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A 3/4approximation algorithm for maximum ATSP with weights zero and one
 Proc. of the 7th Int. Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), volume 3122 of Lecture Notes in Computer Science
, 2004
"... We present a polynomial time 3/4approximation algorithm for the maximum asymmetric TSP with weights zero and one. As applications, we get a 5/4approximation algorithm for the (minimum) asymmetric TSP with weights one and two and a 3/4approximation algorithm for the Maximum Directed Path Packing ..."
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We present a polynomial time 3/4approximation algorithm for the maximum asymmetric TSP with weights zero and one. As applications, we get a 5/4approximation algorithm for the (minimum) asymmetric TSP with weights one and two and a 3/4approximation algorithm for the Maximum Directed Path Packing Problem.
Improved approximation algorithms for metric maximum ATSP and maximum 3cycle cover problems
 In Proc. of the 9th Workshop on Algorithms and Data Structures (WADS
, 2005
"... We consider an APXhard variant (∆MaxATSP) and an APXhard relaxation (Max3DCC) of the classical traveling salesman problem. We present a 31 40approximation algorithm for ∆MaxATSP and a 3 4approximation algorithm for Max3DCC with polynomial running time. The results are obtained via a new ..."
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We consider an APXhard variant (∆MaxATSP) and an APXhard relaxation (Max3DCC) of the classical traveling salesman problem. We present a 31 40approximation algorithm for ∆MaxATSP and a 3 4approximation algorithm for Max3DCC with polynomial running time. The results are obtained via a new way of applying techniques for computing undirected cycle covers to directed problems.
On approximating restricted cycle covers
 In Workshop on Approximation and Online Algorithms (WAOA
, 2005
"... A cycle cover of a graph is a set of cycles such that every vertex is part of exactly one cycle. An Lcycle cover is a cycle cover in which the length of every cycle is in the set L. The weight of a cycle cover of an edgeweighted graph is the sum of the weights of its edges. We come close to settli ..."
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A cycle cover of a graph is a set of cycles such that every vertex is part of exactly one cycle. An Lcycle cover is a cycle cover in which the length of every cycle is in the set L. The weight of a cycle cover of an edgeweighted graph is the sum of the weights of its edges. We come close to settling the complexity and approximability of computing Lcycle covers. On the one hand, we show that for almost all L, computing Lcycle covers of maximum weight in directed and undirected graphs is APXhard. Most of our hardness results hold even if the edge weights are restricted to zero and one. On the other hand, we show that the problem of computing Lcycle covers of maximum weight can be approximated within a factor of 2 for undirected graphs and within a factor of 8/3 in the case of directed graphs. This holds for arbitrary sets L. 1
Approximation algorithms for restricted cycle covers based on cycle decompositions
 Proc. of the 32nd Int. Workshop on GraphTheoretical Concepts in Computer Science (WG
"... Abstract. A cycle cover of a graph is a set of cycles such that every vertex is part of exactly one cycle. An Lcycle cover is a cycle cover in which the length of every cycle is in the set L ⊆ N. For most sets L, the problem of computing Lcycle covers of maximum weight is NPhard and APXhard. We ..."
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Abstract. A cycle cover of a graph is a set of cycles such that every vertex is part of exactly one cycle. An Lcycle cover is a cycle cover in which the length of every cycle is in the set L ⊆ N. For most sets L, the problem of computing Lcycle covers of maximum weight is NPhard and APXhard. We devise polynomialtime approximation algorithms for Lcycle covers. More precisely, we present a factor 2 approximation algorithm for computing Lcycle covers of maximum weight in undirected graphs and a factor 20/7 approximation algorithm for the same problem in directed graphs. Both algorithms work for arbitrary sets L. To do this, we develop a general decomposition technique for cycle covers. Finally, we show tight lower bounds for the approximation ratios achievable by algorithms based on such decomposition techniques. 1
Minimumweight Cycle Covers and Their Approximability
, 2008
"... A cycle cover of a graph is a set of cycles such that every vertex is part of exactly one cycle. An Lcycle cover is a cycle cover in which the length of every cycle is in the set L ⊆ N. We investigate how well Lcycle covers of minimum weight can be approximated. For undirected graphs, we devise no ..."
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A cycle cover of a graph is a set of cycles such that every vertex is part of exactly one cycle. An Lcycle cover is a cycle cover in which the length of every cycle is in the set L ⊆ N. We investigate how well Lcycle covers of minimum weight can be approximated. For undirected graphs, we devise nonconstructive polynomialtime approximation algorithms that achieve constant approximation ratios for all sets L. On the other hand, we prove that the problem cannot be approximated with a factor of 2 − ε for certain sets L. For directed graphs, we devise nonconstructive polynomialtime approximation algorithms that achieve approximation ratios of O(n), where n is the number of vertices. This is asymptotically optimal: We show that the problem cannot be approximated with a factor of o(n) for certain sets L. To contrast the results for cycle covers of minimum weight, we show that the problem of computing Lcycle covers of maximum weight can, at least in principle, be approximated arbitrarily well.
MultiCriteria TSP: Min and Max Combined
, 2009
"... We present randomized approximation algorithms for multicriteria traveling salesman problems (TSP), where some objective functions should be minimized while others should be maximized. For the symmetric multicriteria TSP (STSP), we present an algorithm that computes (2/3 − ε, 4 + ε) approximate Pa ..."
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We present randomized approximation algorithms for multicriteria traveling salesman problems (TSP), where some objective functions should be minimized while others should be maximized. For the symmetric multicriteria TSP (STSP), we present an algorithm that computes (2/3 − ε, 4 + ε) approximate Pareto curves. Here, the first parameter is the approximation ratio for the objectives that should be maximized, and the second parameter is the ratio for the objectives that should be minimized. For the asymmetric multicriteria TSP (ATSP), we present an algorithm that computes (1/2 − ε, log 2 n + ε) approximate Pareto curves. In order to obtain these results, we simplify the existing approximation algorithms for multicriteria MaxSTSP and MaxATSP. Finally, we give algorithms with improved ratios for some special cases.
Approximability of Minimumweight Cycle Covers
, 2006
"... A cycle cover of a graph is a set of cycles such that every vertex is part of exactly one cycle. An Lcycle cover is a cycle cover in which the length of every cycle is in the set L ⊆ N. For most sets L, computing Lcycle covers of minimum weight is NPhard and APXhard. While computing Lcycle cove ..."
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A cycle cover of a graph is a set of cycles such that every vertex is part of exactly one cycle. An Lcycle cover is a cycle cover in which the length of every cycle is in the set L ⊆ N. For most sets L, computing Lcycle covers of minimum weight is NPhard and APXhard. While computing Lcycle covers of maximum weight admits constant factor approximation algorithms (both for undirected and directed graphs), almost nothing is known so far about the approximability of computing Lcycle cover of minimum weight. We investigate how well Lcycle covers of minimum weight can be approximated. For undirected graphs, we give a positive answer: We devise a polynomialtime algorithm for approximating the Lcycle cover problem in undirected graphs. Our algorithm achieves an approximation ratio of 4 and works for all sets L. For directed graphs, we give a negative answer by proving an unconditional inapproximability results: If the set L is immune, then the problem of computing Lcycle covers of minimum weight in directed graphs cannot be approximated within a factor of o(n) where n is the number of vertices. Finally, we present an improved approximation algorithm for computing Lcycle covers of maximum weight in directed graphs. This algorithm achieves an approximation ratio of 8/3.
On the Complexity of a Matching Problem with Asymmetric Weights
"... We present complexity results regarding a matchingtype problem related to structural controllability of dynamical systems modelled on graphs. Controllability of a dynamical system is the ability to choose certain inputs in order to drive the system from any given state to any desired state; a grap ..."
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We present complexity results regarding a matchingtype problem related to structural controllability of dynamical systems modelled on graphs. Controllability of a dynamical system is the ability to choose certain inputs in order to drive the system from any given state to any desired state; a graph is said to be structurally controllable if it represents the structure of a controllable system. We define the Orientation Control Matching problem (OCM) to be the problem of orienting an undirected graph in a manner that maximizes its structural controllability. A generalized version, the Asymmetric Orientation Control Matching problem (AOCM), allows for asymmetric weights on the possible directions of each edge. These problems are closely related to 2matchings, disjoint path covers, and disjoint cycle covers. We prove using reductions that OCM is polynomially solvable, while AOCM is much harder; we show that it is NPcomplete as well as APXhard. 1
Appliations of discrepancy theory in . . .
, 2011
"... We apply a multicolor extension of the BeckFiala theorem to show that the multiobjective maximum traveling salesman problem is randomized 1/2approximable on directed graphs and randomized 2/3approximable on undirected graphs. Using the same technique we show that the multiobjective maximum satis ..."
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We apply a multicolor extension of the BeckFiala theorem to show that the multiobjective maximum traveling salesman problem is randomized 1/2approximable on directed graphs and randomized 2/3approximable on undirected graphs. Using the same technique we show that the multiobjective maximum satisfiabilty problem is 1/2approximable.