48 research outputs found

### Finding a most biased coin with fewest flips

We study the problem of learning a most biased coin among a set of coins by
tossing the coins adaptively. The goal is to minimize the number of tosses
until we identify a coin i* whose posterior probability of being most biased is
at least 1-delta for a given delta. Under a particular probabilistic model, we
give an optimal algorithm, i.e., an algorithm that minimizes the expected
number of future tosses. The problem is closely related to finding the best arm
in the multi-armed bandit problem using adaptive strategies. Our algorithm
employs an optimal adaptive strategy -- a strategy that performs the best
possible action at each step after observing the outcomes of all previous coin
tosses. Consequently, our algorithm is also optimal for any starting history of
outcomes. To our knowledge, this is the first algorithm that employs an optimal
adaptive strategy under a Bayesian setting for this problem. Our proof of
optimality employs tools from the field of Markov games

### Improving the smoothed complexity of FLIP for max cut problems

Finding locally optimal solutions for max-cut and max-$k$-cut are well-known
PLS-complete problems. An instinctive approach to finding such a locally
optimum solution is the FLIP method. Even though FLIP requires exponential time
in worst-case instances, it tends to terminate quickly in practical instances.
To explain this discrepancy, the run-time of FLIP has been studied in the
smoothed complexity framework. Etscheid and R\"{o}glin showed that the smoothed
complexity of FLIP for max-cut in arbitrary graphs is quasi-polynomial. Angel,
Bubeck, Peres, and Wei showed that the smoothed complexity of FLIP for max-cut
in complete graphs is $O(\phi^5n^{15.1})$, where $\phi$ is an upper bound on
the random edge-weight density and $n$ is the number of vertices in the input
graph.
While Angel et al.'s result showed the first polynomial smoothed complexity,
they also conjectured that their run-time bound is far from optimal. In this
work, we make substantial progress towards improving the run-time bound. We
prove that the smoothed complexity of FLIP in complete graphs is $O(\phi
n^{7.83})$. Our results are based on a carefully chosen matrix whose rank
captures the run-time of the method along with improved rank bounds for this
matrix and an improved union bound based on this matrix. In addition, our
techniques provide a general framework for analyzing FLIP in the smoothed
framework. We illustrate this general framework by showing that the smoothed
complexity of FLIP for max-$3$-cut in complete graphs is polynomial and for
max-$k$-cut in arbitrary graphs is quasi-polynomial. We believe that our
techniques should also be of interest towards addressing the smoothed
complexity of FLIP for max-$k$-cut in complete graphs for larger constants $k$.Comment: 36 page

### Deciding Orthogonality in Construction-A Lattices

Lattices are discrete mathematical objects with widespread applications to
integer programs as well as modern cryptography. A fundamental problem in both
domains is the Closest Vector Problem (popularly known as CVP). It is
well-known that CVP can be easily solved in lattices that have an orthogonal
basis \emph{if} the orthogonal basis is specified. This motivates the
orthogonality decision problem: verify whether a given lattice has an
orthogonal basis. Surprisingly, the orthogonality decision problem is not known
to be either NP-complete or in P.
In this paper, we focus on the orthogonality decision problem for a
well-known family of lattices, namely Construction-A lattices. These are
lattices of the form $C+q\mathbb{Z}^n$, where $C$ is an error-correcting
$q$-ary code, and are studied in communication settings. We provide a complete
characterization of lattices obtained from binary and ternary codes using
Construction-A that have an orthogonal basis. We use this characterization to
give an efficient algorithm to solve the orthogonality decision problem. Our
algorithm also finds an orthogonal basis if one exists for this family of
lattices. We believe that these results could provide a better understanding of
the complexity of the orthogonality decision problem for general lattices

### 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)