C. H. Papadimitriou and K. Steiglitz, Combinatorial Optimization, Prentice-Hall, Englewood Cli#s, NJ, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....leaf represented exactly once. Sometimes, a constraint that no image or model feature may be used twice in a matching is imposed. This can be expressed as a maximal bipartite graph matching problem between image and model features ( Cass, 1988] which is solvable in low order polynomial time ([Papadimitriou and Steiglitz, 1982]) Because of the way features are extracted from grey level images, imposing this constraint is rarely necessary in the 2D case. 4 2D and 3D Recognition We have already discussed in detail the geometry of constraints and transformations in the case of matching 2D images against 2D models. With ....

Papadimitriou C. H., Steiglitz K., 1982, Combinatorial Optimization, Prentice Hall. 13

.... A is called positive semide nite, A 0, if 8z 2 R n0 : z A z 0) For positive semide nite W the QLP is a linear constraint convex quadratic optimization problem, which is solvable by ellipsoid algorithms in polynomial time within an arithmetic error precision , see [4] but also [17] ch. 15 question 16. Mathematician Khachian was the rst, who provided a real polynomial algorithm for linear optimization problems, which then were transformed rather mechanically to semide nite optimization, 14] We will use the solution to this modi ed program as basis for randomized ....

C. H. Papadimitriou and K. Steiglitz. Combinatorial Optimization. Dover Publications, Mineola, New York, 1998.

....corresponds to a document, and m nodes on the right, where each node corresponds to a unit time interval. The edge ij between node i on the left and node j on the right has cost c ij . Finding a schedule now corresponds to finding a minimum cost matching in the graph, which can be solved in O(m [8]. This solution naturally extends to the case of multiple broadcast channels [7] B. Different sized documents broadcast only Unlike the case of unit size documents, scheduling different sized documents on a broadcast link is NP Hard. We show that this case is a generalization of the minimum ....

C. H. Papadimitriou and K. Steiglitz, Combinatorial Optimization. Prentice-Hall, Englewood Cliffs, NJ, 1982.

....(e.g. hard, pre placed, soft, and rectilinear modules) directly and efficiently. In contrast, other representation cannot handle some special modules well. To deal with soft modules using sequence pair, for example, previous work in [8] transformed it to the convex problem (which is NP hard [13]) processed the soft modules, and then transformed back to the sequence pair. B trees do not need to construct constraint graphs for area cost evaluation. In contrast, the sequence pair and O tree representa Permission to make digital hardcopy of all or part of this work for personal or ....

C. H. Papadimitriou, and K. Steiglitz, Combinatorial Optimization, Prentice Hall, 1982.

....coefficients of the network. Though it is very easy to calculate T for a given set of G = G i =1, n , we have to check every possible G to find the optimal T , which takes O(n ) time. As is well known, this kind of enumeration approach is intractable. The branch and bound method [12] and heuristic approximation algorithms are hence adopted here. Find more details for these techniques in the preceding reports [8] 9] 10] 3.2 HEURISTIC ALGORITHMS This section introduces five simple heuristic algorithms. As space is limited, only the core idea of each algorithm is described. ....

....not allocated yet are represented by g. Q j gets larger when the block is large or the communication is suppressed (if allocated on the same processor) Q j gets smaller if the communication emerges by dispatching Q j to P i . In addition to these approximation algorithms, local search technique [12] is examined. After dispatching all blocks by approximation algorithm, we can examine whether the possible swap of two blocks improves the total execution time. A local search iterates such swaps while improvement is derived. Figures 4 and 5 illustrate various aspects of the proposed ....

C. H. Papadimitriou and K. Steiglitz. Combinatorial Optimization. Dover Publications, 1998.

....is significantly better than the quality of the solutions obtained after applying the MCF heuristic only. In this section, we impose without loss of generality an upper bound u on the x variables. 3. 1 Solution by LP Column Generation The basic column generation approach is described in [PS82, Chapter 4] The idea behind LP column generation is the following. Instead of using all decision variables, we use only a small subset and calculate an optimal LP solution for this restricted formulation. The initial subset should be chosen in such a way that the restricted problem is feasible ....

Papadimitriou, C. H. and Steiglitz, K. Combinatorial Optimization. Prentice Hall, Inc., 1982.

....Requirement of Acydic Code Block = V,E) Maximize II qf(i) subject to f(j) f(i) O, for each (vi, v) E, f(0) f(n 1) 1, and f(i) integer, for all v i E V. Linear program 7 is the dual of a min cost flow problem [13 Since polynomial time algorithms exist to solve rain cost flow problem [20], this linear program can be solved in polynomial time. More generally. the constraint matrix in the integer linear prog,an derived for an aeyeli block without conditionals is totally unimodular 16. This allows the integrality constraint to be ignored, because for a linear program with a totally ....

....unimodular constraint matrix and integral right hand side, every basic feasible solution is integral We call the linear program obtained from an integer 16A matrix is toy unimodular fievery noo singular submatrix has dec nnin I or I. Suffient conditions for total unimodularity can be found in [20]. page 317. 3 Ahab sis of A. c c 61 costs.I I I . I I 62 Analysis of Acydic Blocks 5.3 token storage requirements, in the acyclic case. However, determining the max cut of a graph is NP complete, in general [16] The key observation is that legal configurations correspond to a ....

Papadimitriou, C. and Steiglitz, K.. Combinatorial Optimization. Prendce Hall, Inc., Englewood Cliffs, New Jersey, 1.982.

....tasks are discussed in Sections IV A and IV B. Finally, one test on a feedback system for non linear control is analyzed in Section IV C. II. REACTIVE TABU SEARCH: A METHOD DESIGNED TO DISCOURAGE CYCLES Let us define our notation. An instance of a combinato rial optimization (CO) problem [38] is a pair (F, E) where F is a set of feasible points with finite cardinality (we do not consider the case of a countably infinite set) and E is the cost function, i.e. a mapping: E : F R . A solution f is globally optimal if E(f) E(y) for all ycr For many interesting CO problems the ....

C.H. Papadimitriou and K. Steiglitz, Combinatorial Optimiza- tion. Englewood Cliffs, N J: Prentice-Hall, 1982.

....(the only exception is theorem 6.5) It is evident that MinDis(Halfspaces) belongs to NP because there is always a polynomial guess for the (rational) coefficients of an optimal separating hyperplane. We omit the details and refer the reader to standard books on linear programming such as [22]. Theorem 3.1 MinDis(Halfspaces) is NP complete. We give a proof of this theorem in the next section by proving a strengthened result. A. Blum [5] independently suggested another proof of Theorem 3.1 using a polynomial transformation from the NP complete problem Open Hemisphere (problem MP6 in ....

C. H. Papadimitriou and K. Steiglitz, "Combinatorial Optimization," Prentice-Hall, Englewood Cliffs, NJ, 1982.

....existentially quantified variable. The following lemma called Farkas lemma helps us achieve this goal. Lemma: 4.5.4 [Farkas] Either f x 2 n : A: x bg 6= OE or ( exclusively ) 9 y 2 m , such that, A T : y 0 T and y T : b = Gamma1. Proof: See [Sch87, PS82, NW88] Figure (4.3) provides a graphical illustration of the lemma. 2 A T y0 This is the dual space i.e. the space of the dual variables. The axes correspond to y 1 ,y 2 , y n ( dual cone ) w = y T b ( polar cone ) Figure 4.3: Farkas lemma Lemma (4.5.4) has the following interpretation: ....

....of all the variables x 2 ; x n . If a b, we conclude that the system is infeasible. Algorithm (B.1.1) s a formal description of the above procedure. Observation: B.1.1 Since determining the feasibility of a system of linear inequalities in polynomially equivalent to solving System (B. 1) PS82] the FourierMotzkin procedure can be modified in polynomial time, to solve System (B.1) Though elegant, this syntactic procedure suffers from an exponential growth in the constraint set, as it progresses. This growth has been observed both in theory 1 Most of the material discussed in this ....

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C. H. Papadimitriou and K. Steiglitz. Combinatorial Optimization. Prentice Hall, 1982.

....Traveling Salesman Problem (see Example 3. 4) polynomial time approximation algorithms with constant performance ratio can only exist if the edge weights satisfy the triangle inequality [26] In this case, a 2 approximative algorithm can be constructed using a minimum spanning tree in the graph [40]. Christofides algorithm [21] also starts with a minimum spanning tree. For the nodes with odd degree in this tree a shortest perfect matching is computed. Then, a tour following a Eulerian walk in the multi graph formed by the spanning tree and the perfect matching is constructed. The solution ....

C. H. Papadimitriou and K. Steiglitz, Combinatorial optimization, Prentice-Hall, Inc., 1982.

....promises and limitations of quantum computers (sec. 3.1) These notes are directed at physicists with no or little knowledge of computational complexity. Of course this is not the first informal introduction into the field, see e.g. 1, 2] or the corresponding chapters in textbooks on algorithms [3, 4]. For a deeper understanding of the field you are referred to the classical textbooks of Garey and Johnson [5] and Papadimitriou [6] 1.2 The measure of complexity 1.2.1 Algorithms The computational complexity of a problem is a measure of the computational resources, typically time, required to ....

....ASSIGNMENT is the problem of minimizing the total cost. There are n possible assignments of n tasks to n workers, hence exhaustive enumeration again is prohibitive. In contrast to the TSP, however, ASSIGNMENT can be solved in polynomial time, for example using the O(n 3 ) hungarian algorithm [4]. Compared to MST, the algorithm and the underlying mathematical insight are a bit more involved and will not be discussed here. 2 Complexity classes 2.1 Decision problems So far we have discussed optimization problems: solving MINIMUM SPANNING TREE, TSP or ASSIGNMENT implies that we compare an ....

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C.H. Papadimitriou and K. Steiglitz. Combinatorial Optimization. Prentice-Hall, Englewood Cliffs, New Jersey, 1982. 25

....N j=1 b j n. In this section we adapt the CC step to these constraints, and we write the corresponding optimization problem as a combinatorial one. Moreover, we show that its solution is determined by the solution of a linear 34 optimization problem known by the name of Hitchock problem, cf. [16], which is motivated by the following situation: we have s sources of some commodity, each with a supply of a i units, i 2 1: s, and t terminals, each of which has a demand of b j units, j 2 1: t. Furthermore, we know that the cost of sending the commodity form source i to terminal j is c ij . How ....

Papadimitriou, C. and K. Steiglitz. 1998. Combinatorial Optimization. Dover Publ., Mineola, New York.

....this is of the order O(M) O n log oe(n) 3.1) In each cell C i an optimal matching is produced. We assume that we can solve an Euclidean matching problem with k points in time f(k) Dk p with constants Birgit Anthes and Ludger Ruschendorf 5 p, D. This is fulfilled for p = 3 by Papadimitriou and Steiglitz (1982, Theorem 11.3, Problem 14) Improvements of this order to 0(n 2:5 (log n) 4 ) are given in Vaidya (1989) to some more general class of geometric algorithms in dimension two. By the assumption that X 1 ; X n are independent, uniformly distributed on [0; 1] d the number n i of points ....

Papadimitriou, C. H. and K. Steiglitz (1982). Combinatorial Optimization, Algorithms and Complexity. Prentice Hall N.J.

....satisfied. l 1 l 2 l 4 ) l 1 l 2 l 4 ) l 2 l 3 l 6 ) l 1 l 2 l 4 ) l 4 l 5 l 6 ) l 1 l 4 l 6 ) l 1 l 5 l 6 ) l 1 l 2 l 6 ) l 1 l 3 l 6 ) l 4 l 5 l 6 ) l 2 l 4 l 5 ) l 2 l 4 l 6 ) 14) The transformation is derived from [4]. A branch and bound process would involve solving 31 LP subproblems for this example. Although the number of simplex routine calls may depend on the strategies employed in the branch 2 50 100 150 200 250 0.5 0.25 0 0.25 0.5 0.75 1 1.25 1.5 50 100 150 200 250 0.5 0.25 0 0.25 ....

C. H. Papadimitriou and K. Steiglitz, Combinatorial Optimization. Prentice-Hall, 1982.

....that are close to optimal. Space time tradeoffs with optimality favor greedy algorithms as good heuristics. Characterizing the class of easy problems could aid in the discovery of more efficient algorithms for several problems that have not yet been proven to be hard. Many studies of greed [4, 6, 8, 9, 11, 15] have focused on greedy algorithms for specific classes of optimization problems. A common element among these studies is formalization of some notion of exchange, with which a hill climbing approach produces optimal solutions. Conceptually, a greedy solvable problem is solved by repeated ....

C.H. Papadimitriou and K.S. Stieglitz. Combinatorial Optimization. Prentice-Hall, Englewood Cliffs, NJ, 1982.

....average cell delay can be much lower with dynamic arbitration. This is why in a second stage a dynamic arbitration is performed on all unmatched ports. A number of algorithms exist to solve the dynamic matching optimally, either as a maximum size matching (MSM) or a maximum weight matching (MWM) [12]. MSM finds the maximum number of input output connections. MWM maximizes the weight sum of the match. It has been shown that 100 throughput can be achieved for all admissible i.i.d. arrivals [3] with MWM and w i;o = q i;o (LQF) Algorithms for solving the MWM problem are computationally complex ....

C. H. Papadimitriou and K. Steiglitz, Combinatorial Optimization. Prentice-Hall, Inc., 1982.

....in i 1. In each time slot the arbiter selects unique pairs of input and output ports (a match (i; o) based on weight information sent to it from the input ports. For this bipartite graph matching problem [15] fig 2) the optimum maximum size matching (MSM) or a maximum weight matching (MWM) [18] algorithms exist. With a weight chosen as w i;o = q i;o the algorithm is called longest queue first (LQF) in [14] and it has been shown that 100 throughput can be achieved for all admissible i.i.d. arrivals [14] with MWM. Without weight information, as with MSM, instability and unfairness are ....

C. H. Papadimitriou and K. Steiglitz. Combinatorial Optimization. Prentice-Hall, Inc., 1982.

....of the complement problem. The following lemma called Farkas lemma [NW88, Sch87] is crucial to understanding and analyzing the dual. Lemma 4.1 Either f x 2 R n : A x bg 6= OE or ( exclusively ) 9 y 2 R m , such that, y T A 0 T and y T : b = Gamma1. Proof 4. 1 See [Sch87, PS82, NW88]. The lemma is interpreted as follows: If the primal system is infeasible, then there exists a proof of this infeasibility. This proof takes the form of a witness vector which is unbounded in the dual space. Applying the lemma to our problem, we note that query (10) requires the system G: s ....

C. H. Papadimitriou and K. Steiglitz. Combinatorial Optimization. Prentice Hall, 1982.

....(e.g. hard, pre placed, soft, and rectilinear modules) directly and efficiently. In contrast, other representation cannot handle some special modules well. To deal with soft modules using sequence pair, for example, previous work in [8] transformed it to the convex problem (which is NP hard [13]) processed the soft modules, and then transformed back to the sequence pair. ffl B trees do not need to construct constraint graphs for area cost evaluation. In contrast, the sequence pair and O tree representa tions require encoding module sequences for tree operations and constructing ....

C. H. Papadimitriou, and K. Steiglitz, Combinatorial Optimization, Prentice Hall, 1982.

....the complement problem. The following lemma called Farkas lemma [NW88, Sch87] is crucial to understanding and analyzing the dual. Lemma 5.1 Either f x 2 R n : A x bg 6= OE or ( exclusively ) 9 y 2 R m , such that, y T A 0 T and y T : b = Gamma1. Proof 5. 1 See [Sch87, PS82, NW88]. The lemma is interpreted as follows: Either the primal system viz. fA: x] b; x 0g is feasible, in which case the associated polyhedron is non empty or ( exclusively ) the vector b lies in the polar cone of of the dual space viz. f y T :A 0, y 0g. In the latter case, the ....

....associated polyhedron is non empty or ( exclusively ) the vector b lies in the polar cone of of the dual space viz. f y T :A 0, y 0g. In the latter case, the function y T : b is unbounded below and its minimum is Gamma1. For a geometric interpretation of the lemma, refer [PS82]. Query (9) requires the system G: s g to be infeasible for a particular e 2 E. Farkas lemma assures us that this is possible only if 9 y 0 2 R m , such that y 0 T :G 0; y 0 T : b Gamma B: e) Gamma1 (10) which implies that G T : y 0 0; y 0 ....

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C. H. Papadimitriou and K. Steiglitz. Combinatorial Optimization. Prentice Hall, 1982.

....performed on the energy function. As the temperature is gradually reduced, the doubly substochastic correspondence matrix approaches a permutation matrix (with binary outlier rejection) In more formal terms, the energy function in (4) is a linear assignment or weighted bipartite matching problem [15] with respect to the correspondences. Softassign used within deterministic annealing has been shown to nd the optimal solution to linear assignment [12] However, while softassign and the TPS solution are independently optimal, the combination is not. Our approach is only guaranteed to nd a ....

C. Papadimitriou and K. Steiglitz. Combinatorial Optimization. Prentice-Hall, Inc., Englewood Clis, NJ, 1982.

....two edges in share an endpoint. The weight of a matching is the sum of the weights of the edges in the matching. A maximum weight matching is a matching of maximum weight. Maximum weight matchings in bipartite graphs can be found in polynomial time ( for example, using the Hungarian algorithm [15]. Given a matching in , define . That is, EQ 15) Lemma 3: The graph consists of only directed paths, meaning that if , then there is no such that or . Proof: Suppose that . This means that . Then we cannot have such that . Indeed, if were true, then this would mean that and would contradict ....

C. Papadimitriou, K. Steiglitz, Combinatorial Optimization, Dover, 1998.

....and output a locally optimal solution. This notion of applying local improvement heuristics to hard optimization problems was around [30] long before the invention of NP completeness [83] It has been applied heuristically to solve a variety of NP hard problems in combinatorial optimization [121]. As we mentioned in Section 1.10, a recent breakthrough in this area was the work of Furer and Raghavachari [50] who applied this technique to the minimum degree spanning tree problem. They showed a version of a local optimization algorithm runs in polynomial time to produce near optimal ....

....sections, we prove Theorems 7.1.1 and 7.1.2. We close this chapter with some remarks on related problems. 107 7.2 Related Work The technique of local optimization has been applied to provide heuristic solutions for a variety of hard problems in combinatorial optimization. Chapter 19 of [121] surveys a few applications of this technique. Some notable examples are its applications to the graph partitioning [84] and the Traveling Salesperson problems [102] A number of complexity results relating to the paradigm of local optimality and the time complexity of computing a locally optimal ....

C. H. Papadimitriou, and K. Steiglitz, "Combinatorial Optimization," Prentice-Hall Inc. (1982).

.... problem (or the maximum weighted matching problem on uniformly weighted graphs) several NC approximation algorithms have been known (see [KR98] for comprehensive reference) The maximum cardinality matching can be characterized by the notions of alternating paths and augmenting paths (see, e.g. [PS82] for the details) The NC approximation algorithms are based on the characterization. However, when the edges are weighted, we can not use the characterization. For example, for a path (e 1 ; e 2 ; e 3 ; e 4 ; e 5 ; e 6 ) when w(e 1 ) w(e 3 ) w(e 4 ) w(e 6 ) 1 and w(e 2 ) w(e 5 ) 3, ....

C.H. Papadimitriou and K. Steiglitz. Combinatorial Optimization. Dover, 1982.

....INRIA Mesh Partitioning Techniques and New Observations for 3 regular Graphs 5 2.1 The Kernighan Lin Algorithm The Kernighan Lin (KL) algorithm is a procedure to locally improve the size of the cut K. It was developed in the early 70 s but it has become a fundamental textbook graph algorithm [8]. Suppose we have a bisection (V 1 ; V 2 ) with cut size K. For any vertex a let D(a) be the out degree (edges going outside of its subset) minus the in degree (edges within its subset) Moving a to the other partition results in a new cutsize. K 0 = K Gamma D(a) and thus exchanging two ....

C.H. Papadimitriou and K. Steiglitz. -- Combinatorial Optimization. -- Prentice-Hall, 1982.

....in standard form: max c: x; s:t: A: x = b; x 0 (4) where c is a n Gammavector, b is a m Gammavector, A is m Theta n rational matrix and x is a n Gammavector. The first algorithm for this problem was proposed in [Dan63] This algorithm has exponential time worst case complexity [PS82, KM72]. Since then a number of polynomial time algorithms have been developed for this problem [Kha79, Vai87, Kar84] 8 B Convex Programming Complexity Issues Consider the standard nonlinear program, expressed as: minf(x) s:t: g i (x) 0; 8i = 1; m (5) 6) where the g i (x) are convex ....

....and the constraint set formed by their intersection is a convex space in the n Gammadimensional Euclidean space R n . If f is a convex funtion [HuL93, Roc70] then the problem is called a convex minimization or convex programming problem. This problem is known to be solvable in polynomial time [HuL93, PS82]. A fast algorithm for this problem is provided in [KV86] C Matrix Reorganization In this section, we focus on manipulating a system of linear inequalities, through the reorganization of its representative matrix. Consider a linear system of inequalities in two variables x and y:A: x; ....

C. H. Papadimitriou and K. Steiglitz. Combinatorial Optimization. Prentice Hall, 1982.

....of other problems including the traveling salesman problem, graph coloring etc. 4] Existing techniques for determining satisfiability are, at their core, some form of combinatorial search [17, 16, 6, 5, 3] Although the satisfiability problem can be converted into a linear programming problem [14, 15], the resulting problem still involves a searching phase to avoid local minimums. On the other hand, the recent results on the statistical behavior of satisfiability [12, 8] the existence of phase transition type behavior might be explainable using the analytic tools developed in this paper. ....

C. H. Papadimitriou and K. Steiglitz. Combinatorial Optimization. Prentice Hal, Englwood Cli#s, NJ, 1982.

....subject to Ax b and, x j = 0 or 1 for j = 1; n; 4.2) where A is a rational m Theta n matrix, b a rational m vector and c a rational n vector. It is well known that IP is NP complete [19] A standard technique used to solve this problem is the branch andbound (see, e.g. Chapter 18 in [16]) The new algorithm to be introduced now can be considered as an alternative to the branch and bound methods. Let P = fx : Ax b and 0 x 1g: Then we have the following lemma. Lemma 4.1. A point x in P is an optimal solution to the IP (4.2) if and only if x maximizes cx over all vertices of ....

C.H. Papadimitriou and K. Steiglitz, Combinatorial Optimization, Printice-Hall, 1982.

....selects unique pairs of input and output ports (a match (i; o) Dynamic arbitration algorithms decide based on information sent to it from the input ports. For this bipartite graph matching problem [2] a number of solution algorithms exist based on maximum size or weight matching (MWM, MSM) [8]. It has been shown that 100 throughput can be achieved for uniform and independent traffic [3] with MWM. A number of approximations for the computationally complex MSM and MWM have been proposed, e.g. MCFF, iMCFF [4] and SIMP [7] or the iterative algorithms PIM [9] and iSLIP [2] Static ....

C. H. Papadimitriou and K. Steiglitz, Combinatorial Optimization. Prentice-Hall, Inc., 1982.

....output ports (a match (i; o) based on information sent to it from the input ports. This task is equivalent to a bipartite graph matching problem [2] as shown in fig 2. A number of algorithms exist to solve it optimally, either as a maximum size matching (MSM) or a maximum weight matching (MWM) [9]. MSM finds a match with the maximum number of input output connection crosspoints. MWM finds a match where the sum of the weights of the crosspoints w i;o are maximum. For the weight chosen as w i;o = q i;o the algorithm is called longest queue first (LQF) in [4] and it has been shown that ....

C. H. Papadimitriou and K. Steiglitz, Combinatorial Optimization. Prentice-Hall, Inc., 1982.

....Brucker and Kramer [1993] Sect. 3.3.1 Brucker and Kramer [1993] Sect. 3.3.1 O2 j fix ij j C max O(n) O2 j r i j C max ,O2 jj L max Brucker and Kramer [1993] Sect. 3.3. 1 Lawler et al. 1981,1982] O3 jj C max Gonzales and Sahni [1976] O2jtree j C max Lenstra [ O2 jj P C i Achugbue and Chin [1982] O j fix ij ; n = k; prec; r i ; stages = rjf max ( P f i ) O(1) O j n = 3jC max Om j fix ij ; n = k; prec; r i jf max ( P f i ) Gonzales and Sahni [1976] Sect. 3.3.1 Sect. 3.3.1 Ojfix ij ; p ij = 1; stages = rjC max , p(log n) Omjfix ij ; p ij = 1jC max Ojfix ij ; p ij = 1; stages = rjf ....

C.H. Papadimitriou, K. Steiglitz [1982] Combinatorial optimization, Prentice-Hall, Englewood Cliffs, New Jersey.

....for E 388 the binary decision array C, the value which minimizes function tcost(QS, C) E 385 subject to constraints (1) 2) 3) expressed in Section 4.1. VFP is a 0 1 integer E 372 linear programming problem of kind set covering with additional constraints, and is E 392 known to be NP hard [14]. In this section we propose a branch and bound approach to E 391 solve it optimally. E 87 The essential ingredients of a branch and bound procedure for a discrete E 370 optimization problem such as VFP are [1] E 211 1. A branching rule for breaking up the problem into subproblems. Let VFP a ....

Papadimitriou, C.H., Steiglitz, K.: Combinatorial optimization. Prentice Hall,<E-380> Englewood Cliffs (1982)<E-139>

....for subsequences with the adjacent sequence interchange (ASI) property. In the first part of this paper we derive sufficient conditions for the existence of such relations. Based on these relations we obtain new conditions which are necessary and sufficient for the optimality of permutations. In Papadimitriou Steiglitz [1982] the notion of exact neighborhoods can be found. These are neighborhoods in which each local optimum is globally optimal. Tovey [1985] has shown that problems with polynomially bounded exact neighborhoods are in NP co NP . An example of a polynomially solvable problem whose minimal exact ....

....These are neighborhoods in which each local optimum is globally optimal. Tovey [1985] has shown that problems with polynomially bounded exact neighborhoods are in NP co NP . An example of a polynomially solvable problem whose minimal exact neighborhood has an exponential size can be found in Papadimitriou Steiglitz [1982]. This implies that a polynomially bounded exact neighborhood does not exist for all polynomially solvable problems. In the second part of this paper we show that polynomially bounded exact neighborhoods exist for all our considered problems. This paper is organized as follows. In Section 2 the ....

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Papadimitriou, C.H., Steiglitz, K. [1982] Combinatorial Optimization, Wiley, New York.

....every flow network with integral capacities admits an integral maximum flow. But even more, there is an integral 6 C. FREMUTH PAEGER AND D. JUNGNICKEL optimum flow in the general setting This results from the total unimodularity of adjacency matrices of digraphs, which we shall not discuss (see [14] for example) Note that all of the known network flow algorithms actually yield integral solutions (At least the preflow push method [12] 13] and all network simplex based algorithms [1] 5] Thus we can determine an integral optimum flow f , say of value v , instead of a balanced ....

C.H. Papadimitriou and K. Steiglitz. Combinatorial Optimization. Prentice Hall, Englewood Cliffs, N.J., 1982.

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C. H. Papadimitriou and K. Steiglitz, Combinatorial Optimization, Prentice-Hall, Englewood Cli#s, NJ, 1982.

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C. H. Papadimitriou and K. Steiglitz. Combinatorial Optimization. Dover Publications, Inc. NY, 1998.

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C. H. Papadimitriou and K. Steiglitz. Combinatorial Optimization. Prentice-Hall, NJ, 1982.

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C. H. Papadimitriou and K. Steiglitz. Combinatorial Optimization. Prentice-Hall, NJ, 1982.

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Papadimitriou, C., Steiglitz, K., 1998. Combinatorial Optimization. Dover Publ., Mineola, New York.

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C. H. Papadimitriou and K. Steiglitz, Combinatorial Optimization. Prentice Hall, 1982.

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C. H. Papadimitriou and K. Steiglitz. Combinatorial Optimization. Prentice-Hall, 1982. 35

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Papadimitriou, C. and K. Steiglitz. 1998. Combinatorial Optimization. Dover Publ., Mineola, New York.

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C. H. Papadimitriou and K. Steiglitz, Combinatorial Optimization, Prentice-Hall, 1982.

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C.H. Papadimitriou and K. Steiglitz. Combinatorial optimization. Prentice Hall, Englewood Cliffs, 1982.

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C. H. Papadimitriou and K. Steiglitz, Combinatorial Optimization, Prentice Hall, 1982.

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C. Papadimitriou, K. Steiglitz, Combinatorial Optimization, Dover, 1998.

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C. H. Papadimitriou and K. Steiglitz, Combinatorial Optimization, Prentice Hall, 1982.

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C. H. Papadimitriou and K. Steiglitz, Combinatorial Optimization, prentice Hall, 1982.

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C.Papadimitriou, K.Steiglitz, Combinatorial Optimization, Dover, 1998.

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