Labeled Traveling Salesman Problems: Complexity and approximation

We consider labeled Traveling Salesman Problems, defined upon a complete graph of n vertices with colored edges. The objective is to find a tour of maximum or minimum number of colors. We derive results regarding hardness of approximation and analyze approximation algorithms, for both versions of the problem. For the maximization version we give a $1/2$-approximation algorithm based on local improvements and show that the problem is APX-hard. For the minimization version, we show that it is not approximable within $n^{1-\epsilon}$ for any fixed $\epsilon>0$. When every color appears in the graph at most $r$ times and $r$ is an increasing function of $n$, the problem is shown not to be approximable within factor $O(r^{1-\epsilon})$. For fixed constant $r$ we analyze a polynomial-time $(r +H_r)/2$ approximation algorithm, where $H_r$ is the $r$-th harmonic number, and prove APX-hardness for $r = 2$. For all of the analyzed algorithms we exhibit tightness of their analysis by provision of appropriate worst-case instances

Least and most colored bases

Consider a matroid M = (E, B), where M denotes the family of bases of A and assign a color c(e) to every element e E E (the same color can go to more than one element). The palette of a subset F of E, denoted by c(F), is the image of F under c. Assume also that colors have prices (in the form of a function pi(l), where e is the label of a color), and define the chromatic price as: pi(F) = Sigma(l is an element of C(F))pi(l). We consider the following problem: find a base B is an element of M such that pi(B) is minimum. We show that the greedy algorithm delivers a In r (M)-approximation of the unknown optimal value, where r(M) is the rank of matroid X. By means of a reduction from SETCOVER, we prove that the In r(M) ratio cannot be further improved, even in the special case of partition matroids, unless NP is an element of DTIME((n)log log n). The results apply to the special case where It is a graphic matroid and where the prices pi(l) are restricted to be all equal. This special case was previously known as the minimum label spanning tree (MLST) problem. For the MLST, our results improve over the In(n - 1) + I ratio achieved by Wan, Chen and Xu in 2002. Inspired by the generality of our results, we study the approximability of coloring problems with different objective function pi(F), where F is a common independent set on matroids M-1,..... M-k and, more generally, to independent systems characterized by the k-for-1 property

Least and most colored bases

Consider a matroid M = (E, B), where M denotes the family of bases of A and assign a color c(e) to every element e E E (the same color can go to more than one element). The palette of a subset F of E, denoted by c(F), is the image of F under c. Assume also that colors have prices (in the form of a function pi(l), where e is the label of a color), and define the chromatic price as: pi(F) = Sigma(l is an element of C(F))pi(l). We consider the following problem: find a base B is an element of M such that pi(B) is minimum. We show that the greedy algorithm delivers a In r (M)-approximation of the unknown optimal value, where r(M) is the rank of matroid X. By means of a reduction from SETCOVER, we prove that the In r(M) ratio cannot be further improved, even in the special case of partition matroids, unless NP is an element of DTIME((n)log log n). The results apply to the special case where It is a graphic matroid and where the prices pi(l) are restricted to be all equal. This special case was previously known as the minimum label spanning tree (MLST) problem. For the MLST, our results improve over the In(n - 1) + I ratio achieved by Wan, Chen and Xu in 2002. Inspired by the generality of our results, we study the approximability of coloring problems with different objective function pi(F), where F is a common independent set on matroids M-1,..... M-k and, more generally, to independent systems characterized by the k-for-1 property