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+\documentclass[11pt]{article}
+\usepackage{url}
+\usepackage[square,comma,numbers]{natbib}
+\usepackage{color} 
+
+% Figure stuff.  To add a new figure named `foo.fig', put
+% `\includexfig{foo}' in the appropriate place and add `foo' to the
+% figures line at the top of the Makefile.
+\usepackage{ifpdf}
+\ifpdf
+ \usepackage[pdftex]{graphicx}
+ \def\includexfig#1{\input{#1.pdf_t}}
+\else
+ \usepackage[dvips]{graphicx}
+ \def\includexfig#1{\input{#1.ps_t}}
+\fi
+
+\usepackage{amssymb, amsmath, amsthm}
+\newtheorem{theorem}{Theorem}
+\newtheorem{lemma}{Lemma}
+
+\usepackage[letterpaper,left=2.6cm,right=2.6cm,top=2.6cm,bottom=2.6cm]{geometry}
+
+\usepackage{floatflt}
+\usepackage{pstricks}
+\usepackage{delarray}
+
+\def\code#1{\texttt{#1}}
+
+\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 ``desirability''
+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 choose to use a single set of scores for both
+desirability and expertise. We believe that making this 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 $r$ to
+the sink, indicating that it needs $r$ 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 proposal $j$, 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 desirability value $d_{ij}$ stated by reviewer $i$ for proposal
+$j$.  For values on the NSF scale of 1 (best) to 40 (worst), we chose the cost
+function $(10 + d_{ij})^2$, 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 desirability
+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 desirability 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 $r = 3$;
+this would be easy to change for other values of $r$.
+
+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 desirability 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 (NOT YET):
+\[\hbox{\url{https://mattmccutchen.net/match/}}\]
+There are currently two branches:
+\begin{itemize}
+\item \code{master} has the tool as originally designed for NSF, with no
+distinction between desirability and expertise.
+\item \code{popl2012} is the basis of the version used for POPL 2012.  The main
+differences are that it has separate desirability 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}
