For the composition of a constant factor approximation algorithm for in- stances in which the value of a maximum fractional one-flow is large, we first prove that the additive integrality gap of the max-1FP is less than m. This result follows from basic linear programming theory.
Proposition 4.18. The difference of the value F∗
F of a maximum fractional
one-flow and the value F∗
I of a maximum integral one-flow is less than m.
Proof. Consider an optimal basic solution x to the maximum one-flow LP given in Section 4.2 and letPF be the set of paths P for which xP is fractional.
It suffices to show that |PF| ≤ m, because it then follows that we lose less
than m units of flow by dropping the fractional part of an optimal basic solution. The remaining part yields an integral solution.
Let A = (AP)P ∈PF be the matrix given by the inequalities in (4.1), i.e.,
AP ∈ {0, 1}A is the vector that has 1-entries for all a ∈ P and 0-entries
elsewhere. Further let eP ∈ {0, 1}P be the vector that has a single 1-entry
for P ∈ P and 0-entries elsewhere. Since xP is fractional for P ∈ PF, the slack
variable of the inequality xP ≤ 1 is positive. All vectors corresponding to
positive variables in a basic solution must be linearly independent. It follows that the vectors (AP, eP), (0, eP) ∈ {0, 1}A∪P with P ∈ PF are linearly
indepedent. This implies immediately that the vectors AP with P ∈ PF
must also be linearly independent. Since the dimension of these vectors is m, it follows that |PF| ≤ m.
Combining the insights from Proposition 4.18 and its proof with the FPTAS from the last section yields the following result.
Corollary 4.19. For any ǫ > 0, an integral one-flow of value FI > (1 −
ǫ)F∗
F − m (where FF∗ is the value of an optimal fractional one-flow) can be
computed in time polynomial in the input size, ǫ−1, and F I.
Proof. Consider any instance of the max-1FP. Let F∗
F be the value of a
maximum fractional one-flow. By Theorem 4.12 we can compute a frac- tional one-flow (¯xP)P ∈P of value at least (1− ǫ)FF∗, for any ǫ > 0, in time
polynomial in the input size, ǫ−1, and F∗
F. Based on this, we compute an-
other one-flow of the same or larger value. Consider the system of linear inequalities constructed in the following. We only take the set of paths
¯
the size of the system X P ∈ ¯P xP ≥ X P ∈ ¯P ¯ xP X P ∈ ¯P: a∈P xP ≤ u(a) ∀ a ∈ A 0≤ xP ≤ 1 ∀ P ∈ ¯P
is polynomial in the input size of the original problem, ǫ−1, and F∗
F. It follows
that a basic solution to this system can be computed in polynomial time. The flow value of this solution is at least (1− ǫ)F∗
F. For the fractional part of this
basic solution, we can argue as in the proof of Proposition 4.18 that there are at most m paths carrying a fractional amount of flow. (Note that all vectors only have an additional 1 in the first entry.) Thus, we obtain an integral one-flow of value FI > (1− ǫ)FF∗ − m when dropping the fractional part of
the considered basic solution. Since F∗
F < (1− ǫ)−1(FI + m), the described
procedure can be implemented to run in time polynomial in the input size, ǫ−1, and F
I.
As an immediate consequence, we obtain the following approximation result for the integral max-1FP.
Theorem 4.20. There exists a constant factor approximation algorithm for the integral max-1FP whose runtime is polynomial in input plus ouput size, if we restrict to instances whose maximum fractional flow value is larger than some constant c > 1 times their numbers of arcs.
In the remainder of this section we only consider instances of the max- 1FP whose minimum arc capacity is Ω(log m). Starting with a nearly optimal fractional solution we use randomized rounding to transform the fractional one-flow into an integral one. Randomized rounding was first introduced by Raghavan and Thompson [88]. We adapt a revision by Kleinberg [61] to the problem considered here. This yields a constant factor approximation with high probability.
For a given µ ∈ [0, 1] and a nearly optimal fractional one-flow (xP)P ∈P,
randomized rounding routes one unit of flow along path P ∈ P with prob- ability µxP. Obviously, this produces an integral one-flow which does not
necessarily obey all arc capacities. With an analysis similar to one by Klein- berg [61] we show that the probability of violation of any arc capacity is at most 1/m. For this proof we need the following lemma [61].
Lemma 4.21. Let µ ∈ (0, 1] and Ψ1, . . . , ΨN be completely independent
Bernoulli trials with E[Ψi] = µpi, for all i∈ {1, . . . , N} and some pi∈ [0, 1].
Then it holds for Ψ :=PN
i=1Ψi and p :=
PN
i=1pi that
P r[Ψ > p] < (eµ)p .
Using Lemma 4.21, we prove that, for thoroughly chosen µ, the probabil- ity for a violation of any arc capacity is low.
Theorem 4.22. If umin ≥ c log m, for some constant c ∈ R+, and µ :=
e−14−1/c, the probability, that any capacity constraint is violated after ran-
domized rounding, will be less than 1/m.
Proof. This proof is similar to one given by Kleinberg [61].
We assume to have a capacity function u and µ as given in the theorem. If we show for each arc that after randomized rounding its capacity is violated with probability less than 1/m2, we are done. Hence, consider any arc a =
(v, w)∈ A.
Note that randomized rounding works independently from the fact that all paths share common start and end nodes. Thus, we may assume without loss of generality that the flow along a in the fractional one-flow equals u(a). Otherwise we could add artificial paths with corresponding flow across a.
For all P ∈ P, let ΨP be the random 0/1-variable that indicates if P
sends one unit of flow across a in the computed integral one-flow. Further, let xP(a) be 0, if a /∈ P , and xP otherwise. Then E[ΨP] = µxP(a) and
P
P ∈PxP(a) = u(a). Using Lemma 4.21 it follows for Ψ :=
P
P ∈P ΨP that
P r[Ψ > u(a)] < (eµ)u(a) = 1 41/c u(a) ≤ 1 4log m = 1 m2 . (4.5)
Since Ψ is the random variable giving the number of paths that send one unit of flow across a in the computed integral one-flow, the proof is complete.
It remains to prove, that randomized rounding can be used to find a constant factor approximation with high probability.
Theorem 4.23. Let k ∈ N. For any instance of the integral max-1FP with umin = Ω(log m), randomized rounding needs O(k) iterations to find a con-
stant factor approximation with probability at least 1− 2−k.
Proof. We may assume without loss of generality that m > 2µ−1− 1, other- wise we can efficiently solve the problem exactly.
Let vI be the random variable indicating the amount of flow that is routed
by the integral one-flow obtained from one application of randomized round- ing. It follows that E[vI] is exactly µ times the amount of flow vF that is
routed by the fractional flow. Thus, E[vI] is only a constant factor smaller
than the optimal value of an integral one-flow.
Let p be the probability for the event, that vI is at least µ2vF. Since
E[vI] = µvF, it holds that
(1− p)µ
2vF + p· vF ≥ µvF, which is equivalent to p≥ 2−µµ .
Thus, vI is at most some constant factor away from the value of an optimal
one-flow with probability at least 2−µµ .
It can still happen that the computed one-flow is infeasible. A feasible constant factor approximation is obtained with probability at least q := 2−µµ −
1
m > 0. For some constant ℓ ∈ N that only depends on µ, it holds that
(1− q)ℓ < 1/2, which means that the probability of a flop will be less than
1/2 after ℓ iterations of the algorithm. Thus, we get that after ℓk iterations for some k ∈ N the probability of success will be at least 1−(1−q)ℓk > 1−2−k,
which completes the proof.