1. What is the expected number of edges deleted due to same color in MODIFIED GREEDY?
In the MODIFIED GREEDY algorithm, the expected number of edges deleted due to the same color as the matching edge is calculated using the formula EQUATION. This formula takes into account the number of vertices (n), the number of unused colors (q), and the average degree of the unrevealed graph (l). The probability of the first chosen vertex having degree k for 1 <= k <= log2(n) is given by EQUATION. Conditional on a total of <= 2 log2(n) edges being adjacent to the two chosen vertices, the expected number of edges deleted is determined by the formula EQUATION. The big-O term on the last line follows from the approximations (14), (15), and (16). Substituting n/n = N, u/n = M, and using (13) with k = q(0)/n, t = t/n, the equations EQUATION and EQUATION are derived. By substituting (21) into (19), we get M' = BN^2(N + t + k - 1). Setting t = t0 so that N(t) = 0, we obtain 0 >= 1 - 2t0 + t0 >= 1/2 - 1c(1 - e -c). From (23), it follows asymptotically that u <= 1 - 1c(1 - e -c)^2 - 1c(1 - e -c)n.
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2. What progress has been made in understanding greedy algorithms for rainbow matchings?
Significant progress has been made in understanding the performance of simple greedy algorithms for finding large rainbow matchings in sparse random graphs. Specifically, the result is precise for GREEDY when k equals 1/2. However, this is a challenging question due to its relation to finding large matchings in sparse random 3-uniform hypergraphs. In this context, an edge {u, v} of color c can be thought of as a triple {u, v, c}. The decision problem of whether a 3-uniform hypergraph has a matching of size k is NP-complete, as listed by Karp. The analysis of algorithms GREEDY and MODIFIED GREEDY has been reduced to the analysis of differential equations, which are not easily solvable. Further study is needed to gain a better understanding of these equations. The researchers also briefly explored a colored version of the Karp-Sipser algorithm and constructed the appropriate differential equations, but did not pursue this line of study. Proving the conjecture that MODIFIED GREEDY performs better than GREEDY in the color-free setting would be a valuable contribution. Dyer, Frieze, and Pittel have already proven that the Modified Greedy Matching algorithm performs better than Greedy Matching on a random graph, but the rainbow version presents additional complexities due to the lack of explicit elementary solutions to the differential equations.
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