Expected Matching Size

m

¢−1/4

+ ²´

-near uniform.

2.5.2 Expected Matching Size

Now we prove Lemma2.8, i.e. we show thatEmulatePhasecomputes a large matching. In the proof we argue that the expected total size of sets Hiand Fiis not significantly impacted by relatively low-degree vertices classified as heavy or by an unlucky assignment of vertices to

subgraphs resulting in local vertex degrees not corresponding to global degrees. Namely, we show that the expected number of friends a heavy vertex adds is O(1) and at the same time the probability that the vertex gets matched isΩ(1).

Lemma 2.8. Let∆, G?= (V?, E_?), m, andD be parameters forEmulatePhase^{such that}

• D is an independent and ²-near uniform distribution on assignments of vertices V?to [m] for² ∈ [0,1/200],

• _m^∆ ≥ 4000µ⁻²_R ln²n,

• the maximum degree of a vertex in G_?is at most³₂∆.

For each i ∈ [m], let Hi, Fi, and Mibe the sets constructed byLocalPhasefor the i -th induced subgraph. Then, the following relationship holds for their expected sizes:

X

i ∈[m]

E[|Hi∪ Fi|] ≤ n⁻⁹+ 1200 X

i ∈[m]

E[|Mi|] .

Proof. We borrow more notation fromEmulatePhaseand the m executions ofLocalPhase initiated by it. For i ∈ [m], Vi is the set inducing the i -th subgraph. Value∆?= _m^∆ is the rescaled threshold passed to the executions ofLocalPhase^{. R}iis the reference set created by LocalPhasefor the i -th induced subgraph.

For each induced subgraph,LocalPhasecomputes a maximal matching Mi in Line5.

While such a matching is always large—its size is at least half the maximum matching size—it is hard to relate its size directly to the sizes of H_iand F_i. Therefore, we first analyze the size of a matching that would be created byMatchHeavy(G_?[Hi∪ Fi], Hi, Fi). We refer to this matching the expected total size of sets H_i⁰. What is the probability that a given vertex v of degree less than¹₈∆ is included in S_{i ∈[m]}Hi? Suppose that v ∈ Vk, where k ∈ [m]. The expected number of v’s neighbors in Rk is at most (1 + ²) · µR·¹₈∆/m ≤₁₆³µR∆?due to the independence and

²-near uniformity of D[C ]. Using the independence, Lemma2.7, and the lower bound on∆?, we obtain the following bound: partition is based on the execution ofMatchHeavyfor the i -th subgraph. In Line1, this algorithm selects for every vertex v ∈ Fi a random heavy neighbor v_?∈ Hi. If v_?∈ H_i⁰, we assign v to F_i⁰. Analogously, if v_?∈ H_i⁰⁰, we assign v to F_i⁰⁰. Obviously, a heavy vertex in H_i⁰ can be selected only if H_i⁰ is non-empty. By Markov’s inequality and the upper bound on Pi ∈[m]E£¯¯H_i⁰¯

¯¤, the probability that at least one set H_i⁰is non-empty is at most n⁻¹¹. Even if

for all i ∈ [m], all vertices in Fiselect a heavy neighbor in H_i⁰whenever it is available, the total expected number of vertices in sets F_i⁰is at mostP

i ∈[m]Eh¯

Before we proceed to bounding sizes of the remaining sets, we prove that with high probability, all vertices have a number of neighbors close to the expectation. Letϕ : V?→ [m]

be the assignment of vertices to subgraphs. We defineE as the event that for all v ∈ V?,

¯ This in particular implies thatE£¯¯N(v) ∩Vϕ(v)¯

¯¤ ≤ ¡³₂+₄₀₀³ ¢ As a result, with this probability, we have

¯

By the union bound, this bound holds for all vertices in V_?simultaneously—and henceE occurs—with probability at least 1 − n · n⁻¹²= 1 − n⁻¹¹. at most n⁻¹¹· n = n⁻¹⁰to the expected size of each of these quantities. Suppose now thatE occurs. Consider an arbitrary v ∈ H_i⁰⁰for some i . The number of neighbors of v in V_i lies in the

This occurs if one of v’s neighbors w is added to Fiand selects v as w_?, and additionally, v and w are colored blue and red, respectively. The number of v’s neighbors is at least₁₆¹∆?. Since each vertex w in Vihas at most 2∆?neighbors, the number of heavy neighbors of w is bounded by the same number. This implies that in the process of selecting Fi, only the first branch in the definition ofµFis used and each vertex w is included with probability exactly equal to the number of its neighbors in Hi divided by 4∆t +1. Then each heavy neighbor of w is selected as w_?with probability one over the number of heavy neighbors of w . What this implies is that each neighbor w of v is selected for Fi and selects v as w_?with probability

exactly (4∆?)⁻¹. Hence the probability that v is not selected as w_?by any of its at least₁₆¹∆?

neighbors w can be bounded by µ 1/100. Then with probability 1/4, these two vertices have appropriate colors and this or an-other edge incident to v with the same properties is added toMf_i. In summary, the probability that an edge (v, w ) for some w as described is added toMfiis at least 1/400. Since we do not count any edge in the matching twice for two heavy vertices, by the linearity of expectation E£¯¯Mf_i¯ In general, without conditioning onE ,

X

We now combine bounds on all terms to finish the proof of the lemma.

X

Next we prove Lemma2.9. We start with an auxiliary lemma that gives a simple criterion under which an independent distribution remains independent after conditioning on a random event. Consider a random vector with independently distributed coordinates. Suppose that for any value of the vector, a random eventE occurs when all coordinates “cooperate”, where each coordinate cooperates independently with probability that depends only on the value of that coordinate. We then show that the distribution of the vector’s coordinates givenE remains independent.

Lemma 2.11. Let k be a positive integer and A an arbitrary finite set. Let X = (X1, . . . , X_k) be a random vector in A^kwith independently distributed coordinates. LetE be a random event of non-zero probability. If there exist functions pi : A → [0,1], for i ∈ [k], such that for any x = (x1, . . . , x_k) ∈ A^kappearing with non-zero probability,

Pr[E |X = x] =Y^k

i =1

pi(xi),

then the conditional distribution of coordinates in X givenE is independent as well.