483 research outputs found
Solving a "Hard" Problem to Approximate an "Easy" One: Heuristics for Maximum Matchings and Maximum Traveling Salesman Problems
We consider geometric instances of the Maximum Weighted Matching Problem
(MWMP) and the Maximum Traveling Salesman Problem (MTSP) with up to 3,000,000
vertices. Making use of a geometric duality relationship between MWMP, MTSP,
and the Fermat-Weber-Problem (FWP), we develop a heuristic approach that yields
in near-linear time solutions as well as upper bounds. Using various
computational tools, we get solutions within considerably less than 1% of the
optimum.
An interesting feature of our approach is that, even though an FWP is hard to
compute in theory and Edmonds' algorithm for maximum weighted matching yields a
polynomial solution for the MWMP, the practical behavior is just the opposite,
and we can solve the FWP with high accuracy in order to find a good heuristic
solution for the MWMP.Comment: 20 pages, 14 figures, Latex, to appear in Journal of Experimental
Algorithms, 200
Approximation of Euclidean k-size cycle cover problem
For a fixed natural number k, a problem of k collaborating salesmen servicing the same set of cities (nodes of a given graph) is studied. We call this problem the Minimumweight k-size cycle cover problem (or Min-k-SCCP) due to the fact that the problem has the following mathematical statement. Let a complete weighted digraph (with loops) be given; it is required to find a minimum-weight cover of the graph by k vertex-disjoint cycles. The problem is a simple generalization of the well-known Traveling Salesman Problem (TSP). We show that Min-k-SCCP is strongly NP-hard in the general case. Metric and Euclidean special cases of the problem are intractable as well. We also prove that the Metric Min-k-SCCP belongs to the APX class and has a 2-approximation polynomial-time algorithm. For the Euclidean Min-2-SCCP in the plane, we present a polynomial-time approximation scheme extending the famous result obtained by S. Arora for the Euclidean TSP. Actually, for any fixed c > 1, the scheme finds a (1 + 1/c)-approximate solution of the Euclidean Min-2-SCCP in O(n^3(log n)^O(c)) time
Target Assignment in Robotic Networks: Distance Optimality Guarantees and Hierarchical Strategies
We study the problem of multi-robot target assignment to minimize the total
distance traveled by the robots until they all reach an equal number of static
targets. In the first half of the paper, we present a necessary and sufficient
condition under which true distance optimality can be achieved for robots with
limited communication and target-sensing ranges. Moreover, we provide an
explicit, non-asymptotic formula for computing the number of robots needed to
achieve distance optimality in terms of the robots' communication and
target-sensing ranges with arbitrary guaranteed probabilities. The same bounds
are also shown to be asymptotically tight.
In the second half of the paper, we present suboptimal strategies for use
when the number of robots cannot be chosen freely. Assuming first that all
targets are known to all robots, we employ a hierarchical communication model
in which robots communicate only with other robots in the same partitioned
region. This hierarchical communication model leads to constant approximations
of true distance-optimal solutions under mild assumptions. We then revisit the
limited communication and sensing models. By combining simple rendezvous-based
strategies with a hierarchical communication model, we obtain decentralized
hierarchical strategies that achieve constant approximation ratios with respect
to true distance optimality. Results of simulation show that the approximation
ratio is as low as 1.4
Polynomial Time Approximation Scheme for the Minimum-weight k-Size Cycle Cover Problem in Euclidean space of an arbitrary fixed dimension
We study the Min-k-SCCP on the partition of a complete weighted digraph by k vertex-disjoint cycles of minimum total weight. This problem is the generalization of the well-known traveling salesman problem (TSP) and the special case of the classical vehicle routing problem (VRP). It is known that the problem Min-k-SCCP is strongly NP-hard and remains intractable even in the geometric statement. For the Euclidean Min-k-SCCP in Rd, we construct a polynomial-time approximation scheme, which generalizes the approach proposed earlier for the planar Min-2-SCCP. For any fixed c > 1, the scheme finds a (1 + 1/c)-approximate solution in time of O(nd+1(k log n)(O (√dc))d-1 2k). © 201
Comb inequalities for typical Euclidean TSP instances
We prove that even in average case, the Euclidean Traveling Salesman Problem
exhibits an integrality gap of for when the
Held-Karp Linear Programming relaxation is augmented by all comb inequalities
of bounded size. This implies that large classes of branch-and-cut algorithms
take exponential time for the Euclidean TSP, even on random inputs.Comment: 19 pages, 4 figure
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