to t in the graph with capacities given by u e + i(e) for edges e is maximized. We further distinguish the problem MaxFlowImp according to the valid v

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1 Flow Improvement and Network Flows with Fixed Costs S. O. Krumke, Konrad-Zuse-Zentrum fur Informationstechnik Berlin H. Noltemeier, S. Schwarz, H.-C. Wirth, Universitat Wurzburg R. Ravi, Carnegie Mellon University, Pittsburgh PA Summary: We investigate the complexity and approximability of network ow improvement problems. In these problems, one incurs costs for increasing the capacity of an edge, while the goal is to achieve a ow of maximum value through the network. We study several improvement strategies. Furthermore, we investigate the relationship of network ow improvement problems to network xed cost ow problems, where one incurs a xed charge for each edge with nonzero ow which does not depend on the amount of ow sent over the edge. All xed cost problems studied in this paper are NP-hard to solve. We present various approximation algorithms for the problems under study. 1. Introduction Presented at OR'98 Zurich Minimum cost ows play an important role in operations research, They can be used to model a large number of transportation, logistics and communication network design problems. Usually, the ow costs are linear or convex, that is, one is charged c e (t) units of money if sending t units of ow over edge e, where each c e is a linear or convex function. However, linear or convex cost functions do not reect the problem nature appropriately in a number of applications. Often, there are either start-up costs or xed costs involved in production planning [GP90]. Consider for instance the problem of leasing communication lines in a network. A xed charge occurs for each link that is rented, no matter how much trac actually passes through this link. In this paper we investigate such network ow problems with xed costs, where a cost arises for each edge with nonzero ow and which does not depend on the amount of ow on that edge. We also investigate ow problems where a budget can be used to increase capacities in the network. Here, the goal is to improve the network such that the maximum ow with respect to the new capacities is maximized. 2. Problem Denition and Preliminaries Unless otherwise stated, by G = (V; E) we denote a directed graph with node set V and edge set E. We write n := jv j and m := jej. By s and t, we denote two distinguished nodes, the source and the sink, respectively. Denition 1 (MaxFlowImp) Let G = (V; E) be a graph. For each edge e 2 E, let u e 0 be its capacity, U e u e its maximum capacity, and b e be the cost of increasing its capacity be one unit. A valid improvement is a function i: E! Q 0 such that u e + i(e) U e for each edge e, and the total cost P e2e i(e)b e is bounded by budget value B. Find a valid improvement i, such that the ow from s supported by Deutsche Forschungsgemeinschaft (DFG), Grant NO 88/15-1

2 to t in the graph with capacities given by u e + i(e) for edges e is maximized. We further distinguish the problem MaxFlowImp according to the valid values of the improvement strategy. For Continuous-MaxFlowImp, improvement i(e) can take any rational number within the capacity constraints. For Integer-MaxFlowImp, the improvement is restricted to integer numbers. Finally, for 0/1-MaxFlowImp, improvement i(e) is restricted to the two values 0 and U e? u e. It turns out that Continuous-MaxFlowImp and even Integer-MaxFlowImp can be solved in polynomial time, while 0/1-MaxFlowImp is NP-hard. Furthermore, we will show the equivalence between 0/1-MaxFlowImp and the following problem: Denition 2 (MaxFlowFixedCost) Given a graph G = (V; E) with nonnegative capacities u e and nonnegative costs c e for the edges e 2 E, nd an edge subset A E of cost P e2a c e B, such that in (V; A) the ow from the source s to the sink t is maximized. This problem is a bicriteria optimization problem where the cost is constrained and the ow is maximized under that constraint. In this context a generic bicriteria maximization problem = (f; g;?) on a weighted graph is dened by specifying two polynomial time computable objectives, f and g and a membership requirement in a class of weighted subgraphs? (not necessarily weighted exactly the same way as the original graph). An instance of the problem species a budget value B as upper bound for the objective g. The goal is to nd a subgraph from the set f x 2? : g(x) B; i = 2; : : : ; k g having maximum possible value for f. Denition 3 A polynomial time algorithm for a bicriteria maximization problem = (f; g;?) is said to have performance (; ), if it has the following property: For any instance of the algorithm 1. either produces a solution x from the subgraph class? for which the value of objective g is at most times the specied budget and OPT f (x), where OPT is the minimum value of a solution from? that satises the budget constraint, or 2. correctly provides the information that there is no subgraph from? which satises the budget constraint on g. An approximation algorithm with performance (; 1) is usually also said to have performance and referred to as an -approximation algorithm. The dual problem 0 to the maximization problem = (f; g;?) is a minimization problem obtained by exchanging the two criteriae. The problem 0 consists of nding a subgraph from the set f x 2? : f(x) B 0 g having minimum possible value for g. In our particular case, the dual of MaxFlowFixedCost is dened as follows: Denition 4 (MinCostFixedFlow) Given a graph G = (V; E) with nonnegative capacities u e and nonnegative costs c e for the edges e 2 E, nd a minimum cost subset A E of the edges of G such that in (V; A) the ow from the source s to the sink t is at least F.

3 The denition of an (; )-approximation algorithm for the minimization problem 0 is similar to the one given in Denition 3. As shown in [MR + 95, Kru96] there is a tight relation between the complexity of a bicriteria problem and its dual 0 : The problem is solvable in polynomial time if and only if 0 is. Moreover, the problems are closely related with respect to their approximability. We refer to [RM + 93, MR + 95, Kru96] for a more detailed treatment of bicriteria problems. We show that MinCostFixedFlow is NP-hard even on series-parallel graphs. On the other hand, we provide a FPAS for that graph class. For general graphs, we provide a simple algorithm with approximation factor F, where F is the amount of ow to achieve. 3. Polynomial Time Solvable Flow Improvement The problem Continuous-MaxFlowImp can be solved optimally in polynomial time by the following easy extension of the well known Linear Programming formulation for the maximum ow problem. subject to maximize F 8 P (v;w)2e f(v; w)? P >< (u;v)2e f(u; v) = >: 0 f(e) u e + i(e) for all e 2 E F if v = s 0 for all v 2 V n fs; tg?f if v = t 0 u e + i(e) U e for all e 2 E P e2e i(e)b e B: Our main result of this section is to show that MaxFlowImp can also be solved eciently if we require the improvement strategy to be integral. Theorem 5 Integer-MaxFlowImp can be solved optimally in polynomial time by O(log(nU)) minimum cost ow computations on a graph with 2m edges, where is the maximum capacity occurring in the input. Proof: U := maxf U e : e 2 E g (1) The crux of the proof is to show that Integer-MaxFlowImp can be transformed in polynomial time into a budget-constrained minimum cost ow problem. Let i be an optimal improvement strategy and f be a corresponding maximal ow. For each edge e, since b e is nonnegative, we have i (e) = maxf0; f (e)? u e g; otherwise the strategy would waste money. We can model this behavior by a ow cost function c e dened as follows. As long as one sends ow along an edge e of value at most u e, there are no costs. But to send more ow along this edge, one has to pay b e units of money for each unit of ow exceeding the old capacity u e. This results in the following piecewise-linear ow cost function c e (e 2 E): c e (f) = 8 < : 0 for 0 f u e (2) b e (f? u e ) for u e <f U e

4 Although the cost functions c e are not linear, they have the nice property to be convex. We will show in the sequel how to exploit this fact. But rst note that our MaxFlowImp problem is equivalent to nding a maximum ow of cost at most B in the graph with upper capacities U e (e 2 E) and nonlinear ow cost functions c e as dened above. We call this equivalent problem a budget-constrained minimum cost ow problem. Our next goal is to relinearize the cost functions. To achieve this goal we use a method described in [AMO93]. Each edge e 2 E is replaced by two parallel edges e 0 and e 1. The capacities u e and linear ow costs c e of these edges are set as follows: u e0 := u e u e1 := U e? u e c e0 := 0 c e1 := b e (3) The validity of the transformation follows easily from the convexity of the ow cost functions c e. Let F be the maximum ow value achievable for a budget of B. The value F can be determined by a binary search which nds the largest integer F 2 [0; nu] such that there exists a ow of value F with costs at most B. This binary search needs O(log(nU)) minimum cost ow computations Approximate Solution Instead of performing a binary search on the interval [0; nu] we can search the interval only in multiplicative steps of 1 + ", where " > 0 is a xed accuracy parameter. More formally, we nd the maximum value F 0 2 f1; 1 + "; : : : ; (1 + ") k g; where k = dlog (1+") (nu)e; (4) such that there exists ow of value F 0 of cost at most B. The value F 0 found by this modied binary search satises F 0 F =(1 + "). We thus obtain the following theorem: Theorem 6 For any xed " > 0, a (1+")-approximation for Integer-MaxFlowImp can be found by O(log log 1+" (nu)) minimum cost ow computations on a graph with 2m edges Nonlinear Cost Functions It should be noted that the techniques presented above can be extended to the case when the original cost functions b e given in the specication of MaxFlowImp are piecewise-linear convex functions instead of linear functions. Then, again the cost functions c e constructed in the rst transformation are also convex piecewise-linear and in the nal step we can replace each edge by a set of parallel edges (each with linear cost function) where each of the parallel edges corresponds to one piece of the cost function c e. Thus, Theorem 5 and 6 carry over to the more general case of piecewise-linear convex cost functions b e. However, each of the minimum cost ow computations must now be carried out on a graph with O(gm) edges, where g is the maximum number of breakpoints occurring in the piecewise-linear cost functions.

5 4. Flow Improvement and its Relation to Flows with Fixed Costs In this section, we examine the remaining variant 0/1-MaxFlowImp of the problem under study. Since it is more convenient, we will deal with the dual version 0/1-MinImpFlow of the problem. As pointed out in the introduction, hardness and approximation results then carry over to 0/1- MaxFlowImp. Denition 7 (0/1-MinImpFlow) Given a graph G = (V; E) with edge capacities u e 0, maximum capacities U e u e, and capacity improvement costs b e, nd an improvement strategy i: E! f0; U e? u e g of minimum cost P e2e i(e) b e, such that the graph with edge capacities given by c e + i(e) admits a ow of value F from s to t. We rst show the equivalence of 0/1-MinImpFlow to MinCostFixedFlow. Theorem 8 0/1-MinImpFlow is equivalent to MinCostFixedFlow. Proof: Let I = (G; u; U; b; F ) be an instance of 0/1-MinImpFlow. We construct an instance I 0 = (G 0 ; u 0 ; c 0 ; F 0 ) for MinCostFixedFlow in the following way. To obtain G 0 from G, replace each edge e by two parallel edges e 0 1 and e 0 2. Set the capacities of the edges to u 0 (e 0 1) := u e and u(e 0 2) := U e? u e, and the costs to c 0 (e 0 1) := 0 and c 0 (e 0 2) := b e (U e? u e ), respectively, and let F 0 := F. Then a solution of I 0 for MinCostFixedFlow with cost B implies a solution of I for 0/1-MinImpFlow with the same cost. Conversely, let I 0 = (G 0 ; u 0 ; c 0 ; F 0 ) be given. Set G := G 0, F := F 0, u := 0. For each edge e 0, set U e := u 0 (e 0 ) and b e := c 0 (e 0 )=u 0 (e). Then, the equivalence is immediate. 2 As we will show in Section 6., MinCostFixedFlow is NP-hard. Hence we can not expect to nd any polynomial time algorithm which solves the problem to optimality. Thus in the next section we will concentrate on designing approximation algorithms for MinCostFixedFlow. Due to the results of Theorem 8, our approximation results for MinCostFixedFlow carry over to 0/1-MinImpFlow. 5. Approximation Algorithms for Flows with Fixed Costs 5.1 An FPAS for MinCostFixedFlow on series-parallel graphs A family fa " g " of approximation algorithms for a problem is called a fully polynomial approximation scheme or FPAS, if algorithm A " is a (1 + ")-approximation algorithm for and its running time is polynomial in the size of the input and 1=". In this section we rst present a pseudo-polynomial algorithm for MinCostFixedFlow on seriesparallel directed graphs. Then, we show how to convert this algorithm into a FPAS by scaling techniques similar to those given in [SK98]. First we recall the recursive denition of series-parallel graphs (cf. [BLW87]): Denition 9 (Series-Parallel Graph) The directed graph G with vertex set fa; bg and edge set f(a; b)g is series-parallel with terminals a and b. If G 1 = (V 1 ; E 1 ) and G 2 = (V 2 ; E 2 ) are seriesparallel graphs, with terminals a 1, b 1 and a 2, b 2, respectively, then

6 1. The graph obtained by identifying a 2 and b 1 is a series-parallel graph, with a 1 and b 2 as its terminals. This graph is the series composition of G 1 and G The graph obtained by identifying a 1 and a 2 and also b 1 and b 2 is a series-parallel graph, the parallel composition of G 1 and G 2. This graph has a 1 (= a 2 ) and b 1 (= b 2 ) as its terminals. In [VTL82] the authors present a linear time algorithm to decide whether a given digraph is seriesparallel, and if this is true, produce a parse tree (or decomposition tree) specifying how G is constructed using the above rules. The size of the parse tree is linear in the size of the input graph. The following lemma can be proven by an easy induction on the size of the parse tree of a given graph. Lemma 10 Let G = (V; E) be a series-parallel graph. Let s; t 2 V be two dierent vertices and denote by G 0 the graph obtained from G by removing all the vertices that are not reachable from s or from which one can not reach t. Then G 0 is series-parallel with terminals s and t. 2 In view of Lemma 10 we will assume in the sequel without loss of generality that for the given series-parallel graph G the two terminals coincide with the source s and the sink t between which the ow is to be maximized. Let C = max e2e c e be the maximum cost of an edge in the graph and let B 2 [0; mc] be an integral budget value allowed for the xed costs of a ow. The value B will act as \guess value" for the optimum cost in the nal algorithm. Notice that the optimum xed cost is an integer between 0 and mc. For 0 b B we dene C G (b) to be the maximum ow that can be achieved by using edges of total cost no more than b. In our algorithm we rst use the algorithm from [VTL82] to obtain a decomposition tree for the input graph G in time O(n + m). We then use dynamic programming and the decomposition tree to compute all the values C G (b), f = 0; : : : ; B in O(mB 2 ) time. Clearly, if G consists of just the two vertices s and t joined by an edge (s; t), we can trivially compute all the values C G (b) in O(B) time. On the other hand, if G is the series composition of G 1 and G 2, then for b = 0; : : : ; B we have Similarly, if G is the parallel composition of G 1 and G 2, then C G (b) = max 0ib minfc G 1 (i); C G2 (b? i)g: (5) C G (b) = max 0ib C G 1 (i) + C G2 (b? i): (6) Since the size of the parse tree for the series-parallel graph G was assumed to be O(m), the dynamic programming algorithm using the recurrences (5) and (6) terminates in O(mB 2 ) time having correctly computed all the values C G (b), b = 0; : : : ; B. By also keeping track of the respective edge sets we can also obtain the corresponding edge sets. Let A(G; u; c; B) be the algorithm from above that returns a set A of edges of cost c(a) at most B such that the ow in (V; A) with capacities given by u (restricted to A) is maximized. Let G, u, c and F be as specied for an instance of MinCostFixedFlow and let C again denote the maximum capacity. Let " > 0 be a given accuracy requirement. Now consider the following test

7 for a parameter M 2 [1; nc]: First we scale all edge costs in the graph by the factor M"=m, i.e., we set c M e := & ce M" m ' = mce : (7) M" We then run A(G; u; c M ; (1+1=")m). We call the test successful if the algorithm gives the information that the ow value F can be achieved by edges of cost at most (1 + 1=")m. Observe that running time of A(G; u; c M ; (1 + 1=")m) is O(m 3 (1 + 1=") 2 ). Denote by OPT the minimum xed cost in the original graph and let A be a corresponding edge set. We now show that the test is successful if M OPT. For such a value of M we have X X c M mce (e) M" + 1 m=" + ja j (1 + 1=")m: (8) e2a e2a Thus the edge set A is a feasible solution for the scaled instance yielding a ow of value F. Consequently, the test will be successful. We now use a binary search to nd the minimum integer M 0 2 [0; mc] such that the test described above succeeds. Our arguments from above show that the value M 0 found this way satises M 0 OPT. Let A 0 be the corresponding edge-set found by A(G; u; c M 0 ; (1 + 1=")m) which yields a ow of value at least F. Then X c e M 0 " m e2a 0 X e2a 0 c M 0 e M 0 " (1 + 1=")m (1 + ")OPT: (9) m Thus, the edge set A 0 found by our algorithm has cost at most 1 + " times the optimum cost. The running time of the algorithm can be bounded as follows: We run O(log mc) tests on scaled instances, each of which needs O(m 3 (1 + 1=")) time. Thus, the total running time is O(m 3 (1 + 1=") log mc), which is bounded by a polynomial in the input size and 1=". following theorem: We summarize our results in the Theorem 11 There is a FPAS for the problem MinCostFixedFlow when restricted to seriesparallel (directed) graphs. 2 By a similar proof, one can obtain the following result for the dual problem: Theorem 12 There is a FPAS for the problems MaxFlowFixedCost and 0/1-MaxFlowImp when restricted to series-parallel graphs An approximation for general graphs In this section we present an approximation algorithm for MinCostFixedFlow on general graphs with performance F. This algorithm works both for the directed as well as for the undirected case. The algorithm works as follows: Compute an integral minimum cost ow ' of value F in the graph G from s to t where the capacities u e of the edges are as given and the cost of one unit of ow over edge e is dened to be c e =u e. Let C ow = P e2e ce u e computed this way. '(e) denote the cost of the integral ow ': E! N

8 First notice that C ow OPT, that is, the ow cost is bounded above by the optimum cost of a set of edges that allow a ow of value F. We now dene the edge set A := f e 2 E : '(e) > 0 g. Then X e2a c e = X e2a c X e c X e c e u e F F '(e) = F C u e u e2a e u e2a e ow F OPT: (10) Thus, the set A has total cost at most F OPT. It remains to show that A is also feasible. For any cut (S) separating the source s and the sink t, we have X e2(s)\a u e X e2(s)\a since ' is a ow of value F. We thus have the following theorem: '(e) F; (11) Theorem 13 An approximation with performance F can be found in time O(m + T MCF ), where T MCF is the time needed to compute a minimum cost ow of value F in the input graph with capacities u e and per unit ow costs of c e =u e Hardness Results We rst show the hardness of MinCostFixedFlow even when restricted to the class of seriesparallel graphs. Notice that for the decision version of the dual problem, MaxFlowFixedCost, the NP-hardness on general graphs is already known [GJ79, Problem ND32]. Theorem 14 MinCostFixedFlow is NP-hard even on series-parallel graphs. Proof: We show the lemma by a reduction from Knapsack. An instance of Knapsack is given by a nite set A = fa 1 ; : : : ; a k g of items, each with weight w(a i ) 0 and value l(a i ) 0, and two integers W and L. It is NP-complete to decide whether there is a subset A 0 w(a 0 ) W and l(a 0 ) L [GJ79, Problem MP9]. A such that Given an instance of Knapsack, we construct a graph with vertex set fs; tg joined by jaj parallel edges. For item a i, edge e i minimum value L of the knapsack. has cost w(a i ) and capacity l(a i ). We set the ow constraint to the It is easy to see that the instance of Knapsack has a solution if and only if there is a solution of MinCostFixedFlow by a selection of edges of cost at most W. 2 Before stating the hardness result we recall the denition of the MinSetCover problem [GJ79, Problem SP5] and cite the hardness result from [Fei96] about the hardness of approximating Min- SetCover. An instance (M; F) of MinSetCover consists of a nite set M of ground elements, a family F of subsets of M 1. The objective is to nd a sub-collection C F of minimum size jcj which contains all the ground elements. Theorem 15 ([Fei96]) Unless NP DTIME(N O(log log N ) ), for any " > 0 there is no approximation algorithm for MinSetCover with a performance of (1? ") ln jmj, where M is the set of ground elements. 2 1 Without loss of generality we assume that each element of Q belongs to at least one subset in F.

9 Theorem 16 ([AS97]) There exists a constant > 0 such that, unless P = NP, there is no approximation algorithm for MinSetCover with a performance of ln jmj, where M is the set of ground elements. 2 Theorem 17 MinCostFixedFlow is strongly NP-hard even on bipartite graphs. Unless NP DTIME(N O(log log N ) ), for any " > 0 there is no approximation algorithm for MinCostFixedFlow on bipartite graphs with performance guarantee of (1? ") ln F, where F is the given ow value to be achieved. Proof: We show the theorem by providing an approximation preserving reduction from the Min- SetCover problem. Given an instance of MinSetCover, we rst construct the natural bipartite graph, one side of the partition for set nodes and the other for element nodes. We insert an edge (Q; q) if q 2 Q. All these edges have capacity 1 and zero costs. We now add a source node s and a sink node t to the graph. The source node is joined to all the set nodes via edges (s; Q) (Q 2 F). We set u (s;q) := jmj and c (s;q) := 1. For each element q there is an edge (q; t) from q to the sink with u (q;t) := 1 and c (q;t) := 0. Let us denote the resulting graph by G. Finally, we set the ow value F to be the size jmj of the ground set. Since the cost of any selection E 0 of edges is exactly the number of edges in E 0 emanating from the source, we can assume without loss of generality that each such set E 0 contains all zero-cost edges between sets and elements and the elements and the sink. It is now easy to see that the sets f Q : (s; Q) 2 E 0 g form a cover of M if and only if the ow in (V; E 0 ) has value jmj. Thus, a set cover of size K transforms into a feasible solution for Min- CostFixedFlow of the same cost and vice versa. 2 Corresponding Author: wirth,noltemeier,schwarz@informatik.uni-wuerzburg.de, krumke@zib.de, ravi+@cmu.edu References [AMO93] R. K. Ahuja, T. L. Magnanti, and J. B. Orlin, Networks ows, Prentice Hall, Englewood Clis, New Jersey, [AS97] S. Arora and M. Sudan, Improved low-degree testing and its applications, Proceedings of the 29th Annual ACM Symposium on the Theory of Computing (STOC'97), 1997, pp. 485{496. [BLW87] M. W. Bern, E. L. Lawler, and A. L. Wong, Linear-time computation of optimal subgraphs of decomposable graphs, Journal of Algorithms 8 (1987), 216{235. [Fei96] U. Feige, A threshold of ln n for approximating set cover, Proceedings of the 28th Annual ACM Symposium on the Theory of Computing (STOC'96), 1996, pp. 314{318. [GJ79] M. R. Garey and D. S. Johnson, Computers and intractability (a guide to the theory of NP-completeness), W.H. Freeman and Company, New York, 1979.

10 [GP90] G. M. Guisewite and P. M. Pardalos, Minimum concave-cost network ow problems: Applications, complexity and algorithms, Annals of Operations Research 25 (1990), 75{100. [Kru96] S. O. Krumke, On the approximability of location and network design problems, Ph.D. thesis, Lehrstuhl fur Informatik I, Universitat Wurzburg, December [MR + 95] M. V. Marathe, R. Ravi, R. Sundaram, S. S. Ravi, D. J. Rosenkrantz, and H. B. Hunt III, Bicriteria network design problems, Proceedings of the 22nd International Colloquium on Automata, Languages and Programming (ICALP'95), Lecture Notes in Computer Science, vol. 944, 1995, pp. 487{498. [RM + 93] R. Ravi, M. V. Marathe, S. S. Ravi, D. J. Rosenkrantz, and H. B. Hunt III, Many birds with one stone: Multi-objective approximation algorithms, Proceedings of the 25th Annual ACM Symposium on the Theory of Computing (STOC'93), May 1993, pp. 438{447. [SK98] S. Schwarz and S. O. Krumke, On budget constrained ow improvement, Information Processing Letters 66 (1998), 291{297. [VTL82] J. Valdes, R. E. Tarjan, and E. L. Lawler, The recognition of series-parallel digraphs, SIAM Journal on Computing 11 (1982), no. 2, 298{313.