# Secretary and Online Matching Problems with Machine Learned Advice

NIPS 2020, 2020.

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Keywords:

matching problemcompetitive ratioonline algorithmonline bipartite matchingbipartite matchingMore(8+)

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Abstract:

The classical analysis of online algorithms, due to its worst-case nature, can be quite pessimistic when the input instance at hand is far from worst-case. Often this is not an issue with machine learning approaches, which shine in exploiting patterns in past inputs in order to predict the future. However, such predictions, although usu...More

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Introduction

- There has been enormous progress in the field of machine learning in the last decade, which has affected a variety of other areas as well.
- This follows from the fact that in this case the authors obtain the so-called vertex-weighted online bipartite matching problem for which there is a deterministic 1/2-competitive algorithm, and no algorithm can do better [1].
- Korula and Pal [25] provided the first constant competitive algorithm for the online bipartite matching problem considered in Section 4, of which the transversal matroid secretary problem is a special case.

Highlights

- There has been enormous progress in the field of machine learning in the last decade, which has affected a variety of other areas as well
- In Section 4, we study the online bipartite matching problem in which the set of nodes L of a bipartite graph G = (L ∪ R, E), with |L| = n and |R| = m, arrives online in a uniformly random order [25, 22]
- As a result of possible independent interest, we show that there exists a deterministic (1/4 − o(1))-competitive algorithm for the graphic matroid secretary problem, which can roughly be seen as a deterministic version of the algorithm of Soto et al [41]
- Korula and Pal [25] provided the first constant competitive algorithm for the online bipartite matching problem considered in Section 4, of which the transversal matroid secretary problem is a special case
- It turns out that this type of predictions closely corresponds to the so-called online vertexweighted bipartite matching problem where every offline node is given a weight wr, and the goal is to select a matching with maximum weight, which is the sum of all weights wr for which the corresponding r is matched in the online algorithm
- Our results can be seen as the first evidence that online selection problems are a promising area for the incorporation of machine learned advice following the frameworks of [32, 38]

Results

- The authors assume that the authors are given, for all offline nodes r ∈ R, a prediction p∗r for the value of the edge weight adjacent to r in some fixed optimal offline matching.
- It turns out that this type of predictions closely corresponds to the so-called online vertexweighted bipartite matching problem where every offline node is given a weight wr, and the goal is to select a matching with maximum weight, which is the sum of all weights wr for which the corresponding r is matched in the online algorithm.
- The authors will present deterministic and randomized algorithms, inspired by algorithms for the online vertex-weighted bipartite matching problem, that can be combined with the algorithm in [22] in order to obtain algorithms that incorporate the predictions and have the desired properties.
- It is well-known that, even for uniformly random arrival order and unit edge weights, one cannot obtain anything better than a 1/2-approximation with a deterministic algorithm [1].8 This means that, with the choice of predictions, the authors cannot do better than a 1/2-approximation in the ideal case in which the predictions are perfect.
- There is a deterministic algorithm for the online bipartite matching problem with uniformly random arrivals that is asymptotically gc,d,λ(η)-competitive in expectation, where gc,d,λ(η) =
- The authors can give better approximation guarantees than the algorithm given in the previous section by using a convex combination of the algorithm of Kesselheim et al [22], and the randomized algorithm of Huang et al [18] for online vertex-weighted bipartite matching with uniformly random arrivals.
- Suppose the authors are given an algorithm A for instances of the online vertex-weighted bipartite matching problem.

Conclusion

- It is well-known that the offline optimal solution of this problem can be found by the greedy algorithm that orders all the edge weights in decreasing order, and selects elements in this order whenever possible.
- If adding emax(v) would have created an cycle in the set of elements selected so far, this yields a unique directed cycle in the graph DM defined in the previous section.
- Is it possible to show that there exists an algorithm under a natural prediction model that is constant-competitive for accurate predictions, and that is still O(1/ log(log(r)))-competitive in the worst case, matching the results in [13, 28]?

Summary

- There has been enormous progress in the field of machine learning in the last decade, which has affected a variety of other areas as well.
- This follows from the fact that in this case the authors obtain the so-called vertex-weighted online bipartite matching problem for which there is a deterministic 1/2-competitive algorithm, and no algorithm can do better [1].
- Korula and Pal [25] provided the first constant competitive algorithm for the online bipartite matching problem considered in Section 4, of which the transversal matroid secretary problem is a special case.
- The authors assume that the authors are given, for all offline nodes r ∈ R, a prediction p∗r for the value of the edge weight adjacent to r in some fixed optimal offline matching.
- It turns out that this type of predictions closely corresponds to the so-called online vertexweighted bipartite matching problem where every offline node is given a weight wr, and the goal is to select a matching with maximum weight, which is the sum of all weights wr for which the corresponding r is matched in the online algorithm.
- The authors will present deterministic and randomized algorithms, inspired by algorithms for the online vertex-weighted bipartite matching problem, that can be combined with the algorithm in [22] in order to obtain algorithms that incorporate the predictions and have the desired properties.
- It is well-known that, even for uniformly random arrival order and unit edge weights, one cannot obtain anything better than a 1/2-approximation with a deterministic algorithm [1].8 This means that, with the choice of predictions, the authors cannot do better than a 1/2-approximation in the ideal case in which the predictions are perfect.
- There is a deterministic algorithm for the online bipartite matching problem with uniformly random arrivals that is asymptotically gc,d,λ(η)-competitive in expectation, where gc,d,λ(η) =
- The authors can give better approximation guarantees than the algorithm given in the previous section by using a convex combination of the algorithm of Kesselheim et al [22], and the randomized algorithm of Huang et al [18] for online vertex-weighted bipartite matching with uniformly random arrivals.
- Suppose the authors are given an algorithm A for instances of the online vertex-weighted bipartite matching problem.
- It is well-known that the offline optimal solution of this problem can be found by the greedy algorithm that orders all the edge weights in decreasing order, and selects elements in this order whenever possible.
- If adding emax(v) would have created an cycle in the set of elements selected so far, this yields a unique directed cycle in the graph DM defined in the previous section.
- Is it possible to show that there exists an algorithm under a natural prediction model that is constant-competitive for accurate predictions, and that is still O(1/ log(log(r)))-competitive in the worst case, matching the results in [13, 28]?

Related work

- This subsection consists of three parts. First we discuss relevant approximation algorithms for the matroid secretary problem without any form of prior information, then we consider models that incorporate additional information, such as the area of prophet inequalities. Finally, we give a short overview of related problems that have been analyzed with the inclusion of machine learned advice following the frameworks in [32, 38], which we study here as well.

Approximation algorithms for the matroid secretary problem. The classical secretary problem was originally introduced by Gardner [15], and solved by Lindley [30] and Dynkin [11], who gave 1/e-competitive algorithms. Babaioff et al [4] introduced the matroid secretary problem, a considerable generalization of the classical secretary problem, where the goal is to select a set of secretaries with maximum total value under a matroid constraint for the set of feasible secretaries. They provided an O(1/ log(r))-competitive algorithm for this problem, where r is the rank of the underlying matroid. Lachish [28] later gave an O(1/ log log(r))-competitive algorithm, and a simplified algorithm with the same guarantee was given by Feldman, Svensson and Zenklusen [13]. It is still a major open problem if there exists a constant-competitive algorithm for the matroid secretary problem. Nevertheless, many constant-competitive algorithms are known for special classes of matroids, and we mention those relevant to the results in this work (see, e.g., [4, 41] for further related work).

Study subjects and analysis

cases: 3

To see this, note that when the vertex arrived, there was the option to match it to r, as this node is assumed not to have been matched in the second phase. So, there are the following three cases: either got matched to r, or it got matched to some other r for which w( , r ) ≥ w( , r) ≥ p∗r − λ or r was matched earlier during the third phase to some other for which w( , r) ≥ p∗r − λ. Looking closely at the analysis of Kesselheim et al [22], see Appendix C.3, it follows that the probability that a fixed node r did not get matched in the second phase satisfies d P(r was not matched in Phase II) ≥ − o(1)

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