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Improved Algorithms For Bipartite Network Flow
, 1994
"... In this paper, we study network flow algorithms for bipartite networks. A network G = (V; E) is called bipartite if its vertex set V can be partitioned into two subsets V 1 and V 2 such that all edges have one endpoint in V 1 and the other in V 2 . Let n = jV j, n 1 = jV 1 j, n 2 = jV 2 j, m = jE ..."
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Cited by 43 (5 self)
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In this paper, we study network flow algorithms for bipartite networks. A network G = (V; E) is called bipartite if its vertex set V can be partitioned into two subsets V 1 and V 2 such that all edges have one endpoint in V 1 and the other in V 2 . Let n = jV j, n 1 = jV 1 j, n 2 = jV 2 j, m = jEj and assume without loss of generality that n 1 n 2 . We call a bipartite network unbalanced if n 1 ø n 2 and balanced otherwise. (This notion is necessarily imprecise.) We show that several maximum flow algorithms can be substantially sped up when applied to unbalanced networks. The basic idea in these improvements is a twoedge push rule that allows us to "charge" most computation to vertices in V 1 , and hence develop algorithms whose running times depend on n 1 rather than n. For example, we show that the twoedge push version of Goldberg and Tarjan's FIFO preflow push algorithm runs in O(n 1 m + n 3 1 ) time and that the analogous version of Ahuja and Orlin's excess scaling algori...
Fast algorithms for parametric scheduling come from extensions to parametric maximum
, 1999
"... Chen [6] develops an attractive variant of the classical problem of preemptively scheduling independent jobs with release dates and due dates. Chen suggests that in practice one can often pay to reduce the processing requirement of a job. This leads to two parametric max flow problems. Serafini [26] ..."
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Cited by 15 (2 self)
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Chen [6] develops an attractive variant of the classical problem of preemptively scheduling independent jobs with release dates and due dates. Chen suggests that in practice one can often pay to reduce the processing requirement of a job. This leads to two parametric max flow problems. Serafini [26] considers scheduling independent jobs with due dates on multiple machines, where jobs can be split among machines so that pieces of a single job can execute in parallel. Minimizing the maximum tardiness again gives a parametric max flow problem. A third problem of this type is deciding how many more games a baseball team can lose partway through a season without being eliminated from finishing first (assuming a best possible distribution of wins and losses by other teams). A fourth such problem is an extended selection problem of Brumelle, Granot, and Liu [4], where we want to discount the costs of “treestructured ” tools as little as possible to be able to process all jobs at a profit. It is tempting to try to solve these problems with the parametric pushrelabel max flow methods of Gallo, Grigoriadis and Tarjan (GGT) [10]. However, all of these applications appear to violate the conditions necessary to apply GGT. We extend GGT in three ways which allow it to be applied to all four of the above applications. We also consider some other applications where these ideas apply. Our extensions to GGT yield faster algorithms for all these applications. 1
The Structure and Complexity of Sports Elimination Numbers
 Algorithmica
, 1999
"... Identifying the teams that are already eliminated from contention for first place of a sports league, is a classic problem that has been widely used to illustrate the application of linear programming and network flow. In the classic setting each game is played between two teams and the first place ..."
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Identifying the teams that are already eliminated from contention for first place of a sports league, is a classic problem that has been widely used to illustrate the application of linear programming and network flow. In the classic setting each game is played between two teams and the first place goes to the team with the greatest total wins. Recently, two papers [Way] and [AEHO] detailed a surprising structural fact in the classic setting: At any point in the season, there is a computable threshold W such that a team is eliminated (has no chance to win or tie for first place) if and only if it cannot win W or more games. They used this threshold to speed up the identification of eliminated teams. In both papers, the proofs of the existence the threshold are tied to the computational methods used to find it. In this paper we show that thresholds exist for a wide range of elimination problems (greatly generalizing the classical setting), via a simpler proof not connected to any partic...
OR on the ball: Applications in sports scheduling and management
 OR/MS Today
, 2004
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A New Property And A Faster Algorithm For Baseball Elimination
 SIAM Journal on Discrete Mathematics
, 1999
"... . In the baseball elimination problem, there is a league consisting of n teams. At some point during the season, team i has w i wins and g ij games left to play against team j.Ateamis eliminated if it cannot possibly finish the season in first place or tied for first place. The goal is to determine ..."
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. In the baseball elimination problem, there is a league consisting of n teams. At some point during the season, team i has w i wins and g ij games left to play against team j.Ateamis eliminated if it cannot possibly finish the season in first place or tied for first place. The goal is to determine exactly which teams are eliminated. The problem is not as easy as many sports writers would have you believe, in part because the answer depends not only on the number of games won and left to play, but also on the schedule of remaining games. In the 1960's, Schwartz showed how to determine whether one particular team is eliminated using a maximum flow computation. This paper indicates that the problem is not as di#cult as many mathematicians would have you believe. For each team i,letg i denote the number of games remaining. We prove that there exists a value W # such that team i is eliminated if and only if w i + g i <W # . Using this surprising fact, we can determine all eliminated team...
Baseball, Optimization and the World Wide Web
, 1999
"... The competition for baseball playoff spots  the fabled "pennant race"  is one of the most closelywatched American sports traditions. Baseball fans, known for their love of statistics, check newspapers and web sites daily looking for measures of their team's progress (or lack ther ..."
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The competition for baseball playoff spots  the fabled "pennant race"  is one of the most closelywatched American sports traditions. Baseball fans, known for their love of statistics, check newspapers and web sites daily looking for measures of their team's progress (or lack thereof !). While traditionallyreported playoff race statistics such as games back and "magic number" are informative, they are overly conservative and ignore the remaining schedule of games. By using optimization techniques, one can model schedule effects explicitly and determine when a team has locked up a playoff spot or is truly "mathematically eliminated" from contention. This paper describes the Baseball Playoff Races web site, a popular site developed at Berkeley that provides automatic daily updates of new, optimizationbased playoff race statistics. During the development of the site, it was found that the firstplace elimination status of all teams in a division can be determined using a single linear prog...
Mathematically Clinching a Playoff Spot in the NHL and the Effect of Scoring Systems
"... Abstract. A problem of intense interest to many sports fans as a season progresses is whether their favorite team has mathematically clinched a playoff spot; i.e., whether there is no possible scenario under which their team will not qualify. In this paper, we consider the problem of determining whe ..."
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Abstract. A problem of intense interest to many sports fans as a season progresses is whether their favorite team has mathematically clinched a playoff spot; i.e., whether there is no possible scenario under which their team will not qualify. In this paper, we consider the problem of determining when a National Hockey League (NHL) team has clinched a playoff spot. The problem is known to be NPComplete and current approaches are either heuristic, and therefore not always announced as early as possible, or are exact but do not scale up. In contrast, we present an approach based on constraint programming which is fast and exact. The keys to our approach are the introduction of dominance constraints and specialpurpose propagation algorithms. We experimentally evaluated our approach on the past two seasons of the NHL. Our method could show qualification before the results posted in the Globe and Mail, a widely read newspaper which uses a heuristic approach, and each instance was solved within seconds. Finally, we used our solver to examine the effect of scoring models on elimination dates. We found that the scoring model can affect the date of clinching on average by as much as two days and can result in different teams qualifying for the playoffs. 1
Sum decompositions of symmetric matrices
, 1994
"... Given a symmetric n X n matrix A and n numbers rl, " " r n, necessary and sufficient conditions for the existence of a matrix B, with a given zero pattern, with row sums r 1,..., r n, and such that A = B + BT are proven. If the pattern restriction ..."
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Given a symmetric n X n matrix A and n numbers rl, " " r n, necessary and sufficient conditions for the existence of a matrix B, with a given zero pattern, with row sums r 1,..., r n, and such that A = B + BT are proven. If the pattern restriction
A Hybrid Constraint Programming and Enumeration Approach for Solving NHL Playoff Qualification and Elimination Problems
"... Many sports fans invest a great deal of time into watching and analyzing the performance of their favorite team. However, the tools at their disposal are primarily heuristic or based on folk wisdom. We provide a concrete mechanism for calculating the minimum number of points needed to guarantee a pl ..."
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Many sports fans invest a great deal of time into watching and analyzing the performance of their favorite team. However, the tools at their disposal are primarily heuristic or based on folk wisdom. We provide a concrete mechanism for calculating the minimum number of points needed to guarantee a playoff spot and the minimum number of points needed to possibly qualify for a playoff spot in the National Hockey League (NHL). Our approach uses a combination of constraint programming, enumeration, network flows and decomposition to solve the problem efficiently. The technique can successfully be applied to any team at any point of the season to determine how well a team must do to make the playoffs.
Lessons Learned from Modelling the NHL Playoff Qualification Problem
"... Abstract. The modelling of complex problems tends to be most effective when modelling is calibrated using a concrete solver and modifications to the model are made as a result. In some cases, the final model is significantly different from the simple model that best fits the constraints of the probl ..."
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Abstract. The modelling of complex problems tends to be most effective when modelling is calibrated using a concrete solver and modifications to the model are made as a result. In some cases, the final model is significantly different from the simple model that best fits the constraints of the problem. While there are several projects that are attempting to create black box solvers with generic modelling languages, for the moment the modelling processes is intimately tied to the solving and search procedure. This paper looks at the changes to the simple model and the search procedures that were made to create an efficient solution to the NHL Playoff Qualification problem. The approaches for improving the model could be extended to other applications as the techniques are not specific to this problem. 1