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 two-edge 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 two-edge 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...

. 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...

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...

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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 NP-Complete 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 special-purpose 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

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

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 * The research of all three authors was supported by the Funda~ao Calouste Gulbenkian, Lisboa. tThe research of this author was carried out within the activity of the Centro de Algebra da

In this paper, network flow algorithms for bipartite networks are studied. A network G = (V. E) is called bipartite if its vertex set V can be partitioned into two subsets V1 and V2 such that all edges have one endpoint in VY and the other in V2. Let n = I., nl = ll I1, n = IV21, m = El and assume without loss of generality that n s < n2. A bipartite network is called unbalanced if n << n and balanced otherwise. (This notion is necessarily imprecise.) It is shown that several maximum flow algorithms can be substantially sped up when applied to unbalanced networks. The basic idea in these improvements is a ntwo-edge push rule that allows one to &quot;charge&quot; most computation to vertices in VI, and hence develop algorithms whose running times depend on n rather than n. For example, it is shown that the two-edge push version of Goldberg and Tarjan's FIFO preflow-push algorithm runs in O(nlm + n3) time and that the analogous version of Ahuja and Orlin's excess scaling algorithm runs in O(n m + n log U) time, where U is the largest edge capacity. These ideas are also extended to dynamic tree implementations, parametric maximum flows, and minimum-cost flows.