(Near) Optimal Adaptivity Gaps for Stochastic Multi-Value Probing

Consider a kidney-exchange application where we want to find a max-matching in a random graph. To find whether an edge e exists, we need to perform an expensive test, in which case the edge e appears independently with a known probability p_e. Given a budget on the total cost of the tests, our goal is to find a testing strategy that maximizes the expected maximum matching size. The above application is an example of the stochastic probing problem. In general the optimal stochastic probing strategy is difficult to find because it is adaptive - decides on the next edge to probe based on the outcomes of the probed edges. An alternate approach is to show the adaptivity gap is small, i.e., the best non-adaptive strategy always has a value close to the best adaptive strategy. This allows us to focus on designing non-adaptive strategies that are much simpler. Previous works, however, have focused on Bernoulli random variables that can only capture whether an edge appears or not. In this work we introduce a multi-value stochastic probing problem, which can also model situations where the weight of an edge has a probability distribution over multiple values. Our main technical contribution is to obtain (near) optimal bounds for the (worst-case) adaptivity gaps for multi-value stochastic probing over prefix-closed constraints. For a monotone submodular function, we show the adaptivity gap is at most 2 and provide a matching lower bound. For a weighted rank function of a k-extendible system (a generalization of intersection of k matroids), we show the adaptivity gap is between O(k log k) and k. None of these results were known even in the Bernoulli case where both our upper and lower bounds also apply, thereby resolving an open question of Gupta et al. [Gupta et al., 2017]

Robust Algorithms for the Secretary Problem

In classical secretary problems, a sequence of n elements arrive in a uniformly random order, and we want to choose a single item, or a set of size K. The random order model allows us to escape from the strong lower bounds for the adversarial order setting, and excellent algorithms are known in this setting. However, one worrying aspect of these results is that the algorithms overfit to the model: they are not very robust. Indeed, if a few "outlier" arrivals are adversarially placed in the arrival sequence, the algorithms perform poorly. E.g., Dynkin’s popular 1/e-secretary algorithm is sensitive to even a single adversarial arrival: if the adversary gives one large bid at the beginning of the stream, the algorithm does not select any element at all. We investigate a robust version of the secretary problem. In the Byzantine Secretary model, we have two kinds of elements: green (good) and red (rogue). The values of all elements are chosen by the adversary. The green elements arrive at times uniformly randomly drawn from [0,1]. The red elements, however, arrive at adversarially chosen times. Naturally, the algorithm does not see these colors: how well can it solve secretary problems? We show that selecting the highest value red set, or the single largest green element is not possible with even a small fraction of red items. However, on the positive side, we show that these are the only bad cases, by giving algorithms which get value comparable to the value of the optimal green set minus the largest green item. (This benchmark reminds us of regret minimization and digital auctions, where we subtract an additive term depending on the "scale" of the problem.) Specifically, we give an algorithm to pick K elements, which gets within (1-ε) factor of the above benchmark, as long as K ≥ poly(ε^{-1} log n). We extend this to the knapsack secretary problem, for large knapsack size K. For the single-item case, an analogous benchmark is the value of the second-largest green item. For value-maximization, we give a poly log^* n-competitive algorithm, using a multi-layered bucketing scheme that adaptively refines our estimates of second-max over time. For probability-maximization, we show the existence of a good randomized algorithm, using the minimax principle. We hope that this work will spur further research on robust algorithms for the secretary problem, and for other problems in sequential decision-making, where the existing algorithms are not robust and often tend to overfit to the model.ISSN:1868-896

Covering random graphs by monochromatic trees

Given an r-edge coloured complete graph Kn , how many monochromatic connected com- ponents does one need in order to cover its vertex set? This natural question is a well-known essentially equivalent formulation of the classical Ryser’s conjecture which, despite a lot of at- tention over the last 50 years, still remains wide open. A number of recent papers considersparse random analogue of this question, asking for the minimum number of monochromatic components needed to cover the vertex set of the random graph G(n, p). Recently, Bucić, Korándi and Sudakov established a connection between this problem andcertain Helly-type local to global question for hypergraphs raised about 30 years ago by Erdős, Hajnal and Tuza. We identify a modified version of the hypergraph problem which controls the answer to the problem of covering random graphs with monochromatic components more pre- cisely. To showcase the power of our approach, we essentially resolve the 3-colour case by showing that (log n/n)1/4 is a threshold at which point 3 monochromatic components are needed to cover all vertices of a 3-edge-coloured random graph, answering a question posed by Kohayakawa, Mendonça, Mota and Schülke

Covering random graphs with monochromatic trees

Given an r-edge-colored complete graph Kn, how many monochromatic connected components does one need in order to cover its vertex set? This natural question is a well-known essentially equivalent formulation of the classical Ryser's conjecture which, despite a lot of attention over the last 50 years, still remains open. A number of recent papers consider a sparse random analogue of this question, asking for the minimum number of monochromatic components needed to cover the vertex set of an r- edge-colored random graph.(n, p). Recently, Bucic, Korandi, and Sudakov established a connection between this problem and a certain Helly-type local to global question for hypergraphs raised about 30 years ago by Erd.os, Hajnal, and Tuza. We identify a modified version of the hypergraph problem which controls the answer to the problem of covering random graphs with monochromatic components more precisely. To showcase the power of our approach, we essentially resolve the 3-color case by showing that (log n/n)(1/4) is a threshold at which point three monochromatic components are needed to cover all vertices of a 3-edge-colored random graph, answering a question posed by Kohayakawa, Mendonca, Mota, and Schulke. Our approach also allows us to determine the answer in the general r-edge colored instance of the problem, up to lower order terms, around the point when it first becomes bounded, answering a question of Buci ' c, Korandi, and Sudakov.ISSN:1042-9832ISSN:1098-241

Asymptotics of the Hypergraph Bipartite Turan Problem

For positive integers s, t, r, let K(r)s,t denote the r-uniform hypergraph whose vertex set is the union of pairwise disjoint sets X,Y1,…,Yt, where |X|=s and |Y1|=⋯=|Yt|=r−1, and whose edge set is {{x}∪Yi:x∈X,1≤i≤t}. The study of the Turán function of K(r)s,t received considerable interest in recent years. Our main results are as follows. First, we show that ex(n,K(r)s,t)=Os,r(t1s−1nr−1s−1)(1) for all s,t≥2 and r≥3, improving the power of n in the previously best bound and resolving a question of Mubayi and Verstraëte about the dependence of ex(n,K(3)2,t) on t. Second, we show that (1) is tight when r is even and t≫s. This disproves a conjecture of Xu, Zhang and Ge. Third, we show that (1) is not tight for r=3, namely that ex(n,K(3)s,t)=Os,t(n3−1s−1−εs) (for all s≥3). This indicates that the behaviour of ex(n,K(r)s,t) might depend on the parity of r. Lastly, we prove a conjecture of Ergemlidze, Jiang and Methuku on the hypergraph analogue of the bipartite Turán problem for graphs with bounded degrees on one side. Our tools include a novel twist on the dependent random choice method as well as a variant of the celebrated norm graphs constructed by Kollár, Rónyai and Szabó.ISSN:0209-9683ISSN:1439-691