Online Social Networks

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Predicting positive and negative links in online social networks

We give new evidence that quantum computers -- moreover, rudimentary quantum computers built entirely out of linear-optical elements -- cannot be efficiently simulated by classical computers. In particular, we define a model of computation in which identical photons are generated, sent through a linear-optical network, then nonadaptively measured to count the number of photons in each mode. This model is not known or believed to be universal for quantum computation, and indeed, we discuss the prospects for realizing the model using current technology. On the other hand, we prove that the model is able to solve sampling problems and search problems that are classically intractable under plausible assumptions. Our first result says that, if there exists a polynomial-time classical algorithm that samples from the same probability distribution as a linear-optical network, then P#P=BPPNP, and hence the polynomial hierarchy collapses to the third level. Unfortunately, this result assumes an extremely accurate simulation.Our main result suggests that even an approximate or noisy classical simulation would already imply a collapse of the polynomial hierarchy. For this, we need two unproven conjectures: the Permanent-of-Gaussians Conjecture, which says that it is #P-hard to approximate the permanent of a matrix A of independent N(0,1) Gaussian entries, with high probability over A; and the Permanent Anti-Concentration Conjecture, which says that |Per(A)|>=√(n!)poly(n) with high probability over A. We present evidence for these conjectures, both of which seem interesting even apart from our application.This paper does not assume knowledge of quantum optics. Indeed, part of its goal is to develop the beautiful theory of noninteracting bosons underlying our model, and its connection to the permanent function, in a self-contained way accessible to theoretical computer scientists.

/pdf/the-computational-complexity-of-linear-optics-28xlewjz7o.pdf

The computational complexity of linear optics

We give new evidence that quantum computers -- moreover, rudimentary quantum computers built entirely out of linear-optical elements -- cannot be efficiently simulated by classical computers In particular, we define a model of computation in which identical photons are generated, sent through a linear-optical network, then nonadaptively measured to count the number of photons in each mode This model is not known or believed to be universal for quantum computation, and indeed, we discuss the prospects for realizing the model using current technology On the other hand, we prove that the model is able to solve sampling problems and search problems that are classically intractable under plausible assumptions Our first result says that, if there exists a polynomial-time classical algorithm that samples from the same probability distribution as a linear-optical network, then P^#P=BPP^NP, and hence the polynomial hierarchy collapses to the third level Unfortunately, this result assumes an extremely accurate simulation Our main result suggests that even an approximate or noisy classical simulation would already imply a collapse of the polynomial hierarchy For this, we need two unproven conjectures: the "Permanent-of-Gaussians Conjecture", which says that it is #P-hard to approximate the permanent of a matrix A of independent N(0,1) Gaussian entries, with high probability over A; and the "Permanent Anti-Concentration Conjecture", which says that |Per(A)|>=sqrt(n!)/poly(n) with high probability over A We present evidence for these conjectures, both of which seem interesting even apart from our application This paper does not assume knowledge of quantum optics Indeed, part of its goal is to develop the beautiful theory of noninteracting bosons underlying our model, and its connection to the permanent function, in a self-contained way accessible to theoretical computer scientists

The Computational Complexity of Linear Optics.

Subexponential time approximation algorithms are presented for the Unique Games and Small-Set Expansion problems. Specifically, for some absolute constant c, the following two algorithms are presented.(1) An exp(kne)-time algorithm that, given as input a k-alphabet unique game on n variables that has an assignment satisfying 1-ec fraction of its constraints, outputs an assignment satisfying 1-e fraction of the constraints.(2) An exp(ne/δ)-time algorithm that, given as input an n-vertex regular graph that has a set S of δn vertices with edge expansion at most ec, outputs a set S' of at most δ n vertices with edge expansion at most e.subexponential algorithm is also presented with improved approximation to Max Cut, Sparsest Cut, and Vertex Cover on some interesting subclasses of instances. These instances are graphs with low threshold rank, an interesting new graph parameter highlighted by this work.Khot's Unique Games Conjecture (UGC) states that it is NP-hard to achieve approximation guarantees such as ours for Unique Games. While the results here stop short of refuting the UGC, they do suggest that Unique Games are significantly easier than NP-hard problems such as Max 3-Sat, Max 3-Lin, Label Cover, and more, which are believed not to have a subexponential algorithm achieving a nontrivial approximation ratio.Of special interest in these algorithms is a new notion of graph decomposition that may have other applications. Namely, it is shown for every e >0 and every regular n-vertex graph G, by changing at most δ fraction of G's edges, one can break G into disjoint parts so that the stochastic adjacency matrix of the induced graph on each part has at most ne eigenvalues larger than 1-η, where η depends polynomially on e. The subexponential algorithm combines this decomposition with previous algorithms for Unique Games on graphs with few large eigenvalues [Kolla and Tulsiani 2007; Kolla 2010].

https://www.boazbarak.org/Papers/ssesubexp.pdf

Subexponential Algorithms for Unique Games and Related Problems

We prove that approximating the max. acyclic subgraph problem within a factor better than 1/2 is unique games hard. Specifically, for every constant epsiv > 0 the following holds: given a directed graph G that has an acyclic subgraph consisting of a fraction (1-epsiv) of its edges, if one can efficiently find an acyclic subgraph of G with more than (1/2 + epsiv) of its edges, then the UGC is false. 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 existence of a rho-approximation algorithmfor rho > 1/2 has been a basic open problem for a while. Our result is the first tight inapproximability result for an ordering problem. The starting point of our reduction isa directed acyclic subgraph (DAG) in which every cut isnearly-balanced in the sense that the number of forward and backward edges crossing the cut are nearly equal; such DAGs were constructed by Charikar et al. Using this, we are able to study max. acyclic subgraph, which is a constraint satisfaction problem (CSP) over an unbounded domain, by relating it to a proxy CSP over a bounded domain. The latter is then amenable to powerful techniques based on the invariance principle. Our results also give a super-constant factor inapproximability result for the feedback arc set problem. Using our reductions, we also obtain SDP integrality gapsfor both the problems.

Beating the Random Ordering is Hard: Inapproximability of Maximum Acyclic Subgraph

The connection between integrality gaps and computational hardness of discrete optimization problems is an intriguing question. In recent years, this connection has prominently figured in several tight UGC-based hardness results. We show in this paper a direct way of turning integrality gaps into hardness results for several fundamental classification problems. Specifically, we convert linear programming integrality gaps for the Multiway Cut, 0-Extension, and and Metric Labeling problems into UGC-based hardness results. Qualitatively, our result suggests that if the unique games conjecture is true then a linear relaxation of the latter problems studied in several papers (so-called earthmover linear program) yields the best possible approximation. Taking this a step further, we also obtain integrality gaps for a semi-definite programming relaxation matching the integrality gaps of the earthmover linear program. Prior to this work, there was an intriguing possibility of obtaining better approximation factors for labeling problems via semi-definite programming.

Sdp gaps and ugc hardness for multiway cut, 0-extension, and metric labeling

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.

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.

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

A permutation constraint satisfaction problem (permCSP) of arity k is specified by a subset Lambda of permutations on $\{1,2,\dots,k\}$. An instance of such a permCSP consists of a set of variables $V$ and a collection of constraints each of which is an ordered $k$-tuple of $V$. The objective is to find a global ordering $\sigma$ of the variables that maximizes the number of constraint tuples whose ordering (under $\sigma$) follows a permutation in $\Lambda$. This is just the natural extension of constraint satisfaction problems over finite domains (such as Boolean CSPs) to the world of ordering problems. The simplest permCSP corresponds to the case when $\Lambda$ consists of the identity permutation on two variables. This is just the Maximum Acyclic Subgraph (\mas) problem. It was recently shown that the \mas\ problem is Unique-Games hard to approximate within a factor better than the trivial $1/2$ achieved by a random ordering [GMR08]. Building on this work, in this paper we show that for *every* permCSP of arity $3$, beating the random ordering is Unique-Games hard. The result is in fact stronger: we show that for every $\Lambda \subseteq \Pi \subseteq S_3$, given an instance of permCSP$(\Lambda)$ that is almost-satisfiable, it is hard to find an ordering that satisfies more than $\frac{|\Pi|}{6} +\eps$ of the constraints even under the relaxed constraint $\Pi$ (for arbitrary $\eps ≫ 0$). A special case of our result is that the *Betweenness* problem is hard to approximate beyond a factor $1/3$. Interestingly, for *satisfiable* instances of Betweenness, a factor $1/2$ approximation algorithm is known. Thus, every permutation CSP of arity up to $3$ resists approximation beyond the trivial random ordering threshold. In contrast, for Boolean CSPs, there are both approximation resistant and non-trivially approximable CSPs of arity $3$.

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Rajsekar Manokaran

Papers

Beating the Random Ordering is Hard: Inapproximability of Maximum Acyclic Subgraph

Sdp gaps and ugc hardness for multiway cut, 0-extension, and metric labeling

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

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

Every Permutation CSP of arity 3 is Approximation Resistant