CS 880: Approximation Algorithms Homework 3
Out: 3/15/07 Due: 4/12/07
1. (Integrality gaps.) In both of the following cases try to find as large a gap as you can.
(a) Give an integrality gap example for the facility location LP that we discussed in class. (That is, give an
example for which the optimal LP solution is much smaller than the optimal integral solution.)
(b) Given an integrality gap example for the Steiner tree LP that we discussed in class.
2. (Integrality of the Min-cut polytope.) Recall the following LP for the s-tmin-cut problem from class:
min X
e∈E
cexesubject to
X
e∈P
xe≥1∀s-tpaths Pin G= (V, E )
xe≥0∀e∈E
Prove that all basic solutions to this LP are integral.
(Hint: Show that any optimal fractional solution can be written as a convex combination of integral cuts.)
3. (Cuts and `1metrics.) A metric dover a set Vis said to be an `1metric if the points can be mapped to points
in Rkfor some k, such that the distance between any two points according to dis the `1distance between their
mappings: d(x, y) = Pi|xi−yi|.
Also, a linear combination over cuts {αS}S⊂Vdefines the following metric µα(verify that this is indeed a
metric):
µα(x, y) = X
S⊂V:|S∩{x,y}|=1
αS∀x6=y∈V
(Note that PS⊂VαSis not necessarily equal to 1.)
In this problem you will show that the above two classes of metrics—`1metrics and linear combinations of
cuts—are in fact equivalent.
(a) Prove that any metric defined by a linear combination of cuts is an `1metric.
(b) Prove that any `1metric can be expressed as a linear combination of cuts. (Hint: Prove this statement for
a unit-dimensional `1metric first, that is, k= 1. Then extend it to multiple dimensions.)
4. (Prize-collecting Steiner tree.) The prize-collecting Steiner tree problem (PCST) is a variant of Steiner tree in
which there are prizes πvon nodes and costs ceon edges, and a special node rcalled the root. The goal is to
construct a Steiner tree containing rthat minimizes the cost of the edges in the tree plus the value of the nodes
not in the tree.
(a) Give an LP relaxation for this problem using xeas an indicator of the extent to which an edge is included
in the solution, and yvas an indicator of the extent to which a node is covered. (It is okay to have an
exponential number of constraints, as for the Steiner forest LP we studied in class.)
(b) Write the dual of the above LP.
(c) Give a primal dual algorithm for this problem based on the one for Steiner tree (forest). (Don’t forget the
pruning step!)
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