Approximate Binary Search Algorithms for Mean Cuts and Cycles

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1 Approximate Binary Search Algorithms for Mean Cuts and Cycles S. Thomas McCormick Faculty of Commerce and Business Administration University of British Columbia Vancouver, BC V6T 1Z2 Canada June 1992, Revised July 1992, June 1993 UBC Faculty of Commerce Working Paper 92-MSC-021 Keywords: network flow, max mean cut, min mean cycle, approximate binary search Abstract Iwano, et al. [8] have given an approximate binary search algorithm for computing max mean cuts. This paper gives a short proof of the correctness of their algorithm. It also shows how their algorithm can be dualized to an approximate binary search algorithm for computing min mean cycles that is as fast as but simpler than Orlin and Ahuja s algorithm [12]. 1 Introduction In [8] Iwano, et al. derive an approximate binary search algorithm for computing max mean cuts. Max mean cuts were introduced by Ervolina and McCormick [3], and their applications can be found there. See [11, 13, 14] for other methods for computing them. The purpose of this note is to give a short, easy proof that Iwano et al. s algorithm is correct, and to show how their algorithm can be dualized to an approximate binary search algorithm for min mean cycles; see [5] for the significance of min mean cycles. See [9, 10] for other methods for computing them. This approximate binary search algorithm for min mean cycles is as fast as but is simpler than Orlin and Ahuja s [12] approximate binary search algorithm. It uses Goldberg s [4] new algorithm for shortest paths with small negative lengths as a subroutine. We are given a directed network D = (N, A) with lower bounds l, upper bounds u, and costs c on its arcs. The largest absolute arc bound is denoted by B, the largest absolute arc cost by C. A cut is given by a non-trivial partition of the nodes as N = S T, and we define (S, T ) = { i j i S, j T }, and { S, T } = (S, T ) (T, S). The value of a cut is V (S, T ) = l ij u ij, and the mean value of (S, T ) is V (S, T )/ { S, T } (we assume that D is connected so this definition makes sense). The cost of a cycle Q is c(q) = c ij c ij, i j forward in Q i j backward in Q This research was partially supported by an NSERC Operating Grant. 1

2 and the mean cost of Q is c(q)/ Q, where Q is the number of arcs in Q. Assuming that there is some positive cut (a cut with V (S, T ) > 0), the maximum mean cut problem is to find a cut with maximum mean value. We denote the mean value of a max mean cut by V. Assuming that there is some negative cycle (a cycle with c(q) < 0), the minimum mean cycle problem is to find a cycle with minimum mean cost. We denote the mean cost of a min mean cycle by c. We recall that the existence of a positive cut can be checked with one max flow feasibility computation; we denote the running time of max flow by MF (m, n). If there are no positive cuts, then the feasible flow x satisfying conservation ( j x ij = k x ki for all i) proves it (this is Hoffman s Circulation Theorem [7]). We call an x satisfying conservation (but not necessarily the bounds) a circulation. The existence of a negative cycle can be checked with one shortest path computation (on a network with negative costs allowed); we denote the running time of such a shortest path algorithm by SP (m, n). If there are no negative cycles, the optimal shortest path potentials π which satisfy π i + c ij π j for all i j prove it. The basic tools used in computing max mean cuts and min mean cycles are the following two theorems. The first refers to the network D(l δ, u + δ, c) with bounds l ij δ, u ij + δ on each arc i j which depend on scalar parameter δ 0. The second refers to the network D(l, u, c + ɛ) with cost c ij + ɛ on each arc i j which depends on scalar parameter ɛ 0. Theorem 1.1 (Ervolina and McCormick [3], Hassin [6]) Assuming that there is some positive cut, V equals the minimum δ such that there are no positive cuts in D(l δ, u + δ, c). Theorem 1.2 (Goldberg and Tarjan [5]) Assuming that there is some negative cycle, c equals the minimum ɛ such that there are no negative cycles in D(l, u, c + ɛ). These theorems immediately suggest search algorithms: We can determine if there is a positive cut in D(l δ, u+δ, c) in MF (m, n) time, and if there is one then we know that δ < V, else we know that δ V. Similarly, it costs SP (m, n) time to check D(l, u, c + ɛ) for negative cycles, and if there is one then we know that ɛ < c, else we know that ɛ c. Our assumptions tell us that V 0 and c 0, and we must have that V B and c C. With integral data, since each mean value (resp. cost) is a ratio of integers with denominator at most m (n), the difference between the unequal mean values (costs) of any two cuts (cycles) is at least 1/m 2 (1/n 2 ), so binary search gives us the correct mean value (cost) in O(log(mB)) = O(log(nB)) iterations, and O(log(nB)M F (m, n)) total time (O(log(nC)) iterations, and O(log(nC)SP (m, n)) total time). 2 Approximate Binary Search for Max Mean Cuts The idea of approximate binary search (due to Zemel [15]) is to use a rougher computation that can still cut down the interval of uncertainty at each iteration by a constant factor (but one which is less than two), thus retaining the O(log(nB)) (O(log(nC)) bound on iterations. However, the rougher computation can be done faster than an exact computation, leading to an overall faster algorithm. Iwano, et al. s approximate binary search algorithm maintains an interval (δ L, δ U ] known to contain V (it initially chooses δ L = 0 and δ U = B). We know that V δ U because we keep a flow x that is feasible in D(l δ U, u + δ U, c) to verify that D(l δ U, u + δ U, c) has no positive cuts. We know that V > δ L because we have found that D(l δ L, u + δ L, c) is infeasible. Define = (δ U δ L )/3 ( for a third), δ 1/3 = δ L +, and δ 2/3 = δ L +2. Suppose that we want to update x by integral multiples of to x = x + x such that x proves that D(l δ 2/3, u + δ 2/3, c) is 2

3 feasible. Since x proves that D(l δ 2/3, u + δ 2/3, c) is feasible if l ij δ 2/3 x ij + x ij u ij + δ 2/3, or l ij δ 2/3 x ij x ij u ij + δ 2/3 x ij, and since x is constrained to be integral, x must be feasible for the modified bounds l ij lij δ 2/3 x ij = and u uij + δ 2/3 x ij ij =. Denote the network with bounds l and u by D. The key lemma which shows that Iwano, et al. s algorithm works is the following. This proof is much simpler and shorter than the one in [8]. Lemma 2.1 If D is feasible with flow x, then x = x + x is feasible in D(l δ 2/3, u + δ 2/3, c). Conversely, if D has positive cut (S, T ), then (S, T ) is also positive for D(l δ 1/3, u + δ 1/3, c). Proof: The first part is clear from the definitions of l and u. For the second part, (S, T ) positive in D means that lij δ 2/3 x ij > uij + δ 2/3 x ij. Now (1/) (l ij δ 1/3 x ij ) = ( ) lij δ 2/3 x ij + 1 > lij δ 2/3 x ij > uij + δ 2/3 x ij > ( ) uij + δ 2/3 x ij 1 = (1/) (u ij + δ 1/3 x ij ). (1) Multiplying the first and last terms of (1) by, and using the fact that x a circulation implies that x ij = x ij, we get (l ij δ 1/3 ) > (u ij + δ 1/3 ). Thus (S, T ) is also positive for D(l δ 1/3, u + δ 1/3, c). Now we can use Lemma 2.1 to construct an approximate binary search algorithm for computing max mean cuts. At each iteration we use one max flow to check feasibility of D. If D is feasible, then we can replace δ U by δ 2/3 and x by the feasible flow; otherwise, we can replace δ L by δ 1/3. Since the length of the δ L, δ U interval is decreased by one third in each iteration, this algorithm enjoys the same O(log(nB)) iteration bound as plain binary search. To make approximate binary search faster than plain binary search we need to be able to decide if D is feasible with something faster than a general max flow algorithm. Note that since x is feasible 3

4 in D(l δ U, u + δ U, c), l and u are integers satisfying l ij +1 and u ij 1 for all i j. If we choose the initial flow x 0 defined by { 1 if u ij = 1, x 0 ij = +1 if l ij = +1, 0 otherwise (this x 0 satisfies the bounds l, u but might violate conservation), the sum of the positive excesses w.r.t. x 0 is at most m. Thus the max flow routine can push a maximum of O(m) units of flow. We can use Dinic s max flow algorithm [2] to solve this max flow problem; since Dinic s algorithm runs in time O(mn + nf ), where f is the maximum flow value, we can decide if D is feasible in only O(mn) time (the use of Dinic s algorithm was suggested by Jim Orlin). This is indeed a speed-up since O(m, n) < MF (m, n) for the current fastest max flow algorithms [1]. We have shown Theorem 2.2 Approximate binary search solves the maximum mean cut problem in O(mn log(nb)) time. Unless B is very large, this algorithm has the fastest asymptotic running time among known max mean cut algorithms, although an algorithm of Radzik [13, 14] has the same complexity. However, Radzik s algorithm is more difficult to explain than the present one. 3 Approximate Binary Search for Min Mean Cycles We can apply exactly the same ideas to computing min mean cycles. Suppose that we have ɛ U and π such that D(l, u, c + ɛ U ) has no negative cycles, as proven by π; and we have an ɛ L such that D(l, u, c+ɛ L ) does have a negative cycle. Set = (ɛ U ɛ L )/3 and ɛ 1/3 = ɛ L +, ɛ 2/3 = ɛ L +2. If we want to update π by integer multiples of to π = π + π such that π verifies that D(l, u, c+ɛ 2/3 ) has no negative cycles, then π must satisfy π i + c ij π j, where c πi + c ij π j + ɛ 2/3 ij =. Finding such a π is just a shortest path problem on the network D with costs c. Then essentially the same proof as Lemma 2.1 shows that Theorem 3.1 If D has no negative cycles as proven by π, then π = π + π proves that D(l, u, c + ɛ 2/3 ) also has no negative cycles. Conversely, if D has a negative cycle Q, then Q is a negative cycle in D(l, u, c + ɛ 1/3 ). Just as before, we use one shortest path computation to check D for negative cycles. If we find a negative cycle we can replace ɛ L by ɛ 1/3 ; otherwise we can replace ɛ U by ɛ 2/3 and π by the potentials proving that D has no negative cycles. This reduces the length of the interval containing c by one third at each iteration, so we get a bound of O(log(nC)) on the number of iterations just as in plain binary search. As before we need to compute shortest paths in D faster than general shortest path. Since π proves that D(l, u, c+ɛ U ) has no negative cycles, the c are integers satisfying c ij 1 for all i j. Recently, Goldberg [4] came up with a new shortest path algorithm for problems with integral data whose running time is O( nm log(m + 1)), where M is the absolute value of the most negative arc cost. Since M = 1 for the D shortest path problem, Goldberg s algorithm solves the D problem in O( nm) time. This is indeed a speed-up since O( nm) < SP (m, n) for the current fastest shortest path algorithms [1]. This proves that: 4

5 Theorem 3.2 Approximate binary search solves the minimum mean cycle problem in O( nm log(nc)) time. Unless C is very large, this algorithm has the fastest asymptotic running time among known min mean cycle algorithms, although an algorithm in Orlin and Ahuja [12] has the same complexity. However, they embed the min mean cycle problem in a scaled assignment problem, which leads to a more difficult exposition than in the present paper. References [1] Ahuja, R.K., T.L. Magnanti, and J.B. Orlin (1988). Network Flows. Chapter IV of Handbooks in Operations Research and Management Science, Volume 1: Optimization, eds. G.L. Nemhauser, A.H.G. Rinnooy Kan and M.J. Todd, North Holland, pp [2] Dinic, E.A. (1970). Algorithm for Solution of a Problem of Maximum Flow in Networks with Power Estimation. Soviet Math. Dokl. 11, pp [3] Ervolina, T.R. and S.T. McCormick (1991). Two Strongly Polynomial Cut Cancelling Algorithms for Minimum Cost Network Flow. To appear in Discrete Applied Mathematics. [4] Goldberg, A.V. (1992). Scaling Algorithms for the Shortest Paths Problem. Stanford University Computer Science technical report STAN-CS , Stanford, CA. [5] Goldberg, A.V., and R.E. Tarjan (1989). Finding Minimum-Cost Circulations by Canceling Negative Cycles. JACM 33, no. 4, pp [6] Hassin, R. (1991). Algorithms For the Minimum Cost Circulation Problem Based on Maximizing The Mean Improvement. Tel Aviv University Statistics Department Working Paper, Tel Aviv, Israel. [7] Hoffman, A.J. (1960). Some Recent Applications of the Theory of Linear Inequalities to Extremal Combinatorial Analysis. In Bellman, R. and Hall, Jr., M. (eds.) Proceedings of Symposia in Applied Mathematics, Vol. X, Combinatorial Analysis. American Mathematical Society, Providence, RI, pp [8] Iwano, K., S. Misono, S. Tezuka, and S. Fujishige (1990). A New Scaling Algorithm for the Maximum Mean Cut Problem. IBM Research Report RT 0049, Tokyo, Japan; to appear in Algorithmica. [9] Karp, R.M. (1978). A Characterization of the Minimum Cycle Mean in a Digraph. Discrete Math. 23, pp [10] Karp, R.M. and J.B. Orlin (1981). Parametric Shortest Path Algorithms with an Application to Cyclic Staffing. Discrete App. Math. 3, pp [11] McCormick, S. T. and T. R. Ervolina (1990). Computing Maximum Mean Cuts. UBC Faculty of Commerce Working Paper 90-MSC-011, Vancouver, BC. [12] Orlin, J.B. and R.K. Ahuja (1992). New Scaling Algorithms for the Assignment and Minimum Cycle Mean Problems. Mathematical Programming 54, pp [13] Radzik, T. (1992). Minimizing Capacity Violations in a Transshipment Network. Proceedings of the Third Annual ACM-SIAM Symposium on Discrete Algorithms, pp

6 [14] Radzik, T. (1992). Newton s Method for Fractional Combinatorial Optimization. Proc. 33rd IEEE Annual Symp. of Foundations of Computer Science, pp [15] Zemel, E. (1987). A Linear Time Algorithm for Searching Ranked Functions. Algorithmica 2, pp