36 research outputs found

### Approximation of non-boolean 2CSP

We develop a polynomial time $\Omega\left ( \frac 1R \log R \right)$
approximate algorithm for Max 2CSP-$R$, the problem where we are given a
collection of constraints, each involving two variables, where each variable
ranges over a set of size $R$, and we want to find an assignment to the
variables that maximizes the number of satisfied constraints. Assuming the
Unique Games Conjecture, this is the best possible approximation up to constant
factors.
Previously, a $1/R$-approximate algorithm was known, based on linear
programming. Our algorithm is based on semidefinite programming (SDP) and on a
novel rounding technique. The SDP that we use has an almost-matching
integrality gap

### How to Play Unique Games against a Semi-Random Adversary

In this paper, we study the average case complexity of the Unique Games
problem. We propose a natural semi-random model, in which a unique game
instance is generated in several steps. First an adversary selects a completely
satisfiable instance of Unique Games, then she chooses an epsilon-fraction of
all edges, and finally replaces ("corrupts") the constraints corresponding to
these edges with new constraints. If all steps are adversarial, the adversary
can obtain any (1-epsilon) satisfiable instance, so then the problem is as hard
as in the worst case. In our semi-random model, one of the steps is random, and
all other steps are adversarial. We show that known algorithms for unique games
(in particular, all algorithms that use the standard SDP relaxation) fail to
solve semi-random instances of Unique Games.
We present an algorithm that with high probability finds a solution
satisfying a (1-delta) fraction of all constraints in semi-random instances (we
require that the average degree of the graph is Omega(log k). To this end, we
consider a new non-standard SDP program for Unique Games, which is not a
relaxation for the problem, and show how to analyze it. We present a new
rounding scheme that simultaneously uses SDP and LP solutions, which we believe
is of independent interest.
Our result holds only for epsilon less than some absolute constant. We prove
that if epsilon > 1/2, then the problem is hard in one of the models, the
result assumes the 2-to-2 conjecture.
Finally, we study semi-random instances of Unique Games that are at most
(1-epsilon) satisfiable. We present an algorithm that with high probability,
distinguishes between the case when the instance is a semi-random instance and
the case when the instance is an (arbitrary) (1-delta) satisfiable instance if
epsilon > c delta

### On the Expansion of Group-Based Lifts

A $k$-lift of an $n$-vertex base graph $G$ is a graph $H$ on $n\times k$
vertices, where each vertex $v$ of $G$ is replaced by $k$ vertices
$v_1,\cdots{},v_k$ and each edge $(u,v)$ in $G$ is replaced by a matching
representing a bijection $\pi_{uv}$ so that the edges of $H$ are of the form
$(u_i,v_{\pi_{uv}(i)})$. Lifts have been studied as a means to efficiently
construct expanders. In this work, we study lifts obtained from groups and
group actions. We derive the spectrum of such lifts via the representation
theory principles of the underlying group. Our main results are:
(1) There is a constant $c_1$ such that for every $k\geq 2^{c_1nd}$, there
does not exist an abelian $k$-lift $H$ of any $n$-vertex $d$-regular base graph
with $H$ being almost Ramanujan (nontrivial eigenvalues of the adjacency matrix
at most $O(\sqrt{d})$ in magnitude). This can be viewed as an analogue of the
well-known no-expansion result for abelian Cayley graphs.
(2) A uniform random lift in a cyclic group of order $k$ of any $n$-vertex
$d$-regular base graph $G$, with the nontrivial eigenvalues of the adjacency
matrix of $G$ bounded by $\lambda$ in magnitude, has the new nontrivial
eigenvalues also bounded by $\lambda+O(\sqrt{d})$ in magnitude with probability
$1-ke^{-\Omega(n/d^2)}$. In particular, there is a constant $c_2$ such that for
every $k\leq 2^{c_2n/d^2}$, there exists a lift $H$ of every Ramanujan graph in
a cyclic group of order $k$ with $H$ being almost Ramanujan. We use this to
design a quasi-polynomial time algorithm to construct almost Ramanujan
expanders deterministically.
The existence of expanding lifts in cyclic groups of order $k=2^{O(n/d^2)}$
can be viewed as a lower bound on the order $k_0$ of the largest abelian group
that produces expanding lifts. Our results show that the lower bound matches
the upper bound for $k_0$ (upto $d^3$ in the exponent)

### Measuring and Understanding Throughput of Network Topologies

High throughput is of particular interest in data center and HPC networks.
Although myriad network topologies have been proposed, a broad head-to-head
comparison across topologies and across traffic patterns is absent, and the
right way to compare worst-case throughput performance is a subtle problem.
In this paper, we develop a framework to benchmark the throughput of network
topologies, using a two-pronged approach. First, we study performance on a
variety of synthetic and experimentally-measured traffic matrices (TMs).
Second, we show how to measure worst-case throughput by generating a
near-worst-case TM for any given topology. We apply the framework to study the
performance of these TMs in a wide range of network topologies, revealing
insights into the performance of topologies with scaling, robustness of
performance across TMs, and the effect of scattered workload placement. Our
evaluation code is freely available