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CS880: Approximations Algorithms Scribe: Matt Elder Lecturer: Shuchi Chawla Topic: Sparsest Cut and Balanced Cut Date: 3/20/
17.1 Multicut
First, consider the multicut problem. Given a graph G = (V, E), K pairs of terminal vertices {si, ti}, and a cost function on the edges c : E → R, the multicut problem asks for a minimum-cost cut of G that separates si and ti for all i. Last time, we gave a O(log k) approximation for this problem.
The (relaxed) linear program for this problem is as follows; call it “Primal 1”.
minimize
e∈E
cede
where d(si, ti) ≥ 1 ∀i d is a metric
We can rewrite as follows:
minimize
e∈E
cede
where P∑i = {All paths from si to ti}
e∈P
de ≥ 1 ∀i ∀P ∈ Pi
de ≥ 0 ∀e
The dual of this LP, which we’ll call “Dual 1”, is as follows:
maximize
i
P ∈Pi
fi,P
where P∑i = {All paths from si to ti}
i
P ∈Pi P ∋e
fi,P ≤ ce ∀e
fi,P ≥ 0
Dual 1 solves the max-sum multi-commodity flow problem: ce represents the capacity of an edge, and fi,P is the amount of flow directed from si to ti along the path P. The LP tries to maximize the total amount of commodity flow.
Lemma 17.1.1 Multicut is always larger than the corresponding max-sum multi-commodity flow.
Lemma 17.1.2 Multicut is at most O(log K) times the corresponding max-sum multi-commodity flow.
Theorem 17.1.3 When k = 2, multicut equals max-sum multi-commodity flow.
17.2 Maximum Concurrent Multicommodity Flow
A solution to Dual 1 may starve some commodities while routing others. In contrast, max- concurrent multicommodity flow routes equal fractions of all commodities while respecting ca- pacities. Thus, we devise the following LP for max-concurrent multicommodity flow, which we call Dual 2:
maximize t where
i
P ∈Pi P ∋e
fi,P ≤ ce ∀e
∑
P ∈Pi
fi,P ≥ rit ∀i
fi,P ≥ 0 ∀i, ∀P ∈ P
Intuitively, the difference between Dual 1 and Dual 2 is that Dual 1 seeks to maximize the total flow across independent commodities, while Dual 2 seeks to maximize the minimum of a set of weighted flows. This is the maximum concurrent multicommodity flow problem.
Primal 2, the dual of the maximum concurrent multicommodity flow problem, is as follows:
minimize
e
dece
where
e∈P
de ≥ yi ∀i, ∀P ∈ Pi ∑
i
riyi ≥ 1
de ≥ 0 ∀e yi ≥ 0 ∀i
The costs ce are constants of the problem instance, so Primal 2 will seek to minimize the values for de. Thus, they will be no larger than they are constrained to be, so yi = d(si, ti), the shortest distance from si to ti where each edge e has length de. So, we can devise the following linear program, equivalent to Primal 2:
minimize
e
cede
where
i
rid(si, ti) ≥ 1
d is a metric
Up to scaling d, this is the same as the following program:
minimize
∑^ e^ cede i rid(si, ti) where d is a metric,
So, now consider the general case, with an arbitrarily large ratio between ymax and ymin. Define Ix, the set of all yi in a conveniently-defined interval, as:
Ix =
i|yi ∈
( (^) ymax 2 x+^
ymax 2 x
]}
When x is constrained to the integers, it’s clear that every yi is contained in exactly one Ix. For each Ix, our algorithm will construct a multicut instance as in the special case, but it will scale d by 2x+1/ymax instead of 1/ymax. The sparsity for each of these instances is not too large:
α(Ix) ≤ O(log K)
∑^ e^ cede i∈Ix riyi
If there exists a constant β and an x such that
i∈Ix riyi^ ≥^ β
i riyi, then the sparsity of this instance is at least O(log K)β
cede/(
riyi).
We claim that we can ignore all i such that yi < ymax/D^2. Again, D is the total demand
ri. Define the set W containing wee values of yi, W =
i|yi < ymax/D^2
. Ignoring W can result in the loss of at most
i∈W riyi^ <^
i∈W riymax/D (^2) ≤ ymax/D ≤ 1 /D from the denominator of our
algorithm’s sparsity. Assuming D ≥ 2, ignoring W has only a small constant approximation cost. (If D < 2, this is an easy boundary case, which we can effeciently handle in an ad-hoc way.)
Thus, we need to consider only those Ix where 2x^ < D^2. There are at most 2 log D such sets, so β ≥ 1 /(2 log D). This yields a O(log K log D)-approximation.
17.4 Balanced Cut
Given a graph G = (V, E), the balanced cut problem demands the min-cost cut such that each side has at least αn nodes, for some constant value of α ≤ 12. It is known that this problem is inapproximable to n^2 −ǫ/OPT if P 6 = NP. This is an absurdly poor approximation.
So, we consider instead a pseudo-approximation algorithm, in which we approximate both the objective function of the optimal solution and the parameters of its instance. So, in this case, when asked for a cut with a balance of α, we instead output a cut with a balance α′, such that α′^ < α and α′^ ≤ 1 /3. If the optimal cut of balance α has cost Cα, our cut will have cost no greater than O(log n)Cα/(α − α′). Notice that, though we have a reasonable bound on the ratio between the size of our cut and Cα, the ratio between the size of our cut and C α′ may be unbounded.
The algorithm employs a direct reduction to the sparsest cut problem, letting T = V × V and ri = 1. Then, the sparsity of a cut S is, as before, c(E(S, S¯))/(|S|| S¯|). We discuss further details next time.
Even though this algorithm is far from optimal, it is actually useful. This pseudo-approximation has applications in divide-and-conquer algorithms on graphs. It ensures that we can always divide a graph into two pieces, each with size roughly linear in the size of the original graph, such that the cut between the pieces isn’t too large. This yields log-depth recursion, which divide-and-conquer algorithms demand, while bounding the cost of recombining pieces.
