HW 4: due Thurs, March 10
NOTE: No class on Thurs, Feb 24: Tufts is on a MONDAY schedule
Also, do problem 2 first; you won’t be able to do Problem 1 until
after the March 3 lecture.
1. The fault-tolerant version of the k-center problem with triangle inequal-
ity has an additional input α≤kwhich specifies the number of centers
that each vertex must be connected to. In other words, we assume that
up to α−1 centers might be closed, and so the fault-tolerant cost for
a vertex is its distance to its αth closest center. The problem is to
pick kcenters so that the maximum fault-tolerant cost of a vertex is
minimized. A set S⊆Vin an undirected graph H= (V, E) is an α-
dominating set if each vertex v∈Vis adjacent to at least αvertices in
S(we consider a vertex to be adjacent to itself). Let domα(H) denote
the size of a minimum cardinality α-dominating set in H.
(a) Let Ibe an independent set in H2. Show that α|I| ≤ domα(H).
(b) Give a factor 3 approximation algorithm for the fault-tolerant k-
center problem (Hint: Compute a maximal independent set Miin
G2
i, for 1 ≤i≤m. Find the smallest index isuch that |Mi| ≤ b k
αc,
and moreover, the degree of each vertex of Miin Giis ≥α−1.)
2. Let Gbe a complete undirected graph in which all edge lengths are
either 1 or 2.
(a) Show Gsatisfies the triangle inequality.
(b) (∗) Show that you can find a minimum 2-factor in polynomial
time in this graph, where a 2-factor is a subgraph in which every
vertex is incident to exactly two edges in the 2-factor. (Remark:
the definition of k-factor is obvious; a perfect matching can also
be called a 1-factor).
(c) Give a polynomial-time algorithm to find a 4/3-approximation to
the optimum TSP in G. (You may use part b even if you were
not able to do it).