Approximation hardness of deadline-TSP reoptimization

AbstractGiven an instance of an optimization problem together with an optimal solution, we consider the scenario in which this instance is modified locally. In graph problems, e.g., a singular edge might be removed or added, or an edge weight might be varied, etc. For a problem U and such a local modification operation, let lm-U (local-modification-U) denote the resulting problem. The question is whether it is possible to exploit the additional knowledge of an optimal solution to the original instance or not, i.e.,whether lm-U is computationally more tractable than U. While positive examples are known e.g. for metric TSP, we give some negative examples here: Metric TSP with deadlines (time windows), if a single deadline or the cost of a single edge is modified, exhibits the same lower bounds on the approximability in these local-modification versions as those currently known for the original problem

On the Approximability of TSP on Local Modifications of Optimally Solved Instances

Given an instance of TSP together with an optimal solution, we consider the scenario in which this instance is modified locally, where a local modification consists in the alteration of the weight of a single edge. More generally, for a problem U, let LM-U (local-modification-U) denote the same problem as U, but in LM-U, we are also given an optimal solution to an instance from which the input instance can be derived by a local modification. The question is how to exploit this additional knowledge, i.e., how to devise better algorithms for LM-U than for U. Note that this need not be possible in all cases: The general problem of LM-TSP is as hard as TSP itself, i.e., unless P=NP, there is no polynomial-time p(n)-approximation algorithm for LM-TSP for any polynomial p. Moreover, LM-TSP where inputs must satisfy the β-triangle inequality (LM-Δβ-TSP) remains NP-hard for all β>½. However, for LM-Δ-TSP (i.e., metric LM-TSP), we will present an efficient 1.4-approximation algorithm. In other words, the additional information enables us to do better than if we simply used Christofides' algorithm for the modified input. Similarly, for all 1<β<3.34899, we achieve a better approximation ratio for LM-Δ-TSP than for Δβ-TSP. For ½≤β<1, we show how to obtain an approximation ratio arbitrarily close to 1, for sufficiently large input graphs

Reusing optimal TSP solutions for locally modified input instances : Extended abstract

Given an instance of an optimization problem together with an optimal solution, we consider the scenario in which this instance is modified locally. In graph problems, e. g., a singular edge might be removed or added, or an edge weight might be varied, etc. For a problem U and such a local modification operation, let lm-U (local-modification- U) denote the resulting problem. The question is whether it is possible to exploit the additional knowledge of an optimal solution to the original instance or not, i. e., whether lm-U is computationally more tractable than U. Here, we give non-trivial examples both of problems where this is and problems where this is not the case4th IFIP International Conference on Theoretical Computer ScienceRed de Universidades con Carreras en Informática (RedUNCI

Online Coloring of Bipartite Graphs with and without Advice

In the online version of the well-known graph coloring problem, the vertices appear one after the other together with the edges to the already known vertices and have to be irrevocably colored immediately after their appearance. We consider this problem on bipartite, i.e., two-colorable graphs. We prove that at least ⌊1.13746⋅log2(n)−0.49887⌋ colors are necessary for any deterministic online algorithm to be able to color any given bipartite graph on n vertices, thus improving on the previously known lower bound of ⌊log2 n⌋+1 for sufficiently large n. Recently, the advice complexity was introduced as a method for a fine-grained analysis of the hardness of online problems. We apply this method to the online coloring problem and prove (almost) tight linear upper and lower bounds on the advice complexity of coloring a bipartite graph online optimally or using 3 colors. Moreover, we prove that $O(\sqrt{n})$ advice bits are sufficient for coloring any bipartite graph on n vertices with at most ⌈log2 n⌉ colors

Reusing optimal TSP solutions for locally modified input instances : Extended abstract

Given an instance of an optimization problem together with an optimal solution, we consider the scenario in which this instance is modified locally. In graph problems, e. g., a singular edge might be removed or added, or an edge weight might be varied, etc. For a problem U and such a local modification operation, let lm-U (local-modification- U) denote the resulting problem. The question is whether it is possible to exploit the additional knowledge of an optimal solution to the original instance or not, i. e., whether lm-U is computationally more tractable than U. Here, we give non-trivial examples both of problems where this is and problems where this is not the case4th IFIP International Conference on Theoretical Computer ScienceRed de Universidades con Carreras en Informática (RedUNCI

Reoptimization of the Shortest Common Superstring Problem

A reoptimization problem describes the following scenario: given an instance of an optimization problem together with an optimal solution for it, we want to find a good solution for a locally modified instance. In this paper, we deal with reoptimization variants of the shortest common superstring problem (SCS) where the local modifications consist of adding or removing a single string. We show the NP-hardness of these reoptimization problems and design several approximation algorithms for them. First, we use a technique of iteratively using any SCS algorithm to design an approximation algorithm for the reoptimization variant of adding a string whose approximation ratio is arbitrarily close to 8/5 and another algorithm for deleting a string with a ratio tending to 13/7. Both algorithms significantly improve over the best currently known SCS approximation ratio of 2.5. Additionally, this iteration technique can be used to design an improved SCS approximation algorithm (without reoptimization) if the input instance contains a long string, which might be of independent interest. However, these iterative algorithms are relatively slow. Thus, we present another, faster approximation algorithm for inserting a string which is based on cutting the given optimal solution and achieves an approximation ratio of 11/6. Moreover, we give some lower bounds on the approximation ratio which can be achieved by algorithms that use such cutting strategie

On the inapproximability of the metric traveling salesman problem

One of the central tasks in combinatorial optimization is to classify the NP-hard optimization problems according to their approximability. One of the best-known and most important optimization problems is the traveling salesman problem (TSP). In this thesis we consider the metric traveling salesman problem, i.e. the special case of the TSP where the edge costs obey the triangle inequality. Under the assumption P not equal NP we will prove a lower bound of 3813/3812 - epsilon on the polynomial-time approximability of the metric TSP for an arbitrary small epsilon > 0. This improves over the previously known lower bound of 5381/5380 - epsilon (for an arbitrary small epsilon > 0) that was proved by Engebretsen [En99]. Engebretsen used a gap-preserving reduction from a problem about linear equations, the so-called LinEq2-2(3) problem, to the subproblem of the TSP where all edge costs are equal to 1 or 2. We generalize this proof method and consider a reduction from LinEq2-2(3) to those Delta-TSP instances where all edge costs are from {1,2,3}. This modification requires essential changes in the construction of Engebretsen and several new technical considerations. Furthermore we apply our proof method to derive lower bounds on the approximability of the TSP with parameterized triangle inequality