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\title{Assigning Papers to Reviewers}

\author{
Samir Khuller\thanks{Research currently supported by CCF-0728839.
Email:{\tt samir@cs.umd.edu}.}\\
Dept.\ of Computer Science\\
University of Maryland, College Park MD 20742.
\and
Richard Matthew McCutchen\thanks{
The bulk of this work was done while Matt was at the Dept.\ of Computer Science,
University of Maryland, College Park, and supported by REU Supplement to 
CCF-0430650. Current email: {\tt matt@mattmccutchen.net}.}\\
%Department of Computer Science\\
%University of Maryland, College Park, MD 20742.
}

\date{}

\begin{document}

\maketitle

%\begin{abstract}
%
%\end{abstract}

\section{Introduction}
Assignment problems arise in a variety of settings.  
For funding agencies such as NSF program directors 
that co-ordinate panels, assigning proposals to 
reviewers is a major challenge. It is important that each proposal receive
sufficient review by qualified experts, and at the same time we would like to
roughly balance the workload across reviewers and to honor the reviewers'
preferences for which proposals they would like to read. The same
issue arises for a program committee chair, who may have to assign
literally hundreds of papers to a program committee consisting of
thirty to forty program committee members. 

%{\em What does CMT use? What does Easychair use?}

From now on we will focus on the problem of assigning papers to
reviewers. 
We assume that each reviewer is given access to the
list of papers to be reviewed, and gives each paper both a ``preference''
score indicating his/her level of interest in reviewing the paper and an
``expertise'' score indicating how qualified he/she is to evaluate the paper.
(Some organizations may use a single preference score and assume that it
also indicates expertise. We believe that making the distinction may better
model the real-world objective.)
A reviewer may also declare a conflict of interest with a particular paper,
meaning that he/she is forbidden to review the paper.

We do not consider stable marriage type preference lists,
because a strict ranking of papers would be rather tedious
to produce. In this scheme, the papers are essentially grouped
into a few categories.

Let $N$ be the number of papers and $P$ be the number of reviewers.
Suppose that each paper needs $q$ reviews, so a total of $qN$
reviews need to be generated.
Ideally, from the perspective of the papers, we would like to
assign each paper the $q$ most qualified reviewers for the paper.
Of course, this could lead to a load imbalanced solution where
the load on some program committee members is very high, and the
load on others is low. On the other hand, we could insist 
on a perfectly load balanced solution in which the number
of papers assigned to each program committee member does not
exceed $L= \lceil qN/P  \rceil$. However, this may lead to a solution
which is not optimal from the perspective of the papers.

One of our goals is to study precisely this tradeoff, and allow each
reviewer to be assigned up to $L + C$ papers,
where $C$ is the {\em load tolerance}. We consider the question:
is it possible to obtain a high quality
assignment with a fairly low value of $C$? One can also ask whether,
in such an assignment, the reviewers receive the papers that they would
have most liked to review.

{\em Stinkers} are papers that pretty much no-one wanted to review. 
We would like to spread the load of the stinkers as evenly as possible.
 

\section{Formulation as a Min-Cost Max-Flow Problem}

\begin{figure*}
\begin{center}
\def\eg#1{\fbox{#1}}
\centerline{\includexfig{flow}}
\caption{Flow Construction.}
\label{flow-fig}
\end{center}
\end{figure*}

We formulate the assignment problem as a min-cost max-flow problem,
where each unit of flow represents one review.
The construction is somewhat involved in order to incorporate all
the desired incentives.  In several places, it makes
use of sets of ``parallel edges'' of different costs connecting a
single pair of nodes $(x, y)$ to allow flow to be sent from $x$ to $y$
at a cost that grows faster than linear in the amount of the flow.  For
example, if there are five unit-capacity edges from $x$ to $y$ of costs
1, 3, 5, 7, and 9, then any integer amount $f \le 5$ of flow can be
sent from $x$ to $y$ at a cost of $f^2$.

An example of the graph is shown in Figure~\ref{flow-fig}.
We have a source $s$ and a sink $t$.
For each paper $j$ we create a set of nodes $p^1_j, p^2_j,p^3_j$, and for each
reviewer $i$ we create a set of nodes $r^1_i, r^2_i, r^3_i$.  (The rationale
for these sets is discussed below.)
Flow can pass from $s$ through one or more of the nodes $r^t_i$ and
one or more of the nodes $p^t_j$ to the sink to represent a review
by reviewer $i$ of paper $j$.

Each paper has an edge of capacity $q$ to
the sink, indicating that it needs $q$ reviews.  In general, these
edges will constitute the min cut, so any max flow will saturate them
and thereby provide all the required reviews.  We take the min-cost
max flow in order to provide the reviews in the ``best'' possible way.

Each reviewer has a zero-cost edge of capacity $L$ from the source so that
he/she can be assigned $L$ papers.  If that were all, we would get a perfectly
load-balanced solution, but we may be able to improve the quality of the
assignments by allowing some imbalance.
Therefore, we allow each reviewer to be overloaded by up to $C$ papers
($C$ is the load imbalance parameter) via a set
of $C$ additional unit-capacity edges from the source.  We make the cost of the
$l$th edge an increasing function $f(l)$ to provide an incentive to
balance load across reviewers.  Since $2f(1) < f(1) + f(2)$,
a solution that loads two reviewers each by $L+1$ will be preferred
to a solution that loads one reviewer by $L$ and the other by $L+2$
unless the load imbalance in the second solution is outweighed by
other benefits.

For each reviewer $i$ and paper $j$, there is a unit-capacity edge from $i$
to $j$ allowing that pair to be assigned, unless the reviewer declared a
conflict of interest, in which case the edge is not present.  The edge cost is
based on the preference value $a_{ij}$ stated by reviewer $i$ for paper
$j$.  For values on the NSF scale of 1 (best) to 40 (worst), we chose the cost
function $(10 + a_{ij})^2$, in an attempt to provide an incentive to avoid
really bad matched pairs without completely masking the difference between a
good matched pair and an excellent one.  This choice seeks only to achieve a
natural relationship between a linear preference scale as normally interpreted
and the costs to be used in the optimization.  We realize that strategic
reviewers will take the cost function into account in choosing what preference
values to submit, in which case its form matters little.

Alongside these purely additive per-review costs,
we want to avoid an individual reviewer
getting too many papers he/she does not like.
With respect to a reviewer $i$, we classify papers as ``interesting'',
``boring'', or ``very boring'' based on their preference values;
the thresholds for these classes are currently
the same for all reviewers.  The edge for reviewer $i$ and paper
$j$ leaves from $r^1_i$ if $j$ is interesting, $r^2_i$ if $j$ is boring, or
$r^3_i$ if $j$ is very boring.  Since all edges from the source enter $r^1_i$,
flow for boring and very boring papers is forced to pass through a set of
parallel edges from $r^1_i$ to $r^2_i$, and flow for very boring papers
must pass through an additional set of parallel edges from $r^2_i$ to $r^3_i$.
In each of these sets, we make the cost of the $l$th edge an increasing
function of $l$ to provide an incentive to balance the load of boring
papers in the same way as for overload.

Finally, we would like at least one or two of each paper's reviews to be well
qualified, if possible.  The method is the same as that for reviewer fairness.
With respect to a paper $j$, we classify reviewers as ``expert'',
``knowledgeable'', or ``general'' by comparing the expertise values to uniform
thresholds.  (Since this is the only place the expertise values are used, we
effectively assume expertise takes on only these three values.)
Edges representing
expert reviews enter $p^1_j$, edges for knowledgeable reviews enter $p^2_j$,
and edges for general reviews enter $p^3_j$; the edge to the sink leaves
$p^3_j$.  A paper's first knowledgeable (or expert) review scores a bonus $c_2$
by traversing a unit-capacity edge of cost $-c_2$ from $p^2_j$ to $p^3_j$,
and an additional expert review scores another bonus $c_1$ by traversing a
unit-capacity edge of cost $-c_1$ from $p^1_j$ to $p^3_j$.
In addition to the bonus edges,
there are edges of zero cost and unlimited capacity that reviews can follow
from $p^1_j$ to $p^2_j$ and from $p^2_j$ to $p^3_j$ in order to reach the sink.
The choice to offer bonuses for two reviews was based on the value $q = 3$;
this would be easy to change for other values of $q$.

In the example in Figure~\ref{flow-fig},
paper 1 is interesting to reviewer 1 and boring to reviewers 2 and 3.
Reviewer 2 is expert on paper 1, with reviewers 1 and 3 merely knowledgeable.
(Reviewer edges for paper 2 are not shown.)
This illustrates how, in principle,
the preference and expertise relations might differ.
Each is taken into account at a different stage of the construction.

The cost of a flow (assignment) is the sum of its reviewer overload costs,
per-review costs, and reviewer boring / very boring load costs,
minus paper bonuses.  Any one of these components can be traded off against
the others.

\section{Evaluation}

We have implemented the matching algorithm, based on the construction above, in
Haskell.
The construction is intended to illustrate
ways to model real concerns that we find reasonable a priori.
We do not have enough experience with real-world
instances to be confident that each part of the construction serves its intended
purpose or that the parameter values we have chosen are suitable.
Fortunately, the parameter values are easy to change in the source code of our
implementation, and even substantive changes to the graph structure are not too
hard to make.  At a minimum, we believe that min-cost max-flow provides a
reasonable framework for global optimization of an assignment.

We have worked with Michael Hicks, program chair of POPL 2012, to apply our
method to the paper assignment problem for that conference.
The set of POPL 2012 reviewers consisted of the
program committee (PC) and an external review committee (ERC),
where the ERC served two purposes:
\begin{itemize}
\item To provide up to one knowledgeable (or expert) review per paper if
prior knowledge of the topic was hard to find on the PC.
\item To provide all reviews of papers with an author on the PC (``PC papers''),
which were considered to
pose an implicit conflict of interest to all PC members.
\end{itemize}
The special policy for ERC reviews of non-PC papers was realized via a more
complex paper-side gadget, not described here.
Based on an evaluation tool he wrote as well as manual inspection,
Dr.~Hicks was satisfied that the decisions made by the matching tool
closely reflected the best he could have done manually.

We are looking for additional opportunities to apply the matching tool.
Anyone interested is invited to contact us so we can help adapt it
to the scenario and document the experience gained.

%Waiting for data from NSF. 

%Synthetic Data.

\section{Getting the Tool}

A distribution containing the source code for the matching tool as well as this
document may be browsed or downloaded at:
\[\hbox{\url{https://mattmccutchen.net/match/}}\]
There are currently two branches:
\begin{itemize}
\item \code{master} has the tool as originally designed for NSF, with no
distinction between preference and expertise.
\item \code{popl2012} is the basis of the version used for POPL 2012.  The main
differences are that it has separate preference and expertise, support for
``fixing'' previously chosen reviewer-paper pairs (buggy, however),
and the special ERC gadget.
\end{itemize}
We regret that we do not currently have a single well-maintained version of the
tool to recommend.

\begin{thebibliography}{99}

\bibitem{Flow}
R. Ahuja, T. Magnanti and J. Orlin.
Network Flows: Theory and Applications.
{\em Prentice Hall}.



\end{thebibliography}

\end{document}