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
|
|
- Oswin Hodges
- 5 years ago
- Views:
Transcription
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.
On the Complexity of Budgeted Maximum Path Coverage on Trees
On the Complexity of Budgeted Maximum Path Coverage on Trees H.-C. Wirth An instance of the budgeted maximum coverage problem is given by a set of weighted ground elements and a cost weighted family of
More informationMulticriteria approximation through decomposition
Multicriteria approximation through decomposition Carl Burch Sven Krumke y Madhav Marathe z Cynthia Phillips x Eric Sundberg { Abstract We propose a general technique called solution decomposition to devise
More informationThe Maximum Flow Problem with Disjunctive Constraints
The Maximum Flow Problem with Disjunctive Constraints Ulrich Pferschy Joachim Schauer Abstract We study the maximum flow problem subject to binary disjunctive constraints in a directed graph: A negative
More informationApproximation Algorithms for Maximum. Coverage and Max Cut with Given Sizes of. Parts? A. A. Ageev and M. I. Sviridenko
Approximation Algorithms for Maximum Coverage and Max Cut with Given Sizes of Parts? A. A. Ageev and M. I. Sviridenko Sobolev Institute of Mathematics pr. Koptyuga 4, 630090, Novosibirsk, Russia fageev,svirg@math.nsc.ru
More informationRestricted b-matchings in degree-bounded graphs
Egerváry Research Group on Combinatorial Optimization Technical reports TR-009-1. Published by the Egerváry Research Group, Pázmány P. sétány 1/C, H1117, Budapest, Hungary. Web site: www.cs.elte.hu/egres.
More informationThe Complexity of Maximum. Matroid-Greedoid Intersection and. Weighted Greedoid Maximization
Department of Computer Science Series of Publications C Report C-2004-2 The Complexity of Maximum Matroid-Greedoid Intersection and Weighted Greedoid Maximization Taneli Mielikäinen Esko Ukkonen University
More informationMulti-criteria approximation schemes for the resource constrained shortest path problem
Noname manuscript No. (will be inserted by the editor) Multi-criteria approximation schemes for the resource constrained shortest path problem Markó Horváth Tamás Kis Received: date / Accepted: date Abstract
More informationOn-line Bin-Stretching. Yossi Azar y Oded Regev z. Abstract. We are given a sequence of items that can be packed into m unit size bins.
On-line Bin-Stretching Yossi Azar y Oded Regev z Abstract We are given a sequence of items that can be packed into m unit size bins. In the classical bin packing problem we x the size of the bins and try
More informationUpper and Lower Bounds on the Number of Faults. a System Can Withstand Without Repairs. Cambridge, MA 02139
Upper and Lower Bounds on the Number of Faults a System Can Withstand Without Repairs Michel Goemans y Nancy Lynch z Isaac Saias x Laboratory for Computer Science Massachusetts Institute of Technology
More informationIBM Almaden Research Center, 650 Harry Road, School of Mathematical Sciences, Tel Aviv University, TelAviv, Israel
On the Complexity of Some Geometric Problems in Unbounded Dimension NIMROD MEGIDDO IBM Almaden Research Center, 650 Harry Road, San Jose, California 95120-6099, and School of Mathematical Sciences, Tel
More informationApproximation Basics
Approximation Basics, Concepts, and Examples Xiaofeng Gao Department of Computer Science and Engineering Shanghai Jiao Tong University, P.R.China Fall 2012 Special thanks is given to Dr. Guoqiang Li for
More informationThe Constrained Minimum Weighted Sum of Job Completion Times Problem 1
The Constrained Minimum Weighted Sum of Job Completion Times Problem 1 Asaf Levin 2 and Gerhard J. Woeginger 34 Abstract We consider the problem of minimizing the weighted sum of job completion times on
More informationVariants of MAX WEIGHTED SAT and MIN WEIGHTED SAT have been considered in the literature, dened by making restrictions on the size of the clauses in t
On the Complexity of Approximating Weighted Satisability Problems (Extended Abstract) Paola Alimonti yzx Giorgio Ausiello y Loredana Giovaniello z Marco Protasi z Abstract The maximum weighted satisability
More informationMaximum Skew-Symmetric Flows. September Abstract
Maximum Skew-Symmetric Flows Andrew V. Goldberg NEC Research Institute 4 Independence Way Princeton, NJ 08540 avg@research.nj.nec.com Alexander V. Karzanov Institute for Systems Analysis 9, Prospect 60
More informationLecture 14 - P v.s. NP 1
CME 305: Discrete Mathematics and Algorithms Instructor: Professor Aaron Sidford (sidford@stanford.edu) February 27, 2018 Lecture 14 - P v.s. NP 1 In this lecture we start Unit 3 on NP-hardness and approximation
More informationNP-Completeness. f(n) \ n n sec sec sec. n sec 24.3 sec 5.2 mins. 2 n sec 17.9 mins 35.
NP-Completeness Reference: Computers and Intractability: A Guide to the Theory of NP-Completeness by Garey and Johnson, W.H. Freeman and Company, 1979. NP-Completeness 1 General Problems, Input Size and
More informationNP-Completeness. NP-Completeness 1
NP-Completeness Reference: Computers and Intractability: A Guide to the Theory of NP-Completeness by Garey and Johnson, W.H. Freeman and Company, 1979. NP-Completeness 1 General Problems, Input Size and
More information9. Submodular function optimization
Submodular function maximization 9-9. Submodular function optimization Submodular function maximization Greedy algorithm for monotone case Influence maximization Greedy algorithm for non-monotone case
More informationMINIMUM DIAMETER COVERING PROBLEMS. May 20, 1997
MINIMUM DIAMETER COVERING PROBLEMS Esther M. Arkin y and Refael Hassin z May 20, 1997 Abstract A set V and a collection of (possibly non-disjoint) subsets are given. Also given is a real matrix describing
More informationM 2 M 3. Robot M (O)
R O M A TRE DIA Universita degli Studi di Roma Tre Dipartimento di Informatica e Automazione Via della Vasca Navale, 79 { 00146 Roma, Italy Part Sequencing in Three Machine No-Wait Robotic Cells Alessandro
More informationThe minimum G c cut problem
The minimum G c cut problem Abstract In this paper we define and study the G c -cut problem. Given a complete undirected graph G = (V ; E) with V = n, edge weighted by w(v i, v j ) 0 and an undirected
More informationDifferential approximation results for the Steiner tree problem
Differential approximation results for the Steiner tree problem Marc Demange, Jérôme Monnot, Vangelis Paschos To cite this version: Marc Demange, Jérôme Monnot, Vangelis Paschos. Differential approximation
More informationKernelization by matroids: Odd Cycle Transversal
Lecture 8 (10.05.2013) Scribe: Tomasz Kociumaka Lecturer: Marek Cygan Kernelization by matroids: Odd Cycle Transversal 1 Introduction The main aim of this lecture is to give a polynomial kernel for the
More informationScheduling Parallel Jobs with Linear Speedup
Scheduling Parallel Jobs with Linear Speedup Alexander Grigoriev and Marc Uetz Maastricht University, Quantitative Economics, P.O.Box 616, 6200 MD Maastricht, The Netherlands. Email: {a.grigoriev, m.uetz}@ke.unimaas.nl
More informationFRACTIONAL PACKING OF T-JOINS. 1. Introduction
FRACTIONAL PACKING OF T-JOINS FRANCISCO BARAHONA Abstract Given a graph with nonnegative capacities on its edges, it is well known that the capacity of a minimum T -cut is equal to the value of a maximum
More information1 Introduction Given a connected undirected graph G = (V E), the maximum leaf spanning tree problem is to nd a spanning tree of G with the maximum num
Approximating Maximum Leaf Spanning Trees in Almost Linear Time Hsueh-I Lu hil@cs.ccu.edu.tw Department. of CSIE National Chung-Cheng University Taiwan R. Ravi ravi@cmu.edu Graduate School of Industrial
More informationThe maximum edge biclique problem is NP-complete
The maximum edge biclique problem is NP-complete René Peeters Department of Econometrics and Operations Research Tilburg University The Netherlands April 5, 005 File No. DA5734 Running head: Maximum edge
More informationMaximum flow problem
Maximum flow problem 7000 Network flows Network Directed graph G = (V, E) Source node s V, sink node t V Edge capacities: cap : E R 0 Flow: f : E R 0 satisfying 1. Flow conservation constraints e:target(e)=v
More information1 Submodular functions
CS 369P: Polyhedral techniques in combinatorial optimization Instructor: Jan Vondrák Lecture date: November 16, 2010 1 Submodular functions We have already encountered submodular functions. Let s recall
More informationTHE MAXIMAL SUBGROUPS AND THE COMPLEXITY OF THE FLOW SEMIGROUP OF FINITE (DI)GRAPHS
THE MAXIMAL SUBGROUPS AND THE COMPLEXITY OF THE FLOW SEMIGROUP OF FINITE (DI)GRAPHS GÁBOR HORVÁTH, CHRYSTOPHER L. NEHANIV, AND KÁROLY PODOSKI Dedicated to John Rhodes on the occasion of his 80th birthday.
More informationLecture 3. 1 Polynomial-time algorithms for the maximum flow problem
ORIE 633 Network Flows August 30, 2007 Lecturer: David P. Williamson Lecture 3 Scribe: Gema Plaza-Martínez 1 Polynomial-time algorithms for the maximum flow problem 1.1 Introduction Let s turn now to considering
More informationSpanning trees with minimum weighted degrees
Spanning trees with minimum weighted degrees Mohammad Ghodsi Hamid Mahini Kian Mirjalali Shayan Oveis Gharan Amin S. Sayedi R. Morteza Zadimoghaddam Abstract Given a metric graph G, we are concerned with
More informationPrimal vector is primal infeasible till end. So when primal feasibility attained, the pair becomes opt. & method terminates. 3. Two main steps carried
4.1 Primal-Dual Algorithms Katta G. Murty, IOE 612 Lecture slides 4 Here we discuss special min cost ow problems on bipartite networks, the assignment and transportation problems. Algorithms based on an
More informationScheduling unit processing time arc shutdown jobs to maximize network flow over time: complexity results
Scheduling unit processing time arc shutdown jobs to maximize network flow over time: complexity results Natashia Boland Thomas Kalinowski Reena Kapoor Simranjit Kaur Abstract We study the problem of scheduling
More informationPATH PROBLEMS IN SKEW-SYMMETRIC GRAPHS ANDREW V. GOLDBERG COMPUTER SCIENCE DEPARTMENT STANFORD UNIVERSITY STANFORD, CA
PATH PROBLEMS IN SKEW-SYMMETRIC GRAPHS ANDREW V. GOLDBERG COMPUTER SCIENCE DEPARTMENT STANFORD UNIVERSITY STANFORD, CA 94305 GOLDBERG@CS.STANFORD.EDU AND ALEXANDER V. KARZANOV INSTITUTE FOR SYSTEMS ANALYSIS
More information5 Flows and cuts in digraphs
5 Flows and cuts in digraphs Recall that a digraph or network is a pair G = (V, E) where V is a set and E is a multiset of ordered pairs of elements of V, which we refer to as arcs. Note that two vertices
More informationIE 5531: Engineering Optimization I
IE 5531: Engineering Optimization I Lecture 7: Duality and applications Prof. John Gunnar Carlsson September 29, 2010 Prof. John Gunnar Carlsson IE 5531: Engineering Optimization I September 29, 2010 1
More informationCS264: Beyond Worst-Case Analysis Lecture #18: Smoothed Complexity and Pseudopolynomial-Time Algorithms
CS264: Beyond Worst-Case Analysis Lecture #18: Smoothed Complexity and Pseudopolynomial-Time Algorithms Tim Roughgarden March 9, 2017 1 Preamble Our first lecture on smoothed analysis sought a better theoretical
More informationK-center Hardness and Max-Coverage (Greedy)
IOE 691: Approximation Algorithms Date: 01/11/2017 Lecture Notes: -center Hardness and Max-Coverage (Greedy) Instructor: Viswanath Nagarajan Scribe: Sentao Miao 1 Overview In this lecture, we will talk
More informationAn 0.5-Approximation Algorithm for MAX DICUT with Given Sizes of Parts
An 0.5-Approximation Algorithm for MAX DICUT with Given Sizes of Parts Alexander Ageev Refael Hassin Maxim Sviridenko Abstract Given a directed graph G and an edge weight function w : E(G) R +, themaximumdirectedcutproblem(max
More informationThe Complexity and Approximability of Finding. Maximum Feasible Subsystems of Linear Relations. Abstract
The Complexity and Approximability of Finding Maximum Feasible Subsystems of Linear Relations Edoardo Amaldi Department of Mathematics Swiss Federal Institute of Technology CH-1015 Lausanne amaldi@dma.epfl.ch
More informationImproving on the 1.5-Approximation of. a Smallest 2-Edge Connected Spanning Subgraph. September 25, Abstract
Improving on the 1.5-Approximation of a Smallest 2-Edge Connected Spanning Subgraph J. Cheriyan y A. Seb}o z Z. Szigeti x September 25, 1999 Abstract We give a 17 -approximation algorithm for the following
More informationFrom Primal-Dual to Cost Shares and Back: A Stronger LP Relaxation for the Steiner Forest Problem
From Primal-Dual to Cost Shares and Back: A Stronger LP Relaxation for the Steiner Forest Problem Jochen Könemann 1, Stefano Leonardi 2, Guido Schäfer 2, and Stefan van Zwam 3 1 Department of Combinatorics
More informationMaximum-Life Routing Schedule
Maximum-Life Routing Schedule Peng-Jun Wan wan@cs.iit.edu Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 1 / 42 Outline Problem Description Min-Cost Routing Ellipsoid Algorithm Price-Directive
More informationCS264: Beyond Worst-Case Analysis Lecture #15: Smoothed Complexity and Pseudopolynomial-Time Algorithms
CS264: Beyond Worst-Case Analysis Lecture #15: Smoothed Complexity and Pseudopolynomial-Time Algorithms Tim Roughgarden November 5, 2014 1 Preamble Previous lectures on smoothed analysis sought a better
More information1 Introduction Greedy approximation algorithms for various N P -hard problems were proposed in [Joh74, Lov75, Chv79]. A more general framework for usi
Generalized Submodular Cover Problems and Applications Judit Bar-Ilan Guy Kortsarz y David Peleg z October 28, 1998 Abstract The greedy approach has been successfully applied in the past to produce logarithmic
More informationStructure in Approximation Classes 2 able to come back from \good" solutions for B to \good" solutions for A. Surprisingly, the rst denition of this k
Submitted to SIAM Journal on Computing Structure in Approximation Classes? P. Crescenzi 1, V. Kann 2, R. Silvestri 1, and L. Trevisan 1 1 Dipartimento di Scienze dell'informazione Universita degli Studi
More informationThe Budget-Constrained Maximum Flow Problem
9 The Budget-Constrained Maximum Flow Problem In this chapter we consider the following problem which is called the constrained maximum flow problem ( Cmfp) [AO95]: We are given a budget B and we seek
More informationEssential facts about NP-completeness:
CMPSCI611: NP Completeness Lecture 17 Essential facts about NP-completeness: Any NP-complete problem can be solved by a simple, but exponentially slow algorithm. We don t have polynomial-time solutions
More informationRealization of set functions as cut functions of graphs and hypergraphs
Discrete Mathematics 226 (2001) 199 210 www.elsevier.com/locate/disc Realization of set functions as cut functions of graphs and hypergraphs Satoru Fujishige a;, Sachin B. Patkar b a Division of Systems
More informationComplexity analysis of job-shop scheduling with deteriorating jobs
Discrete Applied Mathematics 117 (2002) 195 209 Complexity analysis of job-shop scheduling with deteriorating jobs Gur Mosheiov School of Business Administration and Department of Statistics, The Hebrew
More informationAlgorithms and Theory of Computation. Lecture 11: Network Flow
Algorithms and Theory of Computation Lecture 11: Network Flow Xiaohui Bei MAS 714 September 18, 2018 Nanyang Technological University MAS 714 September 18, 2018 1 / 26 Flow Network A flow network is a
More informationthe results from this paper are used in a decomposition scheme for the stochastic service provision problem. Keywords: distributed processing, telecom
Single Node Service Provision with Fixed Charges Shane Dye Department ofmanagement University of Canterbury New Zealand s.dye@mang.canterbury.ac.nz Leen Stougie, Eindhoven University of Technology The
More informationApproximating Minimum-Power Degree and Connectivity Problems
Approximating Minimum-Power Degree and Connectivity Problems Guy Kortsarz Vahab S. Mirrokni Zeev Nutov Elena Tsanko Abstract Power optimization is a central issue in wireless network design. Given a graph
More informationParametrizing Above Guaranteed Values: MaxSat. and MaxCut. Meena Mahajan and Venkatesh Raman. The Institute of Mathematical Sciences,
ECCC TR97-033 FTP: WWW: Email: ftp.eccc.uni-trier.de:/pub/eccc/ http://www.eccc.uni-trier.de/eccc/ ftpmail@ftp.eccc.uni-trier.de with subject help eccc Parametrizing Above Guaranteed Values: MaxSat and
More informationLexicographic Flow. Dexter Kozen Department of Computer Science Cornell University Ithaca, New York , USA. June 25, 2009
Lexicographic Flow Dexter Kozen Department of Computer Science Cornell University Ithaca, New York 14853-7501, USA June 25, 2009 Abstract The lexicographic flow problem is a flow problem in which the edges
More informationDeciding Emptiness of the Gomory-Chvátal Closure is NP-Complete, Even for a Rational Polyhedron Containing No Integer Point
Deciding Emptiness of the Gomory-Chvátal Closure is NP-Complete, Even for a Rational Polyhedron Containing No Integer Point Gérard Cornuéjols 1 and Yanjun Li 2 1 Tepper School of Business, Carnegie Mellon
More informationALGORITHMS AND COMPLETE FORMULATIONS FOR THE NETWORK DESIGN PROBLEM Trilochan Sastry Indian Institute of Management, Ahmedabad November 1997 Abstract
ALGORITHMS AND COMPLETE FORMULATIONS FOR THE NETWORK DESIGN PROBLEM Trilochan Sastry Indian Institute of Management, Ahmedabad November 1997 Abstract We study the multi commodity uncapacitated network
More informationOn Fixed Cost k-flow Problems
On Fixed Cost k-flow Problems MohammadTaghi Hajiaghayi 1, Rohit Khandekar 2, Guy Kortsarz 3, and Zeev Nutov 4 1 University of Maryland, College Park, MD. hajiagha@cs.umd.edu. 2 Knight Capital Group, Jersey
More informationImproved Fully Polynomial time Approximation Scheme for the 0-1 Multiple-choice Knapsack Problem
Improved Fully Polynomial time Approximation Scheme for the 0-1 Multiple-choice Knapsack Problem Mukul Subodh Bansal V.Ch.Venkaiah International Institute of Information Technology Hyderabad, India Abstract
More informationHamiltonian Cycle in Cubic Hamiltonian Graphs. Abstract. It is a simple fact that cubic Hamiltonian graphs have at least two Hamiltonian
On the Approximation of Finding A(nother) Hamiltonian Cycle in Cubic Hamiltonian Graphs Cristina Bazgan y Miklos Santha z Zsolt Tuza x Abstract It is a simple fact that cubic Hamiltonian graphs have at
More informationTopic: Balanced Cut, Sparsest Cut, and Metric Embeddings Date: 3/21/2007
CS880: Approximations Algorithms Scribe: Tom Watson Lecturer: Shuchi Chawla Topic: Balanced Cut, Sparsest Cut, and Metric Embeddings Date: 3/21/2007 In the last lecture, we described an O(log k log D)-approximation
More informationOn 2-Coverings and 2-Packings of Laminar. Families.
On 2-Coverings and 2-Packings of Laminar Families Joseph Cheriyan 1?, Tibor Jordan 2??, and R. Ravi 3??? 1 Department of Combinatorics and Optimization, University of Waterloo, Waterloo ON Canada N2L 3G1,
More informationON THE COMPLEXITY OF SOLVING THE GENERALIZED SET PACKING PROBLEM APPROXIMATELY. Nimrod Megiddoy
ON THE COMPLEXITY OF SOLVING THE GENERALIZED SET PACKING PROBLEM APPROXIMATELY Nimrod Megiddoy Abstract. The generalized set packing problem (GSP ) is as follows. Given a family F of subsets of M = f mg
More informationDiskrete Mathematik und Optimierung
Diskrete Mathematik und Optimierung Annabell Berger, Winfried Hochstättler: Minconvex graph factors of prescribed size and a simpler reduction to weighted f-factors Technical Report feu-dmo006.06 Contact:
More informationThe English system in use before 1971 (when decimal currency was introduced) was an example of a \natural" coin system for which the greedy algorithm
A Polynomial-time Algorithm for the Change-Making Problem David Pearson Computer Science Department Cornell University Ithaca, New York 14853, USA pearson@cs.cornell.edu June 14, 1994 Abstract The change-making
More informationLectures 6, 7 and part of 8
Lectures 6, 7 and part of 8 Uriel Feige April 26, May 3, May 10, 2015 1 Linear programming duality 1.1 The diet problem revisited Recall the diet problem from Lecture 1. There are n foods, m nutrients,
More informationVol. 2(1997): nr 4. Tight Lower Bounds on the. Approximability of Some. Peter Jonsson. Linkoping University
Linkoping Electronic Articles in Computer and Information Science Vol. 2(1997): nr 4 Tight Lower Bounds on the Approximability of Some NPO PB-Complete Problems Peter Jonsson Department of Computer and
More informationDD2446 Complexity Theory: Problem Set 4
DD2446 Complexity Theory: Problem Set 4 Due: Friday November 8, 2013, at 23:59. Submit your solutions as a PDF le by e-mail to jakobn at kth dot se with the subject line Problem set 4: your full name.
More informationApproximate Map Labeling. is in (n log n) Frank Wagner* B 93{18. December Abstract
SERIE B INFORMATIK Approximate Map Labeling is in (n log n) Frank Wagner* B 93{18 December 1993 Abstract Given n real numbers, the -CLOSENESS problem consists in deciding whether any two of them are within
More informationMIT Algebraic techniques and semidefinite optimization February 14, Lecture 3
MI 6.97 Algebraic techniques and semidefinite optimization February 4, 6 Lecture 3 Lecturer: Pablo A. Parrilo Scribe: Pablo A. Parrilo In this lecture, we will discuss one of the most important applications
More informationMulticommodity Flows and Column Generation
Lecture Notes Multicommodity Flows and Column Generation Marc Pfetsch Zuse Institute Berlin pfetsch@zib.de last change: 2/8/2006 Technische Universität Berlin Fakultät II, Institut für Mathematik WS 2006/07
More informationA Note on Perfect Partial Elimination
A Note on Perfect Partial Elimination Matthijs Bomhoff, Walter Kern, and Georg Still University of Twente, Faculty of Electrical Engineering, Mathematics and Computer Science, P.O. Box 217, 7500 AE Enschede,
More informationApproximability and Parameterized Complexity of Consecutive Ones Submatrix Problems
Proc. 4th TAMC, 27 Approximability and Parameterized Complexity of Consecutive Ones Submatrix Problems Michael Dom, Jiong Guo, and Rolf Niedermeier Institut für Informatik, Friedrich-Schiller-Universität
More informationPacking Arborescences
Egerváry Research Group on Combinatorial Optimization Technical reports TR-2009-04. Published by the Egerváry Research Group, Pázmány P. sétány 1/C, H1117, Budapest, Hungary. Web site: www.cs.elte.hu/egres.
More informationShortest paths with negative lengths
Chapter 8 Shortest paths with negative lengths In this chapter we give a linear-space, nearly linear-time algorithm that, given a directed planar graph G with real positive and negative lengths, but no
More informationOptimal Fractal Coding is NP-Hard 1
Optimal Fractal Coding is NP-Hard 1 (Extended Abstract) Matthias Ruhl, Hannes Hartenstein Institut für Informatik, Universität Freiburg Am Flughafen 17, 79110 Freiburg, Germany ruhl,hartenst@informatik.uni-freiburg.de
More informationThe Steiner Network Problem
The Steiner Network Problem Pekka Orponen T-79.7001 Postgraduate Course on Theoretical Computer Science 7.4.2008 Outline 1. The Steiner Network Problem Linear programming formulation LP relaxation 2. The
More informationExercises NP-completeness
Exercises NP-completeness Exercise 1 Knapsack problem Consider the Knapsack problem. We have n items, each with weight a j (j = 1,..., n) and value c j (j = 1,..., n) and an integer B. All a j and c j
More informationA Graph Based Parsing Algorithm for Context-free Languages
A Graph Based Parsing Algorithm for Context-free Languages Giinter Hot> Technical Report A 01/99 June 1999 e-mail: hotzocs.uni-sb.de VVVVVV: http://vwv-hotz.cs.uni-sb. de Abstract We present a simple algorithm
More informationGeometric Steiner Trees
Geometric Steiner Trees From the book: Optimal Interconnection Trees in the Plane By Marcus Brazil and Martin Zachariasen Part 3: Computational Complexity and the Steiner Tree Problem Marcus Brazil 2015
More informationReport No On the Complexity of Testing Membership. in the Core of Min-cost Spanning Tree Games
Report No. 94.166 On the Complexity of Testing Membership in the Core of Min-cost Spanning Tree Games by U. Faigle, S. Fekete, W. Hochstattler, W. Kern 1994 Ulrich Faigle Walter Kern Department of Applied
More informationINVERSE SPANNING TREE PROBLEMS: FORMULATIONS AND ALGORITHMS
INVERSE SPANNING TREE PROBLEMS: FORMULATIONS AND ALGORITHMS P. T. Sokkalingam Department of Mathematics Indian Institute of Technology, Kanpur-208 016, INDIA Ravindra K. Ahuja Dept. of Industrial & Management
More informationAn approximation algorithm for the minimum latency set cover problem
An approximation algorithm for the minimum latency set cover problem Refael Hassin 1 and Asaf Levin 2 1 Department of Statistics and Operations Research, Tel-Aviv University, Tel-Aviv, Israel. hassin@post.tau.ac.il
More informationA necessary and sufficient condition for the existence of a spanning tree with specified vertices having large degrees
A necessary and sufficient condition for the existence of a spanning tree with specified vertices having large degrees Yoshimi Egawa Department of Mathematical Information Science, Tokyo University of
More informationOn Injective Colourings of Chordal Graphs
On Injective Colourings of Chordal Graphs Pavol Hell 1,, André Raspaud 2, and Juraj Stacho 1 1 School of Computing Science, Simon Fraser University 8888 University Drive, Burnaby, B.C., Canada V5A 1S6
More informationApproximation complexity of min-max (regret) versions of shortest path, spanning tree, and knapsack
Approximation complexity of min-max (regret) versions of shortest path, spanning tree, and knapsack Hassene Aissi, Cristina Bazgan, and Daniel Vanderpooten LAMSADE, Université Paris-Dauphine, France {aissi,bazgan,vdp}@lamsade.dauphine.fr
More informationParametric Problems on Graphs of Bounded. Tree-width 3. Department of Computer Science, Iowa State University, Ames, IA April 8, 1992.
Parametric Problems on Graphs of Bounded Tree-width 3 David Fernandez-Baca y and Giora Slutzki Department of Computer Science, Iowa State University, Ames, IA 50011 April 8, 1992 Abstract We consider optimization
More informationinterval order and all height one orders. Our major contributions are a characterization of tree-visibility orders by an innite family of minimal forb
Tree-Visibility Orders Dieter Kratsch y Jean-Xavier Rampon z Abstract We introduce a new class of partially ordered sets, called tree-visibility orders, containing interval orders, duals of generalized
More informationCERNY CONJECTURE FOR DFA ACCEPTING STAR-FREE LANGUAGES
CERNY CONJECTURE FOR DFA ACCEPTING STAR-FREE LANGUAGES A.N. Trahtman? Bar-Ilan University, Dep. of Math. and St., 52900, Ramat Gan, Israel ICALP, Workshop synchr. autom., Turku, Finland, 2004 Abstract.
More informationThe Multiple Traveling Salesman Problem with Time Windows: Bounds for the Minimum Number of Vehicles
The Multiple Traveling Salesman Problem with Time Windows: Bounds for the Minimum Number of Vehicles Snežana Mitrović-Minić Ramesh Krishnamurti School of Computing Science, Simon Fraser University, Burnaby,
More informationNP-hardness of the stable matrix in unit interval family problem in discrete time
Systems & Control Letters 42 21 261 265 www.elsevier.com/locate/sysconle NP-hardness of the stable matrix in unit interval family problem in discrete time Alejandra Mercado, K.J. Ray Liu Electrical and
More information1 Introduction A general problem that arises in dierent areas of computer science is the following combination problem: given two structures or theori
Combining Unication- and Disunication Algorithms Tractable and Intractable Instances Klaus U. Schulz CIS, University of Munich Oettingenstr. 67 80538 Munchen, Germany e-mail: schulz@cis.uni-muenchen.de
More informationRepresentations of All Solutions of Boolean Programming Problems
Representations of All Solutions of Boolean Programming Problems Utz-Uwe Haus and Carla Michini Institute for Operations Research Department of Mathematics ETH Zurich Rämistr. 101, 8092 Zürich, Switzerland
More information7. F.Balarin and A.Sangiovanni-Vincentelli, A Verication Strategy for Timing-
7. F.Balarin and A.Sangiovanni-Vincentelli, A Verication Strategy for Timing- Constrained Systems, Proc. 4th Workshop Computer-Aided Verication, Lecture Notes in Computer Science 663, Springer-Verlag,
More informationMultiple Sequence Alignment: Complexity, Gunnar Klau, January 12, 2006, 12:
Multiple Sequence Alignment: Complexity, Gunnar Klau, January 12, 2006, 12:23 6001 6.1 Computing MSAs So far, we have talked about how to score MSAs (including gaps and benchmarks). But: how do we compute
More informationStrongly chordal and chordal bipartite graphs are sandwich monotone
Strongly chordal and chordal bipartite graphs are sandwich monotone Pinar Heggernes Federico Mancini Charis Papadopoulos R. Sritharan Abstract A graph class is sandwich monotone if, for every pair of its
More informationMultiply Balanced k Partitioning
Multiply Balanced k Partitioning Amihood Amir 1,2,, Jessica Ficler 1,, Robert Krauthgamer 3, Liam Roditty 1, and Oren Sar Shalom 1 1 Department of Computer Science, Bar-Ilan University, Ramat-Gan 52900,
More informationComputer Science & Engineering 423/823 Design and Analysis of Algorithms
Bipartite Matching Computer Science & Engineering 423/823 Design and s Lecture 07 (Chapter 26) Stephen Scott (Adapted from Vinodchandran N. Variyam) 1 / 36 Spring 2010 Bipartite Matching 2 / 36 Can use
More informationThree-dimensional Stable Matching Problems. Cheng Ng and Daniel S. Hirschberg. Department of Information and Computer Science
Three-dimensional Stable Matching Problems Cheng Ng and Daniel S Hirschberg Department of Information and Computer Science University of California, Irvine Irvine, CA 92717 Abstract The stable marriage
More information