In this section, we introduce the implicit verification approach to oracle design. Again, we use the context of the multiplicative weights update method (see Section 2.3.1) but the approach itself is applicable to the fractional packing/covering framework as well.
In Section 6.1, we prove the infeasibility ofαby producing a dual feasible solution. However, the construction of a dual feasible solution is not always easy even if a dual candidate is given. The natural question arises: is it possible to prove the infeasibility without producing an explicit dual solution? The answer to the question is yes. Suppose that we have a 2-approximation algorithm for the problem and the approximation algorithm fails to produce a solution with objective value at leastα/2. This simply proves that α is infeasible. We apply this idea and obtain a general approach to construct an oracle. Algorithm 23 and Algorithm 24 describes such an approach.
Recall the goal of the oracle in Section 6.1. We want to find x∈ {cTx≥α;Ax≤
b}. The primal-dual algorithm maintains u and then, the goal of the oracle is to find x such thatcTx≥α and uT(A−b)≤δP
juj. As a consequence, the problem
reduces into a problem of finding xi such that ci is large andATiu is small.
Algorithm 23 describes the implicit verification approach. In Algorithm 23, the two conditions are summarized into “effective coefficients” inL(x, %) =cTx−%uTAx
Algorithm 23General Oracle Construction (Implicit). 1: LetL(x, %) = cTx−%uTAx. 2: Letγ =uTb. 3: Consider: cTx≥(1−δ)α,uTATx≤uTb,Ax≤ρb (LP7) L(x, %)≥α−%γ,Ax≤ρb (LP8)
4: Letx% be a solution of LP8 for %. If α is feasible, we always find x that satisfies LP8 using Algorithm 24.
5: Use O(1δ) (parallel) calls to LP8 to find %+ and %− that satisfies:
1. %−≤%+ ≤%−+δα2γ, 2. uTAx%+
≤uTb, and
3. cTx%− ≥α.
6: if any call to LP8 returns x% such that L(x%, %)< α−% then
7: Declare a failure. α is infeasible.
8: else
9: Take a linear combination of x%+ and x%− to find xthat satisfies LP7.
10: Return x as a primal witness.
Algorithm 24Algorithm for LP7.
1: If α is feasible, there exists x such that:
L(x, %)≥α−%γ,Ax≤b (LP9)
2: if α−%γ <0 then
3: Return x=0 (a zero vector).
4: end if
5: Find a ρ-approximate solution x0 for LP9, i.e.,
L(x0, %)≥ 1
ρ(α−%γ),Ax
0≤b.
6: Return x=ρx0.
objective value as a consequence, i.e., the linear combination of primal witnesses has an objective value of (1−δ)α (or greater) rather thanα. The Lagrangian multiplier method reduces the problem into a problem of the same format but with a different objective function and relaxed constraints. (See LP8)Algorithm 24 solves the reduced problem using a ρ-approximation algorithm.
Lemma 6.2.1 states that Algorithm 24 is always successful if α is feasible. The consequence of this lemma is that if the algorithm is unsuccessful, i.e., it returns x
such that L(x, %) < α−%γ, we can conclude that the guess α is infeasible, thus, the name implicit verification. Lemma 6.2.2 states that Algorithm 23 returns an admissible primal witness. Combined with Corollary 2.3.3, the overall algorithm finds a feasible primal solution x with objective value (1−δ) mini bib+iδα. Note that
Lemma 6.2.1. If bi > 0 for all constraints and α is feasible, Algorithm 24 always returns xthat satisfies LP8 for any %≥0.
Proof. Suppose that LP9 is feasible. Then, the rest of the proof is straightforward. If
α−%γ is negative, it is obvious that the zero vector would satisfy LP8. Otherwise, the
ρ-approximation algorithm will find x0 that satisfies all the constraints of the original with the objective value at least 1/ρtimes the optimal value which is greater or equal to 1
ρ(α −%γ). Therefore, the simple scaling step (Step 6) gives a desired result if
bi >0 for all constraints.
Let xOP T be the optimal solution. Then, cTxOP T ≥ α and uTAx ≤ uTb since
Aix ≤ bi and ui, bi > 0 for all i. Therefore, for any %, L(xOP T, %) ≥ α−%γ. The
lemma follows.
Lemma 6.2.2. If bi > 0 for all constraints and α is feasible, Algorithm 23 returns an admissible primal witness.
Proof. We already proved the statement in step 4. The remaining parts are: (i) we can find %+ and %− that satisfies the conditions in step 5 and (ii) such x%+
and x%−
produces an admissible primal witness.
Recall that Algorithm 24 returns a zero vector for any % > αγ. The zero vector satisfies the condition 2. On the other hand,x0satisfies the condition 3 sinceL(x,0) =
cTx and α−0·γ = α. If we call LP8 for % = 0,δα
2γ, δα
γ ,· · ·, we will find %
+ and %−
We take a linear combinationx=ax%+ +(1−a)x%−, a∈[0,1] such thatuTAx=γ. Note that aL(x%+, %+) + (1−a)L(x%−, %−)≥α−%−γ−a(%+−%−)γ ≥(1−δ)α−%−γ where we use %+−%− ≤ δα 2γ, γ ≥0. Therefore cTx=aL(x%+, %+) + (1−a)L(x%−, %−) +a%+uTAx%+ + (1−a)%−uTAx%− ≥(1−δ)α−%−γ+a(%+−%−)uTAx%+ +uTA(a%−x%++ (1−a)%−x%−) ≥(1−δ)α−%−γ+ 0 +uTA(a%−x%++ (1−a)%−x%−) (UsinguTAx%+ ≥0 and%+−%−≥0) ≥(1−r)α−%−γ+uTA%−x (Usingx=ax%+ + (1−a)x%−) ≥(1−r)α−%−(γ−uTAx) = (1−r)α (Usingγ =uTAx)
The lemma follows.
The following lemma is a generalized version of the Lagrangian method we use here and will be used later in this thesis. We skip the proof since it is almost identical to the proof of Lemma 6.2.2. The only difference is that the “effective objective” does not have to be greater than β1−%β2. Rather an approximation is sufficient. Note
that A1 and A2 are row vectors not matrices.
β1, β2 ≥0. Consider the system LP10:
{x;A1x≥β1;A2x≤β2;x∈ P} (LP10)
For any r > 0,1≥ q > 0, given LP10 and a subroutine that finds x∈ P such that (A1−%A2)x≥q(β1−%β2)for any%≥0, we can findx∈ P such thatA1x≥q(1−r)β1
and A2x≤β2 with O(1r) (parallel) invocations of the subroutine.
6.2.1
O(
ε12logn)-pass Algorithm
In this section, we demonstrate the implicit verification approach using the bipartite MWM. A disadvantage of the approach is that we do not have a precise control over the primal witness since we use an approximation algorithm as a black box. As a consequence techniques in Sections 6.1.2 and 6.1.3 do not apply and we obtain a worse number of passes for the bipartite MCM and MWM.
However, the implicit verification approach allows us a easier oracle construction as long as we have a constant factor approximation algorithm. Therefore, it suits better in dealing with more complex LPs. For MWM, Feigenbaum et al. gave a one-pass 6-approximation algorithm in the insertion-only model [43]. 2
Algorithm 25 and Algorithm 26 are the oracle for the bipartite MWM problem (LP5). Note that unlike the explicit verification approach, we do not need to design a new algorithm for this specific purpose. We just use Algorithm 23 and Algorithm
2There are other one-pass algorithms that achieve a better approximation factor [95, 42]. How-
Algorithm 25Oracle for the bipartite MWM. 1: LetL(x, %) = P (i,j)∈Ewijxij −% P iu t i P jxij. 2: Letγ =P iuti. 3: Consider: X (i,j)∈E wijxij ≥(1−δ)α,X i utiX j xij≤X i uti,X i X j xij ≤6 (LP11) L(x, %)≥α−%γ,X i X j xij ≤6 (LP12)
4: Use O(1δ) parallel calls to LP12 to find %+, %− such that: 1. %−≤%+ ≤%−+δα 2γ, 2. P iu t i P jxij ≤ P iu t i, and 3. P (i,j)∈Ewijx %− ij ≥α.
5: if any call to LP12 returns x% such thatL(x%, %)< α−% then
6: Declare a failure. α is infeasible.
7: else
8: Take a linear combination of x%+ and x%− to find xthat satisfies LP11. 9: Return x as a primal witness.
10: end if
Algorithm 26Subroutine for MWM.
1: If α is feasible, there exists x such that:
X
i,j
xij(wij−%(uti+utj))≥α−%γ
2: if α−%γ <0 then
3: Return x=0 (a zero vector).
4: end if
5: In one pass, find a 6-approximate matching given “effective weight” wij0 =wij −
%(ut
i+utj) for (i, j)∈E.
6: Return xij = 6 for any (i, j) in the matching and xij = 0 for other edges.
24 with the 6-approximation algorithm as a subroutine. The width parameter of the oracle is 6.
Note that the space-reducing technique in Section 6.1.2 applies to this approach. The oracle uses O(n
ε) space for the computation and returns a vector with O(n)
non-zero coordinates. Overall, we obtain the following result using this oracle.
Theorem 6.2.4. For any ε ≤ 1
2 in T =O( 1
ε2 logn) passes, and O(n(T +
1
ε)) space we can compute a(1−ε)approximation for maximum weighted matching in bipartite graphs (the size can be computed in less space).