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CS880: Approximations Algorithms Scribe: Dave Andrzejewski Lecturer: Shuchi Chawla Topic: Randomized routing, LP-Duality Date: 3/1/
The first part of the lecture shows how a randomized rounding scheme can be used to transform an optimal LP solution to a valid solution of the original problem. The specific problem used for this example is the min-congestion routing problem.
The second part of the lecture introduces the concept of LP-Duality, and the primal-dual interpre- tation of the maxflow-mincut problem.
12.1 Randomized rounding
12.1.1 Min congestion routing problem
GIVEN: a graph G = (V, E) and k pairs of demand vertices (si, ti). DO: find a single path pi from si to ti for every i, trying to minimize the congestion C. The congestion is defined as the number of paths going through the most-used edge in the graph.
C = max e∈E |{i|e ∈ pi}|
We then cast the problem as an LP where xie is defined as the amount of path i flow being sent through edge e.
min t obj fcn ∑
e∈δ+(v)
xie =
e∈δ−(v)
xie ∀i, ∀v 6 = si, ti
∑
e∈δ−(si)
xie =
e∈δ+(ti)
xie = 1 ∀i
∑
i
xie ≤ t ∀e
xie ≥ 0 ∀i, ∀e
The sets δ+(v) and δ−(v) correspond to the incoming and outgoing flow, respectively, for vertex v. The summation constraints then enforce flow conservation, and source/sink assignments.
Since the objective function is to minimize t, which is constrained to be an upper bound for the flow across any edge, t will give us the (possibly fractional) congestion for a solution point of this LP.
In order to recover an integral/unsplittable flow solution from the LP solution, we will consider an equivalent formulation of the LP.
In this alternative formulation xp refers to the flow along path p, and Pi is the set of all possible paths from si to ti.
min t obj fcn ∑
p∈Pi
xp = 1 ∀i ∑
{p|e∈p}
xp ≤ t ∀e
xp ≥ 0 ∀p
These formulations are equivalent in the sense that any solution to one can be converted into a solution of the other.
To convert from the first to the second, find all non-zero xie for a given i. Then, considering only these edges, perform DFS from si to find a path to ti. For this path p, set xp to the minimum xie value on the path, then subtract that amount from all xie on the path. That will effectively remove the minimum edge from our edge set. Repeat this procedure until there are no edges remaining, building a set of xp values as you go.
To convert from the second LP to the first, find all non-zero xp for a given Pi, then simply add that much flow to the edge flow xie for each e ∈ p.
It is important to note that the second formulation is impractical to solve directly, as the number of possible paths in the Pi sets will lead to exponentially many constraints. However the xp quantities will come in useful for our randomized rounding scheme, as we will see.
12.1.2 Randomized rounding transformation
As on previous problems, the optimal objective function value of our LP forms a lower bound on the true optimal solution of the original problem. That is, LP ∗^ ≤ OP T. However, we need a way to do rounding from the flows in our LP solution to legal unsplittable flows for the original problem.
Our approach will be to solve the LP in the first formulation, convert the solution to the second formulation, and then treat the xp values as path selection probabilities. For a given path set Pi, the constraints that all xp must be ≥ 0 and sum to 1 ensure that this is a valid probability distribution. The algorithm is then relatively simple.
- solve original formulation LP,
- convert solution LP ∗^ to second formulation
- for each i, pick a p ∈ Pi with probability xp
Obviously, this algorithm will select exactly one path for each (si, ti) pair, yielding a valid solution. What will the congestion be?
We then manipulate the right-hand side.
eλ (1 + λ)1+λ
)P i μi ≤
eλ λλ
)P i μi
(λ/e)λ
)P i μi
We now pick a value of λ for which the term λλ^ in the denominator becomes nO(1). Specifically we set λ = O( (^) log loglog^ n n ). We then substitute in our definition of λ.
λλ^ = exp(λ log λ)
= exp
log n log log n
(log log n − log log log n
= Θ(nc)
where c is a factor determined by the actual terms in the O() equation used to calculate λ. Setting λ appropriately to get c = 3, we plug this back into the original bound and end up with the result that for any edge e:
P r[Ce > (1 +
log n log log n
)t] < 1 /n^3
We take the union of this bound over all ≤ n^2 edges to get a total probability bound.
P (∃e|Ce > (1 +
log n log log n
)t) < 1 /n
By repeatedly applying our algorithm, we can then achieve an arbitrarily low probability of ex- ceeding an (1 + (^) log loglog^ n n )-approximation to OP T.
12.2 LP-Duality
12.2.1 Definition/Derivation of LP-Duality
Consider the following example linear program. (For more discussion of this example, see [1].)
min x + 4y x + 2y ≥ 5 2 x + y ≥ 4 x, y ≥ 0
How could we obtain a lower bound on the true optimal objective function value for this LP (without actually solving it, that is)?
We can take a non-negative linear combination of the constraint equations. Since x, y ≥ 0, if the x coefficient in our combination is ≤ the x coefficient in the objective function, and the same holds true for the y coefficients, our linear combination of constraints must also be ≤ the objective function for any legal x, y.
When taking linear combinations of the constraints, we will also end up a linear combination of the right-hand side of the constraints. Since it is a lower bound on the left-hand side, the linear combination of the right-hand sides of the constraints is also a lower bound on the objective function.
To see what we mean, call the linear combination coefficients u, v.
min x + 4y (x + 2y)u ≥ 5 u (2x + y)v ≥ 4 v x, y, u, v ≥ 0
Enforcing that the x, y coefficients in the linear combination are ≤ than the x, y coefficients in the original objective function gives us some constraints on u, v. Since we want the tightest lower bound possible, we then want to maximize the right-hand size of the constraint linear combination. This mix of constraints and objective function give us a new LP.
max 5u + 4v u + 2v ≤ 1 2 u + v ≤ 4 u, v ≥ 0
This LP is known as the dual of the original LP, which is called the primal. It is interesting to note that the objective function coefficients of the primal have become the constraint bounds in the dual, while the constraint bounds of the primal have become the objective function coefficients of the dual. Also, the u, v constraint coefficient matrix is the transpose of the x, y constraint coefficient matrix.
The relationship between the primal and dual LPs is very special and useful. The specifics will be spelled out in series of lemmas. For notation, V alP (x, y) and V alD(u, v) are the objective function values of the primal and dual LPs, respectively, with an ∗ denoting the optimal objective function value.
Theorem 12.2.1 Let (x, y) be any feasible primal solution. Let (u, v) be any feasible dual solution. Then V alP (x, y) ≥ V alD (u, v).
This result follows from manipulation of the constraints in the definition of the dual presented above, and is known as the Weak LP-Duality Theorem.
Note that since all ce ≥ 0 and we are trying to minimize the sum over e, in a solution no ye will be assigned a value greater than 1.
What is the intuitive interpretation of the dual of our maxflow LP? Each ye corresponds to ’choosing’ an edge. The constraints state that we must choose ≥ 1 edge from every path, and the objective function tells us to minimize the sum of yece over all e.
Ignoring fractional ye, this means that our optimal LP solution to this dual needs to choose edges so that every path contains one of the chosen edges, and also choose the edges with the smallest total capacity. The smallest capacity set of edges which interesects every path from s to t is, by definition, the minimum weight separating cut between s and t.
As it turns out, all basic points of the primal and dual LPs are integral. This, along with the Strong LP Duality theorem (Theorem 12.2.3), implies the Max-Flow Min-Cut Theorem. The following lemma and theorem formalize this notion.
Lemma 12.2.5 The Mincut LP has integral basic points.
Corollary 12.2.6 Maxflow-Mincut Theorem: The maximum possible flow between s and t is equal to the capacity of the minimum cut separating s and t.
In this way LP-Duality allows us to clearly see the duality of the maxflow and mincut problems.
References
[1] V. Vazirani. Approximation Algorithms. Springer, 2001.
