New Query Lower Bounds for Submodular Function Minimization

We consider submodular function minimization in the oracle model: given black-box access to a submodular set function $f:2^{[n]}\rightarrow \mathbb{R}$, find an element of $\arg\min_S \{f(S)\}$ using as few queries to $f(\cdot)$ as possible. State-of-the-art algorithms succeed with $\tilde{O}(n^2)$ queries [LeeSW15], yet the best-known lower bound has never been improved beyond $n$ [Harvey08]. We provide a query lower bound of $2n$ for submodular function minimization, a $3n/2-2$ query lower bound for the non-trivial minimizer of a symmetric submodular function, and a $\binom{n}{2}$ query lower bound for the non-trivial minimizer of an asymmetric submodular function. Our $3n/2-2$ lower bound results from a connection between SFM lower bounds and a novel concept we term the cut dimension of a graph. Interestingly, this yields a $3n/2-2$ cut-query lower bound for finding the global mincut in an undirected, weighted graph, but we also prove it cannot yield a lower bound better than $n+1$ for $s$-$t$ mincut, even in a directed, weighted graph

Two Problems In Combinatorial Optimization Under Uncertainty

This thesis improves upon existing lower and upper bounds for questions in the paradigm of combinatorial optimization under uncertainty. We address problems in two distinct areas. Firstly, we consider the query complexity of submodular function minimization: given oracle access to an unknown submodular $f: \mathcal{P}([n]) \rightarrow \mathbb{R}$, what is the minimum number of queries needed to deterministically compute $\min_{S} f(S)$ and some element of $\arg \min_{S} f(S)$? In the case that $f$ is symmetric, the previous best lower bound for non-trivial minimization was $n$ queries due to Harvey; we improve this to ${\sim} 1.5n$ via a novel concept we call the cut dimension. We secondly consider questions in the area of prophet inequalities, focusing on problems that arise when feasibility constraints are given by a matroid or an intersection of matroids --- a very important and well-studied setting in the literature. In the case of bipartite matching (an example of feasibility constraints given by the intersection of two matroids), we improve the previous lower bound of 2.25 for the best possible competitive ratio (due to Gravin and Wang) to $7/3$. When feasibility constraints are given by matroids of large girth, we resolve the previously open question of the best possible competitive ratio by showing that it is precisely 2. And finally, in the case where feasibility constraints are given by an intersection of $p$ partition matroids (generalizing bipartite matching), we provide an algorithm which is $(p+1)$-competitive, improving on the previously best-known $({\sim} ep)$-competitive algorithm due to Feldman, Svensson, and Zenklusen

Two Problems In Combinatorial Optimization Under Uncertainty

This thesis improves upon existing lower and upper bounds for questions in the paradigm of combinatorial optimization under uncertainty. We address problems in two distinct areas. Firstly, we consider the query complexity of submodular function minimization: given oracle access to an unknown submodular $f: \mathcal{P}([n]) \rightarrow \mathbb{R}$, what is the minimum number of queries needed to deterministically compute $\min_{S} f(S)$ and some element of $\arg \min_{S} f(S)$? In the case that $f$ is symmetric, the previous best lower bound for non-trivial minimization was $n$ queries due to Harvey; we improve this to ${\sim} 1.5n$ via a novel concept we call the cut dimension. We secondly consider questions in the area of prophet inequalities, focusing on problems that arise when feasibility constraints are given by a matroid or an intersection of matroids --- a very important and well-studied setting in the literature. In the case of bipartite matching (an example of feasibility constraints given by the intersection of two matroids), we improve the previous lower bound of 2.25 for the best possible competitive ratio (due to Gravin and Wang) to $7/3$. When feasibility constraints are given by matroids of large girth, we resolve the previously open question of the best possible competitive ratio by showing that it is precisely 2. And finally, in the case where feasibility constraints are given by an intersection of $p$ partition matroids (generalizing bipartite matching), we provide an algorithm which is $(p+1)$-competitive, improving on the previously best-known $({\sim} ep)$-competitive algorithm due to Feldman, Svensson, and Zenklusen

Two Problems In Combinatorial Optimization Under Uncertainty

This thesis improves upon existing lower and upper bounds for questions in the paradigm of combinatorial optimization under uncertainty. We address problems in two distinct areas. Firstly, we consider the query complexity of submodular function minimization: given oracle access to an unknown submodular $f: \mathcal{P}([n]) \rightarrow \mathbb{R}$, what is the minimum number of queries needed to deterministically compute $\min_{S} f(S)$ and some element of $\arg \min_{S} f(S)$? In the case that $f$ is symmetric, the previous best lower bound for non-trivial minimization was $n$ queries due to Harvey; we improve this to ${\sim} 1.5n$ via a novel concept we call the cut dimension. We secondly consider questions in the area of prophet inequalities, focusing on problems that arise when feasibility constraints are given by a matroid or an intersection of matroids --- a very important and well-studied setting in the literature. In the case of bipartite matching (an example of feasibility constraints given by the intersection of two matroids), we improve the previous lower bound of 2.25 for the best possible competitive ratio (due to Gravin and Wang) to $7/3$. When feasibility constraints are given by matroids of large girth, we resolve the previously open question of the best possible competitive ratio by showing that it is precisely 2. And finally, in the case where feasibility constraints are given by an intersection of $p$ partition matroids (generalizing bipartite matching), we provide an algorithm which is $(p+1)$-competitive, improving on the previously best-known $({\sim} ep)$-competitive algorithm due to Feldman, Svensson, and Zenklusen

Optimal Item Pricing in Online Combinatorial Auctions

We consider a fundamental pricing problem in combinatorial auctions. We are given a set of indivisible items and a set of buyers with randomly drawn monotone valuations over subsets of items. A decision maker sets item prices and then the buyers make sequential purchasing decisions, taking their favorite set among the remaining items. We parametrize an instance by d, the size of the largest set a buyer may want. Our main result asserts that there exist prices such that the expected (over the random valuations) welfare of the allocation they induce is at least a factor 1/ (d+ 1 ) times the expected optimal welfare in hindsight. Moreover we prove that this bound is tight. Thus, our result not only improves upon the 1/ (4 d- 2 ) bound of Dütting et al., but also settles the approximation that can be achieved by using item prices. We further show how to compute our prices in polynomial time. We provide additional results for the special case when buyers’ valuations are known (but a posted-price mechanism is still desired).</p

Optimal Item Pricing in Online Combinatorial Auctions

We consider a fundamental pricing problem in combinatorial auctions. We are given a set of indivisible items and a set of buyers with randomly drawn monotone valuations over subsets of items. A decision maker sets item prices and then the buyers make sequential purchasing decisions, taking their favorite set among the remaining items. We parametrize an instance by d, the size of the largest set a buyer may want. Our main result asserts that there exist prices such that the expected (over the random valuations) welfare of the allocation they induce is at least a factor 1/ (d+ 1 ) times the expected optimal welfare in hindsight. Moreover we prove that this bound is tight. Thus, our result not only improves upon the 1/ (4 d- 2 ) bound of Dütting et al., but also settles the approximation that can be achieved by using item prices. We further show how to compute our prices in polynomial time. We provide additional results for the special case when buyers’ valuations are known (but a posted-price mechanism is still desired).Green Open Access added to TU Delft Institutional Repository 'You share, we take care!' - Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.Learning & Autonomous Contro

Gene-Wide Identification of Episodic Selection

We present BUSTED, a new approach to identifying gene-wide evidence of episodic positive selection, where the non-synonymous substitution rate is transiently greater than the synonymous rate. BUSTED can be used either on an entire phylogeny (without requiring an a priori hypothesis regarding which branches are under positive selection) or on a pre-specified subset of foreground lineages (if a suitable a priori hypothesis is available). Selection is modeled as varying stochastically over branches and sites, and we propose a computationally inexpensive evidence metric for identifying sites subject to episodic positive selection on any foreground branches. We compare BUSTED with existing models on simulated and empirical data. An implementation is available on www.datamonkey.org/busted, with a widget allowing the interactive specification of foreground branches

Gene-Wide Identification of Episodic Selection

We present BUSTED, a new approach to identifying gene-wide evidence of episodic positive selection, where the non-synonymous substitution rate is transiently greater than the synonymous rate. BUSTED can be used either on an entire phylogeny (without requiring an a priori hypothesis regarding which branches are under positive selection) or on a pre-specified subset of foreground lineages (if a suitable a priori hypothesis is available). Selection is modeled as varying stochastically over branches and sites, and we propose a computationally inexpensive evidence metric for identifying sites subject to episodic positive selection on any foreground branches. We compare BUSTED with existing models on simulated and empirical data. An implementation is available on www.datamonkey.org/busted, with a widget allowing the interactive specification of foreground branches