
Fast and NearOptimal Algorithms for Approximating Distributions by Histograms
J. Acharya, I. Diakonikolas, C. Hegde, J. Li, L. Schmidt
Manuscript

Testing Identity of Structured Distributions
[abstract]
[pdf]
I. Diakonikolas, D. Kane, V. Nikishkin
Proceedings of the 26th Annual ACMSIAM Symposium on Discrete Algorithms (SODA 2015)
We study the question of identity testing for structured distributions.
More precisely, given samples from a {\em structured} distribution $q$ over $[n]$ and an explicit distribution $p$ over $[n]$,
we wish to distinguish whether $q=p$ versus $q$ is at least $\epsilon$far from $p$,
in $L_1$ distance. In this work, we present a unified approach that yields new, simple testers, with sample complexity
that is informationtheoretically optimal, for broad classes of structured distributions, including $t$flat distributions,
$t$modal distributions, logconcave distributions, monotone hazard rate (MHR) distributions, and mixtures thereof.

Learning from Satisfying Assignments
[abstract]
[pdf]
A. De, I. Diakonikolas, R. Servedio
Proceedings of the 26th Annual ACMSIAM Symposium on Discrete Algorithms (SODA 2015)
This paper studies the problem of learning ``lowcomplexity"
probability distributions over the Boolean hypercube $\{1,1\}^n$.
As in the standard PAC learning model, a learning problem in our
framework is defined by a class ${\cal C}$ of Boolean functions
over $\{1,1\}^n$, but
in our model the learning algorithm is given uniform random satisfying
assignments of an unknown $f \in \cal{C}$ and its goal is to output
a highaccuracy approximation of the uniform distribution
over $f^{1}(1).$ This distribution learning problem may be viewed as a
demanding variant of standard Boolean function learning, where the learning
algorithm only receives positive examples and  more importantly 
must output a hypothesis function which has small \emph{multiplicative}
error (i.e., small error relative to the size of $f^{1}(1)$).
As our main results, we show that the two most widely studied
classes of Boolean functions in computational learning theory 
linear threshold functions and DNF formulas  have
efficient distribution learning algorithms in our model.
Our algorithm for linear threshold functions runs in
time poly$(n,1/\epsilon)$ and our algorithm for
polynomialsize DNF runs in time quasipoly$(n,1/\epsilon)$.
We obtain both these results via a general approach that
combines a broad range of technical ingredients, including the complexitytheoretic study of approximate counting and uniform generation;
the Statistical Query model from learning theory; and hypothesis testing techniques from statistics. A key conceptual and technical ingredient of
this approach is a new kind of algorithm which we devise called a ``densifier'' and which we
believe may be useful in other contexts.
We also establish limitations on efficient learnability in our model by showing
that the existence of certain types of cryptographic signature schemes
imply that certain learning problems in our framework are computationally
hard. Via this connection we show that
assuming the existence of sufficiently strong unique signature schemes,
there are no subexponential time learning algorithms in our framework for
intersections of two halfspaces, for
degree2 polynomial threshold functions, or for monotone 2CNF formulas.
Thus our positive results for distribution learning come close to the
limits of what can be achieved by efficient algorithms.

NearOptimal Density Estimation in NearLinear Time Using VariableWidth Histograms
[abstract]
[pdf]
S. Chan, I. Diakonikolas, R. Servedio, X. Sun
Advances in Neural Information Processing Systems (NIPS 2014)
Let $p$ be an unknown and arbitrary probability distribution over $[0,1)$. We consider the problem
of density estimation, in which a learning algorithm is given i.i.d. draws from $p$ and must
(with high probability) output a hypothesis distribution that is close to $p$. The main contribution of this paper is
a highly efficient density estimation algorithm for learning using a variablewidth histogram, i.e., a
hypothesis distribution with a piecewise constant probability density function.
In more detail, for any $k$ and $\epsilon$, we give an algorithm that makes $\tilde{O}(k/\epsilon^2)$ draws from $p$, runs in $\tilde{O}(k/\epsilon^2)$ time, and outputs a
hypothesis distribution $h$ that is piecewise constant with $O(k \log^2(1/\epsilon))$ pieces. With high probability the
hypothesis $h$ satisfies $d_{\mathrm{TV}}(p,h) \leq C \cdot \mathrm{opt}_k(p) + \epsilon$, where $d_{\mathrm{TV}}$ denotes the total variation distance
(statistical distance), $C$ is a universal constant,
and $\mathrm{opt}_k(p)$ is the smallest total variation distance between $p$ and any $k$piecewise constant distribution.
The sample size and running time of our algorithm are optimal up to logarithmic factors.
The ``approximation factor'' $C$ in our result is inherent in the problem, as we prove that
no algorithm with sample size bounded in terms of $k$ and $\epsilon$ can achieve $C<2$ regardless
of what kind of hypothesis distribution it uses.

Efficient Density Estimation via Piecewise Polynomial Approximation
[abstract]
[pdf]
S. Chan, I. Diakonikolas, R. Servedio, X. Sun
Proceedings of the 46th Annual ACM Symposium on Theory of Computing (STOC 2014)
We give a highly efficient "semiagnostic" algorithm for learning univariate probability distributions that are well approximated by piecewise polynomial density functions. Let $p$ be an arbitrary distribution over an interval $I$ which is $\tau$close (in total variation distance) to an unknown probability distribution $q$ that is defined by an unknown partition of $I$ into $t$ intervals and $t$ unknown degree$d$ polynomials specifying $q$ over each of the intervals. We give an algorithm that draws $\tilde{O}(t(d+1)/\epsilon^2)$ samples from $p$, runs in time $\mathrm{poly}(t,d,1/\epsilon)$, and with high probability outputs a piecewise polynomial hypothesis distribution $h$ that is $(O(\tau)+\epsilon)$close (in total variation distance) to $p$. This sample complexity is essentially optimal; we show that even for $\tau=0$, any algorithm that learns an unknown $t$piecewise degree$d$ probability distribution over $I$ to accuracy $\epsilon$ must use $\Omega({\frac {t(d+1)} {\mathrm{poly}(1 + \log(d+1))}} \cdot {\frac 1 {\epsilon^2}})$ samples from the distribution, regardless of its running time. Our algorithm combines tools from approximation theory, uniform convergence, linear programming, and dynamic programming.
We apply this general algorithm to obtain a wide range of results for many natural problems in density estimation over both continuous and discrete domains. These include stateoftheart results for learning mixtures of logconcave distributions; mixtures of $t$modal distributions; mixtures of Monotone Hazard Rate distributions; mixtures of Poisson Binomial Distributions; mixtures of Gaussians; and mixtures of $k$monotone densities. Our general technique yields computationally efficient algorithms for all these problems, in many cases with provably optimal sample complexities (up to logarithmic factors) in all parameters.

Deterministic Approximate Counting for Juntas of Degree$2$ Polynomial Threshold Functions
[abstract]
[pdf]
A. De, I. Diakonikolas, R. Servedio
Proceedings of the 29th Annual IEEE Conference on Computational Complexity (CCC 2014)
Let $g: \{1,1\}^k \rightarrow \{1,1\}$ be any Boolean function
and $q_1,\dots,q_k$ be any degree$2$ polynomials over
$\{1,1\}^n.$ We give a
deterministic algorithm which,
given as input explicit descriptions of
$g,q_1,\dots,q_k$ and an accuracy parameter $\epsilon>0$,
approximates
\[
\mathbf{Pr}_{x \sim \{1,1\}^n}[g(\mathrm{sign}(q_1(x)),\dots,\mathrm{sign}(q_k(x)))=1]
\]
to within an additive $\pm \epsilon$. For any constant $\epsilon > 0$
and $k \geq 1$ the running time of our algorithm is a fixed
polynomial in $n$ (in fact this is true even for some nottoosmall
$\epsilon = o_n(1)$ and nottoolarge $k = \omega_n(1)$).
This is the first fixed polynomialtime algorithm
that can deterministically approximately count
satisfying assignments of a natural
class of depth$3$ Boolean circuits.
Our algorithm extends a recent result \cite{DDS13:deg2count}
which gave a deterministic
approximate counting algorithm for a single degree$2$ polynomial
threshold function $\mathrm{sign}(q(x)),$ corresponding to the $k=1$ case of our
result. Note that even in the $k=1$ case it is NPhard to determine
whether $\mathbf{Pr}_{x \sim \{1,1\}^n}[\mathrm{sign}(q(x))=1]$ is nonzero,
so any sort of multiplicative approximation is almost certainly
impossible even for efficient randomized algorithms.
Our algorithm and analysis requires several novel technical ingredients
that go significantly beyond the tools required to handle the $k=1$ case
in \cite{DDS13:deg2count}. One of these
is a new multidimensional central limit theorem
for degree$2$ polynomials in Gaussian random variables which builds
on recent Malliavincalculusbased results from probability theory. We
use this CLT as the basis of a new decomposition technique for $k$tuples
of degree$2$ Gaussian polynomials and thus obtain an efficient
deterministic approximate counting
algorithm for the Gaussian distribution, i.e., an algorithm for estimating
\[
\mathbf{Pr}_{x \sim \mathcal{N}(0,1)^n}[g(\mathrm{sign}(q_1(x)),\dots,\mathrm{sign}(q_k(x)))=1].
\]
Finally, a third new ingredient
is a ``regularity lemma'' for $k$tuples of degree$d$
polynomial threshold functions. This generalizes both the regularity lemmas
of \cite{DSTW:10,HKM:09}
(which apply to a single degree$d$ polynomial threshold
function) and the regularity lemma of Gopalan et al \cite{GOWZ10}
(which applies to
a $k$tuples of linear threshold functions, i.e., the case $d=1$).
Our new regularity lemma lets us extend our deterministic approximate
counting results from the Gaussian to the Boolean domain.

The Complexity of Optimal Multidimensional Pricing
[abstract]
[pdf]
X. Chen, I. Diakonikolas, D. Paparas, X. Sun, M. Yannakakis
Proceedings of the 25th Annual ACMSIAM Symposium on Discrete Algorithms (SODA 2014)
We resolve the complexity of revenueoptimal deterministic auctions in
the unitdemand singlebuyer Bayesian setting, i.e., the optimal item
pricing problem, when the buyer's values for the items are independent.
We show that the problem of computing a revenueoptimal pricing can be solved
in polynomial time for distributions of support size $2$
and its decision version is NPcomplete for distributions of support size $3$.
We also show that the problem remains NPcomplete for the case of identical distributions.

Optimal Algorithms for Testing Closeness of Discrete Distributions
[abstract]
[pdf]
S. Chan, I. Diakonikolas, G. Valiant, P. Valiant
Proceedings of the 25th Annual ACMSIAM Symposium on Discrete Algorithms (SODA 2014)
Blog post about this work: Property Testing Review
We study the question of closeness testing for two discrete distributions.
More precisely, given samples from two distributions $p$ and $q$ over an $n$element set,
we wish to distinguish whether $p=q$ versus $p$ is at least $\epsilon$far from $q$,
in either $\ell_1$ or $\ell_2$ distance. Batu et al~\cite{BFR+:00, Batu13} gave the first sublinear time algorithms for these problems,
which matched the lower bounds of~\cite{PV11sicomp} up to a logarithmic factor in $n$, and a polynomial factor of $\epsilon.$
In this work, we present simple testers for both the $\ell_1$ and $\ell_2$ settings, with sample complexity
that is informationtheoretically optimal, to constant factors, both in the dependence on $n$, and the dependence on $\epsilon$;
for the $\ell_1$ testing problem we establish that the sample complexity is $\Theta(\max\{n^{2/3}/\epsilon^{4/3}, n^{1/2}/\epsilon^2 \}).$

A Polynomialtime Approximation Scheme for Faulttolerant Distributed Storage
[abstract]
[pdf]
C. Daskalakis, A. De, I. Diakonikolas, A. Moitra, R. Servedio
Proceedings of the 25th Annual ACMSIAM Symposium on Discrete Algorithms (SODA 2014)
Featured in Abstract Talk. Podcast is here
We consider a problem which has received considerable attention in
systems literature because of its applications to routing in delay tolerant networks and replica placement
in distributed storage systems.
In abstract terms the problem can be stated as follows: Given a random variable $X$
generated by a known product distribution over $\{0,1\}^n$ and a target
value $0 \leq \theta \leq 1$, output a nonnegative vector $w$, with
$\w\_1 \le 1$, which maximizes the probability of the event $w \cdot X
\ge \theta$. This is a challenging nonconvex optimization problem for
which even computing the value $\Pr[w \cdot X \ge \theta]$ of a proposed
solution vector $w$ is #Phard.
We provide an additive EPTAS for this problem
which, for constantbounded product distributions,
runs in $ \mathrm{poly}(n) \cdot 2^{\mathrm{poly}(1/\epsilon)}$ time and outputs an
$\epsilon$approximately optimal solution vector $w$ for this problem. Our
approach is inspired by, and extends,
recent structural results from the complexitytheoretic
study of linear threshold functions. Furthermore, in spite of the objective function being nonsmooth,
we give a unicriterion PTAS while previous work for such objective functions has typically
led to a bicriterion PTAS. We believe our techniques may be applicable to get unicriterion PTAS for other nonsmooth objective functions.

Learning Sums of Independent Integer Random Variables
[abstract]
[pdf]
C. Daskalakis, I. Diakonikolas, R. O'Donnell, R. Servedio, LY. Tan
Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2013)
Blog post about this work: MIT theory student blog
Let $\mathbf{S} = \mathbf{X}_1 + \cdots + \mathbf{X}_n$ be a sum of $n$ independent
integer random variables $\mathbf{X}_i$, where each $\mathbf{X}_i$ is supported on
$\{0,1,\dots,k1\}$ but otherwise may have an arbitrary distribution
(in particular the $\mathbf{X}_i$'s need not be identically distributed).
How many samples are required to learn the distribution $\mathbf{S}$ to high
accuracy? In this paper we show
that the answer is
completely independent of $n$, and moreover we give a
computationally efficient algorithm which achieves this low sample
complexity. More precisely, our algorithm learns any such $\mathbf{S}$ to $\epsilon$accuracy (with respect
to the total variation distance between distributions)
using $\mathrm{poly}(k,1/\epsilon)$ samples, independent of $n$. Its running time is
$\mathrm{poly}(k,1/\epsilon)$ in the standard word RAM model. Thus we give
a broad generalization of the main result of~\cite{DDS12stoc}
which gave a similar learning result for the special case $k=2$ (when
the distribution~$\mathbf{S}$ is a Poisson Binomial Distribution).
Prior to this work, no nontrivial results were
known for learning these distributions even in the case $k=3$.
A key difficulty is that, in contrast to the case of $k = 2$,
sums of independent $\{0,1,2\}$valued random variables may
behave very differently from
(discretized) normal distributions, and in fact may be
rather complicated  they are not logconcave, they can
be $\Theta(n)$modal, there is no relationship between
Kolmogorov distance and total variation distance for the class, etc.
Nevertheless, the heart of our learning result is a new limit theorem
which characterizes what the sum of an arbitrary number of arbitrary
independent $\{0,1,\dots, k1\}$valued random variables may look like.
Previous limit theorems in this setting made strong assumptions on the
"shift invariance" of the random variables $\mathbf{X}_i$ in order to
force a discretized normal limit. We believe that our new
limit theorem, as the first result for truly arbitrary sums of
independent $\{0,1,\dots,k1\}$valued random variables,
is of independent interest.

Deterministic Approximate Counting for Degree$2$ Polynomial Threshold Functions
[abstract]
[pdf]
A. De, I. Diakonikolas, R. Servedio
Manuscript, 2013. Available as ECCC technical report [link]
We give a
deterministic algorithm for
approximately computing the fraction of Boolean assignments
that satisfy a degree$2$ polynomial threshold function.
Given a degree$2$ input polynomial $p(x_1,\dots,x_n)$
and a parameter $\epsilon > 0$, the algorithm approximates
$\Pr_{x \sim \{1,1\}^n}[p(x) \geq 0]$
to within an additive $\pm \epsilon$ in time $\mathrm{poly}(n,2^{\mathrm{poly}(1/\epsilon)})$.
Note that it is NPhard to determine whether the above probability
is nonzero, so any sort of multiplicative approximation is almost certainly
impossible even for efficient randomized algorithms.
This is the first deterministic algorithm for this counting problem
in which the running time is polynomial in $n$ for $\epsilon= o(1)$.
For "regular" polynomials $p$ (those in which no individual variable's
influence is large compared to the sum of all $n$
variable influences)
our algorithm runs in $\mathrm{poly}(n,1/\epsilon)$ time.
The algorithm also runs in $\mathrm{poly}(n,1/\epsilon)$ time to approximate
$\Pr_{x \sim \mathcal{N}(0,1)^n}[p(x) \geq 0]$ to within an additive $\pm \epsilon$,
for any degree2 polynomial $p$.
As an application of our counting result, we give a deterministic
multiplicative $(1 \pm \epsilon)$approximation algorithm
to approximate the $k$th absolute moment $\mathbf{E}_{x \sim \{1,1\}^n}[p(x)^k]$
of a degree$2$ polynomial. The algorithm runs in fixed
polynomial time for any constants $k$ and $\epsilon.$

A robust Khintchine Inequality and computing optimal constants in Fourier analysis and highdimensional geometry
[abstract]
[pdf]
A. De, I. Diakonikolas, R. Servedio
Proceedings of the 40th Intl. Colloquium on Automata, Languages and Programming (ICALP 2013)
This paper makes two contributions towards determining some wellstudied
optimal constants in Fourier analysis of Boolean functions
and highdimensional geometry.
(1) It has been known since 1994 \cite{GL:94} that every linear threshold function has squared Fourier mass
at least $1/2$ on its degree$0$ and degree$1$ coefficients.
Denote the minimum such Fourier mass by $\mathbf{W}^{\leq 1}[\mathbf{LTF}]$,
where the minimum is taken over all $n$variable linear threshold functions and all $n \ge 0$.
Benjamini, Kalai and Schramm \cite{BKS:99}
have conjectured that the true value of $\mathbf{W}^{\leq 1}[\mathbf{LTF}]$ is $2/\pi$.
We make progress on this conjecture by proving that $\mathbf{W}^{\leq 1}[\mathbf{LTF}]
\geq 1/2 + c$ for some absolute constant $c>0$.
The key ingredient in our proof is a "robust" version of the wellknown
Khintchine inequality in functional analysis, which we
believe may be of independent interest.
(2) We give an algorithm with the following property: given any $\eta > 0$,
the algorithm runs in time $2^{\mathrm{poly}(1/\eta)}$ and determines the value of
$\mathbf{W}^{\leq 1}[\mathbf{LTF}]$ up to an additive error of $\pm\eta$. We give a similar
$2^{{\mathrm{poly}(1/\eta)}}$time algorithm to determine Tomaszewski's constant
to within an additive error of $\pm \eta$; this is the
minimum (over all origincentered hyperplanes $H$) fraction of points
in $\{1,1\}^n$ that lie within Euclidean distance $1$ of $H$.
Tomaszewski's constant is conjectured to be $1/2$; lower bounds on it
have been given by Holzman and Kleitman \cite{HK92} and
independently by BenTal, Nemirovski and Roos
\cite{BNR02}.
Our algorithms combine tools from anticoncentration
of sums of independent random variables, Fourier analysis, and Hermite
analysis of linear threshold functions.

Learning Mixtures of Structured Distributions over Discrete Domains
[abstract]
[pdf]
S. Chan, I. Diakonikolas, R. Servedio, X. Sun
Proceedings of the 24th Annual ACMSIAM Symposium on Discrete Algorithms (SODA 2013)
Let $\mathfrak{C}$ be a class of probability distributions over the discrete domain $[n] = \{1,\dots,n\}.$
We show that if $\mathfrak{C}$ satisfies a rather general condition  essentially, that each distribution in
$\mathfrak{C}$ can be wellapproximated by a variablewidth
histogram with few bins  then there is a highly efficient (both in terms of running time and sample complexity)
algorithm that can learn any mixture of $k$ unknown distributions from
$\mathfrak{C}.$
We analyze several natural types of distributions over $[n]$,
including logconcave, monotone hazard rate and unimodal distributions,
and show that they have the required structural property of being
wellapproximated by a histogram with few bins.
Applying our general algorithm, we
obtain nearoptimally efficient algorithms for all these mixture
learning problems as described below. More precisely,
Logconcave distributions: We learn any mixture of $k$
logconcave distributions over $[n]$ using $k \cdot
\tilde{O}(1/\epsilon^4)$ samples (independent of $n$) and running in time
$\tilde{O}(k \log(n) / \epsilon^4)$ bitoperations (note that reading a single
sample from $[n]$ takes $\Theta(\log n)$ bit operations).
For the special case $k=1$ we give an efficient
algorithm using $\tilde{O}(1/\epsilon^3)$
samples; this generalizes the main result of \cite{DDS12stoc} from the
class of Poisson Binomial distributions to the much broader class of all
logconcave distributions. Our upper bounds are not far from
optimal since any algorithm for this learning problem requires
$\Omega(k/\epsilon^{5/2})$ samples.
Monotone hazard rate (MHR) distributions:
We learn any mixture of $k$ MHR distributions over $[n]$ using
$O(k \log (n/\epsilon)/\epsilon^4)$ samples and running in time $\tilde{O}(k
\log^2(n) / \epsilon^4)$ bitoperations. Any algorithm for this learning problem must use $\Omega(k \log(n)/\epsilon^3)$ samples.
Unimodal distributions:
We give an algorithm that learns any mixture of $k$ unimodal distributions
over $[n]$ using $O(k \log (n)/\epsilon^{4})$ samples and running in time
$\tilde{O}(k \log^2(n) / \epsilon^{4})$ bitoperations.
Any algorithm for this problem must use $\Omega(k \log(n)/\epsilon^3)$ samples.

Testing $k$modal Distributions: Optimal Algorithms via Reductions
[abstract]
[pdf]
C. Daskalakis, I. Diakonikolas, R. Servedio, G. Valiant, P. Valiant
Proceedings of the 24th Annual ACMSIAM Symposium on Discrete Algorithms (SODA 2013)
We give highly efficient algorithms, and almost matching lower bounds, for a range of basic statistical problems
that involve testing and estimating the $L_1$ (total variation) distance between two $k$modal distributions $p$ and $q$ over the discrete domain $\{1,\dots,n\}$.
More precisely, we consider the following four problems: given sample access to an unknown $k$modal
distribution $p$,
Testing identity to a known or unknown distribution:
(1) Determine whether $p = q$ (for an explicitly given $k$modal distribution $q$) versus
$p$ is $\epsilon$far from $q$;
(2) Determine whether $p=q$ (where $q$ is available via sample access) versus
$p$ is $\epsilon$far from $q$;
Estimating $L_1$ distance (``tolerant testing'') against a known or unknown distribution:
(3) Approximate $d_{TV}(p,q)$ to within additive $\epsilon$ where $q$ is an explicitly
given $k$modal distribution $q$;
(4)Approximate $d_{TV}(p,q)$ to within additive $\epsilon$ where $q$ is available via sample access.
For each of these four problems we give sublogarithmic sample algorithms, that we show are tight up to additive $\mathrm{poly}(k)$
and multiplicative $\mathrm{polylog}\log n+\mathrm{polylog} k$ factors.
Thus our bounds significantly improve the previous results of \cite{BKR:04}, which were for testing identity of distributions (items (1) and (2) above) in the special cases
$k=0$ (monotone distributions) and $k=1$ (unimodal distributions) and required $O((\log n)^3)$ samples.

An Optimal Algorithm for the Efficient Approximation of Convex Pareto Curves
I. Diakonikolas, M. Yannakakis
Manuscript, 2012

On the relation of total variation and Kolmogorov distance between Poisson Binomial distributions
C. Daskalakis, I. Diakonikolas, R. Servedio
Manuscript, 2012

EfficiencyRevenue Tradeoffs in Auctions
[abstract]
[pdf]
I. Diakonikolas, C.H. Papadimitriou, G. Pierrakos, Y. Singer
Proceedings of the 39th Intl. Colloquium on Automata, Languages and Programming (ICALP 2012)
When agents with independent priors bid for a single item, Myerson's optimal auction maximizes expected revenue, whereas Vickrey's secondprice auction optimizes social welfare. We address the natural question of tradeoffs between the two criteria, that is, auctions that optimize, say, revenue under the constraint that the welfare is above a given level. If one allows for randomized mechanisms, it is easy to see that there are polynomialtime mechanisms that achieve any point in the tradeoff (the Pareto curve) between revenue and welfare. We investigate whether one can achieve the same guarantees using deterministic mechanisms. We provide a negative answer to this question by showing that this is a (weakly) NPhard problem. On the positive side, we provide polynomialtime deterministic mechanisms that approximate with arbitrary precision any point of the tradeoff between these two fundamental objectives for the case of two bidders, even when the valuations are correlated arbitrarily. The major problem left open by our work is whether there is such an algorithm for three or more bidders with independent valuation distributions.

The Inverse Shapley Value Problem
[abstract]
[pdf]
A. De, I. Diakonikolas, R. Servedio
Games and Economic Behavior, accepted with minor revisions
Proceedings of the 39th Intl. Colloquium on Automata, Languages and Programming (ICALP 2012)
For $f$ a weighted voting scheme used by $n$ voters to choose between two
candidates, the $n$
ShapleyShubik Indices (or
Shapley values)
of $f$ provide a measure of how
much control each voter can exert over the overall outcome of the vote.
ShapleyShubik indices were introduced by Lloyd Shapley
and Martin Shubik in 1954 \cite{SS54} and are widely studied in
social choice theory as a measure of the "influence" of voters.
The
Inverse Shapley Value Problem is the problem of designing a weighted
voting scheme which (approximately) achieves a desired input vector of
values for the ShapleyShubik indices. Despite much interest in this problem
no provably correct and efficient algorithm was known prior to our work.
We give the first efficient algorithm with provable performance guarantees for
the Inverse Shapley Value Problem. For any constant $\epsilon > 0$
our algorithm runs in fixed poly$(n)$ time (the degree of the
polynomial is independent of $\epsilon$) and has the following
performance guarantee: given as input a vector of desired Shapley values,
if any "reasonable" weighted voting scheme
(roughly, one in which the threshold is not too skewed)
approximately matches the desired vector of values to within
some small error,
then our algorithm explicitly outputs a weighted voting scheme that
achieves this vector of Shapley values to within error $\epsilon.$
If there is a "reasonable" voting scheme in which all
voting weights are integers at most $\mathrm{poly}(n)$ that approximately achieves
the desired Shapley values, then our algorithm runs in time
$\mathrm{poly}(n)$ and outputs a weighted voting scheme that achieves
the target vector of Shapley values to within
error $\epsilon=n^{1/8}.$

Nearly optimal solutions for the Chow Parameters Problem and lowweight approximation of halfspaces
[abstract]
[pdf]
A. De, I. Diakonikolas, V. Feldman, R. Servedio
Journal of the ACM, 61(2), 2014. Invited to Theory of Computing special issue on Analysis of Boolean functions (declined)
Proceedings of the 44th Annual ACM Symposium on Theory of Computing (STOC 2012)
The
Chow parameters of a Boolean function $f: \{1,1\}^n \to \{1,1\}$ are its $n+1$ degree$0$ and
degree$1$ Fourier coefficients. It has been known since 1961 \cite{Chow:61, Tannenbaum:61} that the (exact values of the) Chow parameters of
any linear threshold function $f$ uniquely specify $f$ within the space of all Boolean functions, but until
recently \cite{OS11:chow} nothing was known about efficient algorithms for
reconstructing $f$
(exactly or approximately) from exact or approximate values of its Chow parameters. We refer to this reconstruction problem as the
Chow Parameters Problem.
Our main result is a new algorithm for the Chow Parameters Problem which,
given (sufficiently accurate approximations to) the Chow parameters of any linear threshold function $f$, runs in time
$\tilde{O}(n^2)\cdot (1/\epsilon)^{O(\log^2(1/\epsilon))}$ and
with high probability outputs a representation of an LTF $f'$ that is $\epsilon$close to $f$ in Hamming distance.
The only previous algorithm \cite{OS11:chow} had running time $\mathrm{poly}(n) \cdot 2^{2^{\tilde{O}(1/\epsilon^2)}}.$
As a byproduct of our approach, we show that for any linear threshold function $f$ over $\{1,1\}^n$,
there is a linear threshold function $f'$ which is $\epsilon$close to $f$ and has all weights that are integers of magnitude at most $\sqrt{n} \cdot (1/\epsilon)^{O(\log^2(1/\epsilon))}$.
This significantly improves the previous best result of~\cite{DiakonikolasServedio:09} which gave
a $\mathrm{poly}(n) \cdot 2^{\tilde{O}(1/\epsilon^{2/3})}$ weight bound, and is close to the
known lower bound of
$\max\{\sqrt{n},$ $(1/\epsilon)^{\Omega(\log \log (1/\epsilon))}\}$ \cite{Goldberg:06b,Servedio:07cc}.
Our techniques also yield improved algorithms for related problems in learning theory.
In addition to being significantly stronger than previous work, our results
are obtained using conceptually simpler proofs.
The two main ingredients underlying our results are (1) a new structural
result showing that for $f$ any linear threshold function and $g$ any bounded
function, if the Chow parameters of $f$ are close to the Chow
parameters of $g$ then $f$ is close to $g$; (2) a new boostinglike algorithm
that given approximations to the Chow parameters of a linear threshold function outputs a bounded
function whose Chow parameters are close to those of $f$.

Learning Poisson Binomial distributions
[abstract]
[pdf]
C. Daskalakis, I. Diakonikolas, R. Servedio
Invited to Algorithmica special issue on Machine Learning
Proceedings of the 44th Annual ACM Symposium on Theory of Computing (STOC 2012)
We consider a basic problem in unsupervised learning:
learning an unknown
Poisson Binomial Distribution.
A Poisson Binomial Distribution (PBD) over $\{0,1,\dots,n\}$
is the distribution of a sum of $n$ independent Bernoulli
random variables which may have arbitrary, potentially nonequal,
expectations. These distributions were first studied by S. Poisson in 1837 \cite{Poisson:37} and are a natural
$n$parameter generalization of the familiar Binomial Distribution.
Surprisingly, prior to our work this basic learning problem
was poorly understood, and known results for it were far from
optimal.
We essentially settle the complexity of the learning problem for this
basic class of distributions.
As our main result we give a highly efficient algorithm which learns to
$\epsilon$accuracy using $\tilde{O}(1/\epsilon^3)$ samples independent of $n$.
The running time of the algorithm is quasilinear in the
size of its input data. This is nearly optimal
since any algorithm must use $\Omega(1/\epsilon^2)$ samples.
We also give positive and negative results for some extensions of this learning problem.

Learning $k$modal distributions via testing
[abstract]
[pdf]
C. Daskalakis, I. Diakonikolas, R. Servedio
Theory of Computing, 10 (20), 535570 (2014)
Proceedings of the 23rd Annual ACMSIAM Symposium on Discrete Algorithms (SODA 2012)
Oded Goldreich's Choices #72
A $k$modal probability distribution over the domain $\{1,...,n\}$ is one whose
histogram has at most $k$ "peaks" and "valleys." Such distributions are
natural generalizations of monotone ($k=0$) and unimodal ($k=1$)
probability distributions, which have been intensively studied in probability theory and statistics.
In this paper we consider the problem of learning an unknown $k$modal distribution.
The learning algorithm is given access to independent samples drawn from the $k$modal
distribution $p$, and must output a hypothesis distribution $\hat{p}$ such that with high
probability the total variation distance between $p$ and $\hat{p}$ is at most $\epsilon.$
We give an efficient algorithm for this problem that runs in time $\mathrm{poly}(k,\log(n),1/\epsilon)$.
For $k \leq \tilde{O}(\sqrt{\log n})$, the number of samples used by our algorithm is very close (within an
$\tilde{O}(\log(1/\epsilon))$ factor) to being informationtheoretically optimal. Prior to this
work computationally efficient algorithms were known only for the cases $k=0,1$
\cite{Birge:87b,Birge:97}.
A novel feature of our approach is that our learning algorithm crucially uses a new property testing algorithm as a key subroutine.
The learning algorithm uses the property tester to efficiently
decompose the $k$modal distribution into $k$ (near)monotone distributions, which are easier to
learn.

Noise Stable Halfspaces are Close to Very Small Juntas
[abstract]
[pdf]
I. Diakonikolas, R. Jaiswal, R. Servedio, L.Y.Tan, A. Wan
Manuscript, 2012
Bourgain~\cite{Bourgain:02} showed that any noise stable Boolean function $f$
can be wellapproximated by a junta.
In this note we give an exponential sharpening of the parameters of
Bourgain's result under the additional assumption that $f$ is a halfspace.

Supervised Design Space Exploration by Compositional Approximation of Pareto sets
[abstract]
[pdf]
H.Y. Liu, I. Diakonikolas, M. Petracca, L.P. Carloni
Proceedings of the 48th Design Automation Conference (DAC 2011)
Technology scaling allows the integration of billions of transistors on the same
die but CAD tools struggle in keeping up with the increasing design complexity.
Design productivity for multicore SoCs increasingly depends on
creating and maintaining reusable components and hierarchically
combining them to form larger composite cores.
Characterizing such composite cores with respect to their power/performance
tradeoffs is critical for design reuse across various products and relies
heavily on synthesis tools.
We present $\mathrm{CAPS}$, an online adaptive algorithm that efficiently
explores the design space of any given core and returns an accurate
characterization of its implementation tradeoffs in terms of an approximate
Pareto set.
It does so by supervising the order of the timeconsuming logicsynthesis runs
on the core's components.
Our algorithm can provably achieve the desired precision on the approximation in
the shortest possible time, without having any apriori information on any
component.
We also show that, in practice, $\mathrm{CAPS}$ works even better than what is guaranteed
by the theory.

DisjointPath Facility Location: Theory and Practice
[abstract]
[pdf]
L. Breslau, I. Diakonikolas, N. Duffield, Y.Gu, M.T. Hajiaghayi, D.S. Johnson, H. Karloff, M. Resende, S.Sen
Proceedings of the 13th Workshop on Algorithm Engineering and Experiments (ALENEX 2011)
This paper is a theoretical and experimental study of two related facility
location problems that emanated from networking. Suppose we are given a
network modeled as a directed graph $G = (V,A)$, together with
(notnecessarilydisjoint) subsets $C$ and $F$ of $V$ , where $C$ is a set of
customer locations and $F$ is a set of potential facility locations (and
typically $C\subseteq F$). Our goal is to find a minimum sized subset $F' \subseteq F$
such that for every customer $c \in C$ there are two locations $f_1, f_2 \in F'$
such that traffic from $c$ to $f_1$ and to $f_2$ is routed on disjoint paths
(usually shortest paths) under the network's routing protocols.
Although we prove that this problem is impossible to approximate in the
worst case even to within a factor of $2^{\log^{1\epsilon} n}$ for any $\epsilon>0$
(assuming no NPcomplete language can be solved in quasipolynomial
time), we show that the situation is much better in practice. We
propose three algorithms that build solutions and determine lower
bounds on the optimum solution, and evaluate them on several large real
ISP topologies and on synthetic networks designed to reflect realworld
LAN/WAN network structure. Our main algorithms are (1) an algorithm
that performs multiple runs of a straightforward randomized greedy
heuristic and returns the best result found, (2) a genetic
algorithm that uses the greedy algorithm as a subroutine, and (3) a new
"Double Hitting Set" algorithm. All three approaches perform surprising
well, although, in practice, the most costeffective approach is the
multirun greedy algorithm. This yields results that average within 0.7%
of optimal for our synthetic instances and within 2.9% for our
realworld instances, excluding the largest (and most realistic) one.
For the latter instance, the other two algorithms come into their own,
finding solutions that are more than three times better than those of
the multistart greedy approach. In terms of our motivating monitoring
application, where every customer location can be a facility location,
the results are even better. Here the above Double Hitting Set solution
is 90% better than the default solution which places a monitor at each
customer location. Our results also show that, on
average for our realworld instances, we could save an additional 18%
by choosing the (shortest path) routes ourselves, rather than taking
the simpler approach of relying on the network to choose them for us.

Hardness Results for Agnostically Learning LowDegree Polynomial Threshold Functions
[abstract]
[pdf]
I. Diakonikolas, R. O'Donnell, R. Servedio, Y.Wu
Proceedings of the 22nd Annual ACMSIAM Symposium on Discrete Algorithms (SODA 2011)
Hardness results for maximum agreement problems have close connections to hardness results for
proper learning in computational learning theory.
In this paper we prove two hardness results for the problem of finding a low degree polynomial threshold function (PTF)
which has the maximum possible agreement with a given set of labeled examples in $\mathbf{R}^n \times \{1,1\}.$
We prove that for any constants $d\geq 1, \epsilon > 0$,
(1) Assuming the Unique Games Conjecture, no polynomialtime algorithm can find a degree$d$ PTF that
is consistent with a $(1/2 + \epsilon)$ fraction of a given set of labeled examples in $\mathbf{R}^n \times \{1,1\}$,
even if there exists a degree$d$ PTF that is consistent with a $1\epsilon$ fraction of the examples.
(2) It is NPhard to find a degree$2$ PTF that is consistent with
a $(1/2 + \epsilon)$ fraction of a given set of labeled examples in $\mathbf{R}^n \times \{1,1\}$, even if
there exists a halfspace (degree$1$ PTF) that is consistent with a $1  \epsilon$ fraction of the
examples.
These results immediately imply the following hardness of learning results: (i) Assuming the
Unique Games Conjecture, there is no betterthantrivial proper learning algorithm that agnostically learns degree$d$ PTFs under arbitrary distributions;
(ii) There is no betterthantrivial learning algorithm that outputs degree$2$ PTFs and agnostically learns halfspaces (i.e., degree$1$ PTFs) under arbitrary distributions.

Bounded Independence Fools Degree$2$ Threshold Functions
[abstract]
[pdf]
I. Diakonikolas, D. Kane, J. Nelson
Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS 2010)
Let $x$ be a random vector coming from any $k$wise independent distribution over $\{1,1\}^n$.
For an $n$variate degree$2$ polynomial $p$, we prove that $\mathbf{E}[\mathrm{sgn}(p(x))]$ is determined up to an additive $\epsilon$ for $k =
\mathrm{poly}(1/\epsilon)$. This gives a large class of explicit pseudorandom generators against such functions and answers an open
question of Diakonikolas et al. (FOCS 2009).
In the process, we develop a novel analytic technique we dub multivariate FTmollification. This
provides a generic tool to approximate bounded (multivariate) functions by lowdegree polynomials (with respect to
several different notions of approximation). A univariate version of the method was introduced by Kane et al. (SODA 2010) in
the context of streaming algorithms. In this work, we refine it and generalize it to the multivariate setting. We believe that
our technique is of independent mathematical interest. To illustrate its generality, we note that it implies a multidimensional
generalization of Jackson's classical result in approximation theory due to (Newman and Shapiro, 1963).
To obtain our main result, we combine the FTmollification technique with several linear algebraic and probabilistic
tools. These include the invariance principle of of Mossell, O'Donnell and Oleszkiewicz, anticoncentration bounds for
lowdegree polynomials, an appropriate decomposition of degree$2$ polynomials, and a generalized hypercontractive inequality
for quadratic forms which takes the operator norm of the associated matrix into account. Our analysis is quite modular; it
readily adapts to show that intersections of halfspaces and degree$2$ threshold functions are fooled by bounded independence.
From this it follows that $\Omega(1/\epsilon^2)$wise independence derandomizes the GoemansWilliamson hyperplane rounding scheme.
Our techniques unify, simplify, and in some cases improve several recent results in the literature concerning threshold
functions. For the case of "regular" halfspaces we give a simple proof of an optimal independence bound of
$\Theta(1/\epsilon^2)$, improving upon Diakonikolas et al. (FOCS 2009) by polylogarithmic factors. This yields the first optimal
derandomization of the BerryEsseen theorem and  combined with the results of Kalai et al. (FOCS 2005)  implies a
faster algorithm for the problem of agnostically learning halfspaces.

Average Sensitivity and Noise Sensitivity of Polynomial Threshold Functions
[abstract]
[pdf]
I. Diakonikolas, P. Raghavendra, R. Servedio, L.Y. Tan
SIAM Journal on Computing, 43(1), 231253 (2014)
Proceedings of the 42nd Annual ACM Symposium on Theory of Computing (STOC 2010)
(Conference version merged with this paper by Harsha, Klivans and Meka)
We give the first nontrivial upper bounds on the Boolean average
sensitivity and noise sensitivity of degree$d$ polynomial threshold
functions (PTFs). Our bound on the Boolean average sensitivity of PTFs represents the first progress
towards the resolution of a conjecture of Gotsman and Linial \cite{GL:94}, which states that the symmetric function slicing the
middle $d$ layers of the Boolean hypercube has the highest average
sensitivity of all degree$d$ PTFs. Via the $L_1$ polynomial
regression algorithm of Kalai et al. \cite{KKMS:08}, our bound on
Boolean noise sensitivity yields the first polynomialtime
agnostic learning algorithm for the broad class of constantdegree
PTFs under the uniform distribution.
To obtain our bound on the Boolean average sensitivity of PTFs,
we generalize the "criticalindex" machinery of \cite{Servedio:07cc}
(which in that work applies to halfspaces, i.e., degree$1$ PTFs) to general PTFs.
Together with the "invariance principle" of \cite{MOO10},
this allows us to essentially reduce the Boolean setting
to the Gaussian setting. The main ingredients used to obtain our bound
in the Gaussian setting are tail bounds and anticoncentration bounds on
lowdegree polynomials in Gaussian random variables
\cite{Janson:97,CW:01}. Our bound on Boolean noise sensitivity is achieved
via a simple reduction from upper bounds on average sensitivity of Boolean
PTFs to corresponding bounds on noise sensitivity.

A Regularity Lemma, and Lowweight Approximators, for lowdegree Polynomial Threshold Functions
[abstract]
[pdf]
I. Diakonikolas, R. Servedio, L.Y. Tan, A. Wan
Theory of Computing, 10(2), 2753 (2014)
Proceedings of the 25th Annual IEEE Conference on Computational Complexity (CCC 2010)
We give a "regularity lemma" for degree$d$ polynomial threshold
functions (PTFs) over the Boolean cube $\{1,1\}^n$. Roughly
speaking, this result shows that every degree$d$ PTF can be
decomposed into a constant number of subfunctions such that almost
all of the subfunctions are close to being regular PTFs. Here a "regular" PTF
is a PTF $\mathrm{sign}(p(x))$ where the influence of each variable on the
polynomial $p(x)$ is a small fraction of the total influence of $p.$
As an application of this regularity lemma, we prove that for any constants $d \geq 1, \epsilon > 0$, every degree$d$ PTF over $n$
variables can be approximated to accuracy $\epsilon$ by a constantdegree PTF that has integer weights of total magnitude $O_{\epsilon,d}(n^d).$
This weight bound is shown to be optimal up to logarithmic factors.

How Good is the Chord Algorithm?
[abstract]
[pdf]
C. Daskalakis, I. Diakonikolas, M. Yannakakis
Proceedings of the 21st Annual ACMSIAM Symposium on Discrete Algorithms (SODA 2010)
The Chord algorithm is a popular, simple method for the succinct approximation of curves,
which is widely used, under different names, in a variety of areas, such as,
multiobjective and parametric optimization, computational geometry, and graphics.
We analyze the performance of the Chord algorithm, as compared to the
optimal approximation that achieves a desired accuracy with the minimum number of points.
We prove sharp upper and lower bounds, both in the worst case and average case setting.

Bounded Independence Fools Halfspaces
[abstract]
[pdf]
I. Diakonikolas, P. Gopalan, R. Jaiswal, R. Servedio, E. Viola
SIAM Journal on Computing, 39(8), 34413462 (2010)
Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2009)
We show that any distribution on $\{1, 1\}^n$ that is $k$wise
independent fools any halfspace (a.k.a. linear threshold function) $h : \{1, 1\}^n \to \{1, 1\}$, i.e.,
any function of the form $h(x) = \mathrm{sign}(\sum_{i = 1}^n w_i x_i  \theta)$ where the $w_1,\ldots,w_n,\theta$ are arbitrary real
numbers, with error $\epsilon$ for $k = O(\epsilon^{2}\log^2(1/\epsilon))$. Our result is tight up
to $\log(1/\epsilon)$ factors. Using standard constructions of $k$wise independent distributions, we obtain the first
explicit pseudorandom generators $G : \{1, 1\}^s \to \{1, 1\}^n$ that fool halfspaces.
Specifically, we fool halfspaces with error $\epsilon$ and
seed length $s = k \cdot \log n = O(\log n \cdot \epsilon^{2} \log^2(1/\epsilon))$.
Our approach combines classical tools from real approximation theory with
structural results on halfspaces by Servedio (Comput. Complexity 2007).

Improved Approximation of Linear Threshold Functions
[abstract]
[pdf]
I. Diakonikolas, R. Servedio
Computational Complexity, 22(3), 623677 (2013)
Proceedings of the 24th Annual IEEE Conference on Computational Complexity (CCC 2009)
We prove two main results on how arbitrary linear threshold
functions $f(x) = \mathrm{sign}(w\cdot x  \theta)$ over the $n$dimensional
Boolean hypercube can be approximated by simple threshold functions.
Our first result shows that every $n$variable threshold function
$f$ is $\epsilon$close to a threshold function depending only on
$\mathrm{Inf}(f)^2 \cdot \mathrm{poly}(1/\epsilon)$ many variables, where $\mathrm{Inf}(f)$
denotes the total influence or average sensitivity of $f.$ This is
an exponential sharpening of Friedgut's wellknown theorem
\cite{Friedgut:98}, which states that every Boolean function $f$ is
$\epsilon$close to a function depending only on $2^{O(\mathrm{Inf}(f)/\epsilon)}$
many variables, for the case of threshold functions. We complement
this upper bound by showing that $\Omega(\mathrm{Inf}(f)^2 + 1/\epsilon^2)$
many variables are required for $\epsilon$approximating threshold
functions.
Our second result is a proof that every $n$variable threshold
function is $\epsilon$close to a threshold function with integer
weights at most $\mathrm{poly}(n) \cdot 2^{\tilde{O}(1/\epsilon^{2/3})}.$ This
is an improvement, in the dependence on the error
parameter $\epsilon$, on an earlier result of \cite{Servedio:07cc} which
gave a $\mathrm{poly}(n) \cdot 2^{\tilde{O}(1/\epsilon^{2})}$ bound. Our
improvement is obtained via a new proof technique that uses strong
anticoncentration bounds from probability theory. The new technique
also gives a simple and modular proof of the original
\cite{Servedio:07cc} result, and extends to give lowweight
approximators for threshold functions under a range of probability
distributions other than the uniform distribution.

Efficiently Testing Sparse $GF(2)$ Polynomials
[abstract]
[pdf]
I. Diakonikolas, H. Lee, K. Matulef, R. Servedio, A. Wan
Algorithmica, 61(3), 580605 (2011)
Proceedings of the 35th Intl. Colloquium on Automata, Languages and Programming (ICALP 2008)
We give the first algorithm that is both queryefficient and
timeefficient for testing whether an unknown function $f: \{0,1\}^n \to
\{0,1\}$ is an $s$sparse $GF(2)$ polynomial versus $\epsilon$far from
every such polynomial. Our algorithm makes $\mathrm{poly}(s,1/\epsilon)$
blackbox queries to $f$ and runs in time $n \cdot \mathrm{poly}(s,1/\epsilon)$.
The only previous algorithm for this testing problem \cite{DLM+:07}
used $\mathrm{poly}(s,1/\epsilon)$ queries, but had running time exponential in
$s$ and superpolynomial in $1/\epsilon$.
Our approach significantly extends the "testing by implicit
learning" methodology of \cite{DLM+:07}. The learning component of
that earlier work was a bruteforce exhaustive search over a concept
class to find a hypothesis consistent with a sample of random
examples. In this work, the learning component is a
sophisticated exact learning algorithm for sparse $GF(2)$
polynomials due to Schapire and Sellie \cite{SchapireSellie:96}. A
crucial element of this work, which enables us to simulate the
membership queries required by \cite{SchapireSellie:96}, is an
analysis establishing new properties of how sparse $GF(2)$
polynomials simplify under certain restrictions of "lowinfluence"
sets of variables.

Succinct Approximate Convex Pareto Curves
[abstract]
[pdf]
I. Diakonikolas, M. Yannakakis
Proceedings of the 19th Annual ACMSIAM Symposium on Discrete Algorithms (SODA 2008)
We study the succinct approximation of convex Pareto curves of multiobjective optimization problems.
We propose the concept of $\epsilon$convex Pareto ($\epsilon$CP) set as the appropriate one for the convex setting, and
observe that it can offer arbitrarily more compact representations than $\epsilon$Pareto sets in this context. We characterize
when an $\epsilon$CP can be constructed in polynomial time in terms of an efficient routine $\textrm{Comb}$ for optimizing
(exactly or approximately) monotone linear combinations of the objectives. We investigate the problem of computing minimum size
$\epsilon$convex Pareto sets, both for discrete (combinatorial) and continuous (convex) problems, and present general
algorithms using a $\textrm{Comb}$ routine. For biobjective problems, we show that if we have an exact $\textrm{Comb}$
optimization routine, then we can compute the minimum $\epsilon$CP for continuous problems (this applies for example to
biobjective Linear Programming and Markov Decision Processes), and factor 2 approximation to the minimum $\epsilon$CP for
discrete problems (this applies for example to biobjective versions of polynomialtime solvable combinatorial problems such as
Shortest Paths, Spanning Tree, etc.). If we have an approximate $\textrm{Comb}$ routine, then we can compute factor 3 and 6
approximations respectively to the minimum $\epsilon$CP for continuous and discrete biobjective problems.
We consider also the case of three and more objectives and present some upper and lower bounds.

Testing for Concise Representations
[abstract]
[pdf]
I. Diakonikolas, H. Lee, K. Matulef, K. Onak, R. Rubinfeld, R. Servedio, A. Wan
Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2007)
Oded Goldreich's Choices #39
We describe a general method for testing whether a function on $n$
input variables has a concise representation. The approach combines
ideas from the junta test of Fischer
et al. \cite{FKR+:04} with ideas
from learning theory, and yields property testers that make
poly$(s/\epsilon)$ queries (independent of $n$) for Boolean function
classes such as $s$term DNF formulas
(answering a question posed by Parnas
et al. \cite{PRS02}),
size$s$ decision trees, size$s$ Boolean formulas,
and size$s$ Boolean circuits.
The method can be applied to nonBoolean valued function classes as
well. This is achieved via a generalization of the notion of
variation from Fischer et al. to nonBoolean functions. Using
this generalization we extend the original junta test of Fischer
et al. to work for nonBoolean functions, and give
poly$(s/\epsilon)$query testing algorithms for nonBoolean valued
function classes such as size$s$ algebraic circuits and $s$sparse
polynomials over finite fields.
We also prove an $\tilde\Omega(\sqrt{s})$ query lower bound for
nonadaptively testing $s$sparse polynomials over finite fields of
constant size. This shows that in some instances, our general method
yields a property tester with query complexity that is optimal (for nonadaptive
algorithms) up to a polynomial factor.

Small Approximate Pareto Sets for Biobjective Shortest Paths and Other Problems
[abstract]
[pdf]
I. Diakonikolas, M. Yannakakis
SIAM Journal on Computing, 39(4), 13401371 (2009)
Proceedings of the 10th Intl. Workshop on Approximation, Randomization, and Combinatorial Optimization (APPROX 2007)
Honorable Mention, George Nicholson student paper competition
INFORMS society, 2009
We investigate the problem of computing a minimum set of solutions that approximates within a specified accuracy $\epsilon$ the
Pareto curve of a multiobjective optimization problem. We show that for a broad class of biobjective problems (containing
many important widely studied problems such as shortest paths, spanning tree, matching and many others), we can compute in
polynomial time an $\epsilon$Pareto set that contains at most twice as many solutions as the minimum such set. Furthermore we
show that the factor of $2$ is tight for these problems, i.e., it is NPhard to do better. We present upper and lower bounds
for three or more objectives, as well as for the dual problem of computing a specified number $k$ of solutions which provide a
good approximation to the Pareto curve.