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Serna Iglesias, María José

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Randomized parallel approximations to max flow

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Randomized Parallel Approximations to Max Flow, (1991; Serna,Spirakis)Maria Serna, Technical University of Catalonia, www.cs.upc.edu/˜mjsernaINDEX TERMS: Maximum flow, Maximum Weight Matching, Randomized algorithm, Ap-proximation, RNC class, FNCAS, PRAM algorithm.SYNONIMS: Approximate Maximum Flow construction.1 PROBLEM DEFINITIONThe work of Serna and Spirakis provides a parallel approximation schema for the Maximum flowproblem. An approximate algorithm provides a solution whose cost is within a factor of the optimalsolution. The notation and definitions are the standard ones for networks and flows (see for example[7], [2]).A network N = (G, s, t, c) is a structure consisting of a directed graph G = (V,E), two distin-guished vertices, s, t ∈ V (called the source and the sink), and c : E → Z+, an assignment of aninteger capacity to each edge in E. A flow function f is an assignment of a nonnegative numberto each edge of G (called the flow into the edge) such that first at no edge does the flow exceed thecapacity, and second for every vertex except s and t, the sum of the flows on its incoming edgesequals the sum of the flows on its outgoing edges. The total flow of a given flow function f isdefined as the net sum of flow into the sink t. The Maximum flow problem can be stated asName Maximum flowInput A network N = (G, s, t, c)Output Find a flow f for N for which the total flow is maximum.Maximum Flows and Matchings The Maximum Flow Problem is closely related to the Max-imum Matching problem on bipartite graphs.Given a graph G = (V,E) and a set of edges M ⊆ E is a matching if in the subgraph (V,M)all vertices have degree at most one. A maximum matching for G is a matching with maximumnumber of edges. For a graph G = (V,E) with weight w(e), the weight of a matching M is the sumof the weights of the edges in M . The problem can be stated as follows:Name Maximum Weight MatchingInput A graph G = (V,E) and a weight w(e) for each edge e ∈ EOutput Find a matching of G with the maximum possible weight.There is a standard reduction from the Maximum Matching problem for bipartite graphs to theMaximum Flow problem ([7], [8]). In the general weighted case one have just to look at each edgewith capacity c > 1 as c edges joining the same points each with capacity one, and transform themultigraph obtained as shown before. Notice that to perform this transformation it is required ac value which is polynomially bounded. The whole procedure was introduced by Karp, Upfal andWigderson [5] providing the following results1Theorem 1. The Maximum Matching problem for bipartite graphs is NC equivalent to the Maxi-mum Flow problem on networks with polynomial capacities. Therefore, Maximum Flow with poly-nomial capacities problem belongs to the class RNC.2 KEY RESULTSThe first contribution is an extension of Theorem 1 to a generalization of the problem, namely theMaximum Flow on networks with polynomially bounded maximum flow. The proof is based onthe construction (in NC) of a second network which has the same maximum flow but for which themaximum flow and the maximum capacity in the network are polynomially related.Lemma 2. Let N = (G, s, t, c). Given any integer k, there is an NC algorithm that decides whetherf(N) ≥ k or f(N) < km.Since Lemma 2 applies even to numbers that are exponential in size, they getLemma 3. Let N = (G, s, t, c) be a network. there is an NC algorithm that computes an integervalue k such that 2k ≤ f(N) < m 2k+1.The following lemma establishes the NC-reduction from the Maximum Flow problem withpolynomial maximum flow to the Maximum Flow problem with polynomial capacities.Lemma 4. Let N = (G, s, t, c) be a network, there is an NC algorithm that constructs a secondnetwork N1 = (G, s, t, c1) such thatlog(Max(N1)) ≤ log(f(N1)) +O(log n)and f(N) = f(N1).Lemma 4 shows that the Maximum Flow problem restricted to networks with polynomiallybounded maximum flow is NC-reducible to the Maximum Flow problem restricted to polynomiallybounded capacities, the latter problem is a simplification of the former one, so the following resultsfollows.Theorem 5. For each polynomial p, the problem of constructing a maximum flow in a network Nsuch that f(N) ≤ p(n) is NC-equivalent to the problem of constructing a maximum matching in abipartite graph, and thus it is in RNC.Recall that [5] gave us an O(log2 n) randomized parallel time algorithm to compute a maximummatching. The combination of this with the reduction from the Maximum Flow problem to theMaximum Matching leads to the following result.Theorem 6. There is a randomized parallel algorithm to construct a maximum flow in a directednetwork, such that the number of processors is bounded by a polynomial in the number of verticesand the time used is O((log n)α log f(N)) for some constant α > 0.The previous theorem is the first step towards finding a approximate maximum flow in a networkN by an RNC algorithm. The algorithm, given N and an  > 0, outputs a solution f ′ such thatf(N)/f ′ ≤ 1 + 1/. The algorithm uses a polynomial number of processors (independent of ) andparallel time O(logα n(log n+ log )), where α is independent of . Thus the algorithm is an RNCone as long as  is at most polynomial in n. (Actually  can be O(nlogβ n) for some β.) Thus beinga Fully RNC approximation scheme (FRNCAS).The second ingredient is a rough NC approximation to the Maximum Flow problem.2Lemma 7. Let N = (G, s, t, c) be a network. Let k ≥ 1 be an integer, then there is an NC algorithmto construct a network M = (G, s, t, c1) such that k f(M) ≤ f(N) ≤ k f(M) + km.Putting all together and allowing randomization the algorithm can be sketched as follows:FAST-FLOW(N = (G, s, t, c), )1. Compute k such that 2k ≤ F (N) ≤ 2k+1m.2. Construct a network N1 such thatlog(Max(N1)) ≤ log(F (N1)) +O(log n).3. If 2k ≤ (1 + )m then F (N) ≤ (1 + )m2 so use Algorithm ?? to solve the Maximum Flowproblem in N as a Maximum Matching and return4. Let β =⌊2k(1 + )m⌋. Construct N2 from N1 and β using the construction in Lemma 7.5. Solve the Maximum Flow problem in N2 as a Maximum Matching.6. Output F ′ = βF (M2) and for all e ∈ E, f ′(e) = βf(e).Theorem 8. Let N = (G, s, t, c) be a network. Then, algorithm FAST-FLOW is an RNC algorithmsuch that for all  > 0 at most polynomial in the number of network vertices, the algorithm computesa legal flow of value f ′ such thatf(N)f ′≤ 1 + 1.Furthermore, the algorithm uses a polynomial number of processors and runs in expected paralleltime O(logα n(log n+ log )), for some constant α, independent of .3 APPLICATIONSThe rounding/scaling technique is used in general to deal with problems that are hard due to thepresence of large weights in the problem instance. The technique modifies the problem instance inorder to produce a second instance that has no large weights, and thus can be solved efficiently. Theway in which a new instance is obtained consists in computing first an estimate of the optimal value(when needed) in order to discard unnecessary high weights. Then the weights are modified, scalingthem down by an appropriate factor that depends on the estimation and the allowed error. Therounding factor is determined in such a way that the so obtained instance can be solved efficiently.Finally a last step consisting in scaling up the value of the “easy” instance solution is performedin order to meet the corresponding accuracy requirements.It is known that in the sequential case, the only way to construct FPTAS uses rounding/scalingand interval partition [6]. In general, both techniques can be parallelized, although sometimes thedetails of the parallelization are non-trivial [1].The Maximum Flow problem has a long history in computer Science. Here are recorded someresults about its parallel complexity. Goldschlager, Shaw and Staples showed that the MaximumFlow problem is P-complete [3]. The P-completeness proof for Maximum Flow uses large capacitieson the edges; in fact the values of some capacities are exponential in the number of network vertices.If the capacities are constrained to be no greater than some polynomial in the number of networkvertices the problem is in ZNC. In the case of planar networks it is known that the Maximum Flowproblem is in NC, even if arbitrary capacities are allowed [4].34 OPEN PROBLEMSThe parallel complexity of the Maximum Weight Matching problem when the weight of the edgesare given in binary is still an open problem. However, as mentioned earlier, there is a randomizedNC algorithm to solve the problem in O(log2 n) parallel steps, when the weights of the edges aregiven in unary. The scaling technique has been used to obtain fully randomized NC approximationschemes, for the Maximum Flow and Maximum Weight Matching problems (see [10]). The resultappears to be the best possible in regard of full approximation, in the sense that the existence ofan FNCAS for any of the problems considered is equivalent to the existence of an NC algorithmfor perfect matching which is also still an open problem.5 EXPERIMENTAL RESULTSNone is reported.6 DATA SETSNone is reported.7 URL to CODENone is reported.8 CROSS REFERENCESMaximum Flow. Maximum Matching. Online Graph problems (Coloring, Matching). OnlineMatching.9 RECOMMENDED READING[1] J. Dı´az, M. Serna, P.G. Spirakis, and J. Tora´n Paradigms for fast parallel approximation.Cambridge International Series on Parallel Computation, 8. Cambridge University Press,Cambridge, 1997.[2] S. Even. Graph Algorithms. Computer Science Press, Potomac, Md, 1979.[3] L.M. Goldschlager, R.A. Shaw, and J. Staples. The maximum flow problem is log-spacecomplete for P. Theoretical Computer Science, 21:105–111, 1982.[4] D.B. Johnson and S.M. Venkatesan. Parallel algorithms for minimum cuts and maximumflows in planar networks. Journal of the ACM, 34:950–967, 1987.[5] R.M. Karp, E. Upfal, and A. Wigderson. Constructing a perfect matching is in Random NC.Combinatorica, 6:35–48, 1986.[6] B. Korte and R. Schrader. On the existence of fast approximation schemes. NonlinearProgramming, 4:415–437, 1980.[7] E.L. Lawler. Combinatorial Optimization: Networks and Matroids. Holt, Rinehart andWinston, NY, 1976.4[8] C. Papadimitriou. Computational Complexity. Addison-Wesley, Reading, Mass., 1994.[9] J.G. Peters and L. Rudolph. Parallel aproximation schemes for subset sum and knapsackproblems. Acta Informatica, 24:417–432, 1987.[10] P. Spirakis. PRAM models and fundamental parallel algorithm techniques: Part II. InA. Gibbons and P. Spirakis, editors, Lectures on Parallel Computation, pages 41–66. CambrigeUniversity Press, 1993.5

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