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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}
+
+\title{Assigning Reviewers to Proposals}
+
+\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 report 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 $r$ reviews, so a total of $rN$
+reviews need to be generated.
+Ideally, from the perspective of the papers, we would like to
+assign each paper the $r$ 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 $L= \lceil rN/P \rceil$. 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 Flow Problem}
+
+Our main approach is to formulate this as a min-cost flow problem.
+The construction is somewhat involved in order to incorporate all
+the appropriate incentives. 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 $f^2$.
+
+The construction is done as follows: 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.) See Figure~\ref{flow-fig} for an
+example. 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 assignment to be made, unless the reviewer declared a
+conflict of interest, in which case the edge is not present. The edge has
+cost $(10 + p_{ij})^2$, where $p_{ij}$ is the preference value
+expressed by reviewer $i$ for proposal $j$.
+We assume $p_{ij}$ is a value between 1 and 40 (as used by NSF).
+The cost function was chosen to provide an incentive to avoid really bad
+assignments without completely masking the difference between a good assignment
+and an excellent assignment.
+
+These purely additive assignment costs make the algorithm prefer better
+assignments in general but do nothing to promote fairness among reviewers or
+among papers. To do that, we introduce additional reviewer-side costs and
+paper-side bonuses. With respect to a
+reviewer $i$, we classify papers as interesting (preference value 1 to 15), boring
+(16 to 25), or very boring (26 to 40). 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 assignments is forced to pass through a set of
+parallel edges from $r^1_i$ to $r^2_i$, and flow for very boring assignments
+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
+assignments in the same way as for overload.
+
+Similarly, we wish to guarantee each paper at least one or two good reviews.
+With respect to a paper $j$, we classify reviewers as expert (preference value
+1 to 10), knowledgeable (11 to 20), or general (21 to 40). 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 $r$.
+
+The cost of a flow is the sum of its reviewer overload costs,
+assignment costs, and reviewer boring / very boring load costs,
+minus paper bonuses. Any one of those components can be traded off against
+the others. We attempted to assign reasonable weights to each component,
+but it is difficult to know without testing the algorithm on real data.
+In any event, all the parameters are easy to tune to realize the priorities
+of a particular application.
+
+\begin{figure*}
+\begin{center}
+\centerline{\includexfig{flow}}
+\caption{Flow Construction.}
+\label{flow-fig}
+\end{center}
+\end{figure*}
+
+
+\section{Experimental Results}
+Waiting for data from NSF.
+
+Synthetic Data.
+
+\section{Conclusions}
+
+The source code for the current version of the proposal matcher may be
+browsed or downloaded at:
+\[\hbox{\url{TODO}}\]
+
+\begin{thebibliography}{99}
+
+\bibitem{Flow}
+R. Ahuja, T. Magnanti and J. Orlin.
+Network Flows: Theory and Applications.
+{\em Prentice Hall}.
+
+
+
+\end{thebibliography}
+
+\end{document}
Matt McCutchen's Web Site / match / match.git / blobdiff