A Barrier to Obtaining a Non-Adaptive (1 − )-Approximation Algorithm

The approximation ratio of our non-adaptive algorithm is (1

2 −) as opposed to the near- optimal ratio of (1−) achieved by our adaptive algorithm. A natural question, first raised by [25], is if one can obtain a (1−)-approximation using a non-adaptive algorithm, even by allowing arbitrary dependence on pand . In the following, we highlight a possible barrier to obtain such a result.

Consider a bipartite graph G(L, R, E) constructed as follows: (i) the vertex sets are L = V1∪V3, R = V2 ∪V4 and |Vi| = N for i ∈ [4], (ii) there is a perfect matching between V1 and V2, and a perfect matching between V3 and V4, and (iii) there is a gadget graph

b

G(V2, V3,Eb), to be determined later, betweenV₂ and V₃ (see Fig 3.a).

Suppose we want to design a non-adaptive (1−)-approximation algorithm for the instance G(V, E) with the parameter p = 2/3. In this case, for any graph Gp, w.h.p., there is a

V₁ V₂ V₃ V₄

Gadget Graph

(a) Input graphG(V, E)

V₁ V₂ V₃ V₄ Gadget Graph B A (b) A realizationGp(V, Ep)

Figure 3: An example of a barrier to (1−)-approximation non-adaptive algorithms. The edges in the gadget graph arenot presented in this figure. In part (b), solid red edges (resp. dashed edges) are the edges that are realized (resp. not realized).

matchingM1 betweenV1 andV2, and another matchingM2 betweenV3andV4, each of size (2/3)N −o(N). Hence,

opt(Gp)≥ |M1|+|M2|+ (2/3)·m(A, B)−o(N) (3.5)

where A (resp. B) is the set of vertices in V2 (resp. V3) that are not matched by M1 (resp. M2), and m(A, B) denotes the size of a maximum matching between A and B. A few observations are in order. First, picking edges of M1 and M2 is crucial for having any large matching inGp, and second, for a uniformly at random chosen realization ofM1 and

M2, the set Aand B are chosen uniformly at random from V2 and V3 (see Fig 3.b). Based on these observations, we define the following problem.

Problem 1. Given a bipartite graph G(L, R, E), choose a subgraph H(L, R, Q) such that given two subsets A ⊆L and B ⊆R, if m(A, B) ≥N/3−o(N) in G, then H contains at least Ω(N) edges betweenA andB.

The goal is to solve Problem 1 using a graph H with small number of edges. The previ-

ous discussion implies that any non-adaptive (1−)-approximation algorithm has to solve

Problem 1 for the gadget graph Gb, when the two sets A and B are chosen uniformly at

matching size OPT(Gp) in Gp, we have:

ALG(Gp)≤ |M1|+|M2|+o(N)≤(4/3)N +o(N)

OPT(Gp)≥(4/3)N + (2/3)(N/3)−o(N) = (14/9)N −o(N) (by Eq (3.5))

Hence, the approximation ratio of the algorithm on this instance is at most 6/7 +o(1), bounded away from being a (1−) approximation for <1/7.

Although for randomly chosen subsetsAand B, no lower bound on the size ofH is known, we show in the following that ifA and B are chosen adversarially, then there exist graphs

for which solving Problem 1 requires storing a subgraph with super linear in n number

of edges. Note that the number of queries of any non-adaptive algorithm is at least the

number of edges in H and hence this bound on the number of edges in H implies that

ω(n) queries are needed, or in other words, the number of per-vertex query needs to be a

function of n. The existence of such a graph indicates a barrier to obtain a non-adaptive (1−)-approximation algorithm.

To continue, we need a few definitions. For a graph G(V, E), a matching M is called an

induced matching, if there is no edge between the vertices matched inM, i.e.,V(M), except for the edges in M. A graph G(V, E) is called an (r, t)-Ruzsa-Szemer´edi graph ((r, t)-RS graph for short), if the edge setE can be partitioned intotinduced matchings each of size r. Note than the number of edges in any (r, t)-RS graph isr·t.

SupposeG(L, R, E) is an (r, t)-RS graph with the parameter r=N/3 and induced match- ings M1, . . . , Mt. For each i ∈ [t], define Ai (resp. Bi) as V(Mi)∩L (resp. V(Mi)∩R).

Suppose we choose the pair (A, B) only from the set of pairsF :={(A1, B1), . . . ,(At, Bt)}.

Note that between any pair in F, there is a matching of size r = N/3, and moreover all edges of G are partitioned between these matchings. If the subgraph H(V, Q) has only o(r ·t) edges, a simple counting argument suggests that for 1−o(1) fraction of pairs in

Problem 1.

To complete the argument, we point out that there are (r, t)-RS graphs on 2N vertices with parameters r =N/3 and t=NΩ(1/log logN) _{[48, 55]. These constructions certify that} to solve Problem 1 when the sets A and B are chosen adversarially, one needs to store a subgraph with n1+Ω(1/log logn) _{= ω(n·polylog(n)) edges. In conclusion, while this result} does not rule out the possibility of a non-adaptive (1−)-approximation algorithm where the number of per-vertex queries is independent of n, it suggests that any such algorithm has to crucially overcome Problem 1 using the fact that the two sets A and B are chosen randomly instead of adversarially.

3.8. Conclusions and Future Work

In this chapter, we presented our study on the stochastic matching problem. We showed that there exists an adaptive (1−)-approximation algorithm for this problem withO(log(1_p/p)) per-vertex queries and degree of adaptivity. We further presented a non-adaptive (1₂ −)- approximation algorithm with O(log(1_p/p)) per-vertex queries. These results represent an exponential improvement over the previous best bounds of [25], answering an open problem in that work.

An interesting direction for future work is to design a non-adaptive algorithm that obtains a better than 1₂-approximation while maintaining the property that the number of per-vertex queries is independent of n. Toward this direction, we highlighted a potential barrier to achieve a (1−)-approximation non-adaptively and we believe that overcoming this barrier would play a key role in this line of research.