Sam Chiu-wai Wong

A Faster Cutting Plane Method and its Implications for Combinatorial and Convex Optimization

In this paper we improve upon the running time for finding a point in a convex set given a separation oracle. In particular, given a separation oracle for a convex set K ⊂ Rⁿ that is contained in a box of radius R we show how to either compute a point in K or prove that K does not contain a ball of radius ϵ using an expected O(n log(nR/ϵ)) evaluations of the oracle and additional time O(n³ log^O(1)(nR/ϵ)). This matches the oracle complexity and improves upon the O(n^ω+1 log(nR/ϵ)) additional time of the previous fastest algorithm achieved over 25 years ago by Vaidya [91] for the current value of the matrix multiplication constant w <; 2.373 [98], [36] when R/ϵ = O(poly(n)). Using a mix of standard reductions and new techniques we show how our algorithm can be used to improve the running time for solving classic problems in continuous and combinatorial optimization. In particular we provide the following running time improvements: · Submodular Function Minimization: n is the size of the ground set, M is the maximum absolute value of function values and EO is the time for function evaluation. Our weakly and strongly polynomial time algorithms have a running time of O(n² log nM · EO + n³ log^O(1) nM) and O(n³ log² n · EO + n⁴ log^O(1) n), improving upon the previous best of O((n⁴ · EO + n⁵)logM) and O(n⁵ · EO + n⁶) respectively. · Submodular Flow: n = |V|, m = |E|, C is the maximum edge cost in absolute value and U is maximum edge capacity in absolute value. We obtain a faster weakly polynomial running time of O(n² log nCU · EO + n³ logO(1) nCU), improving upon the previous best of O(mn⁵ log nU · EO) and O (n⁴h min {log C, log U}) from 15 years ago by a factor of Õ(n⁴). We also achieve faster strongly polynomial time algorithms as a consequence of our result on submodular minimization. · Matroid Intersection: n is the size of the ground set, r is the maximum size of independent sets, M is the maximum absolute value of element weight, T_rank and T_ind are the time for each rank and independence oracle query. We obtain a running time of O((nr log² nT_rank+n³ log^O(1) n) log nM) and O((n² log nT_ind+n³ log^O(1) n) log nM), achieving the first quadratic bound on the query complexity for the independence and rank oracles. In the unweighted case, this is the first improvement since 1986 for independence oracle. · Semidefinite Programming: n is the number of constraints, m is the number of dimensions and S is the total number of non-zeros in the constraint matrices. We obtain a running time of O(n(n² + m^ω + S)), improving upon the previous best of Õ(n(n^ω + m^ω + S)) for the regime S is small.

Tight algorithms for vertex cover with hard capacities on multigraphs and hypergraphs

In this paper we give a f-approximation algorithm for the minimum unweighted Vertex Cover problem with Hard Capacity constraints (VCHC) on f-hypergraphs. This problem generalizes standard vertex cover for which the best known approximation ratio is also f and cannot be improved assuming the unique game conjecture. Our result is therefore essentially the best possible. This improves over the previous 2.155 (for f = 2) and 2f approximation algorithms by Cheung, Goemans and Wong (CGW). At the heart of our approach is to apply iterative rounding to a natural LP relaxation that is slightly different from prior works which used (non-iterative) rounding. Our algorithm is significantly simpler and offers an intuitive explanation why f-approximation can be achieved for VCHC. We also present faster implementations of our method based on iteratively rounding the solution to certain CGW-style covering LPs.

Multiplicative bidding in online advertising

In this paper, we initiate the study of the multiplicative bidding language adopted by major Internet search companies. In multiplicative bidding, the effective bid on a particular search auction is the product of a base bid and bid adjustments that are dependent on features of the search (for example, the geographic location of the user, or the platform on which the search is conducted). We consider the task faced by the advertiser when setting these bid adjustments, and establish a foundational optimization problem that captures the core difficulty of bidding under this language. We give matching algorithmic and approximation hardness results for this problem; these results are against an information-theoretic bound, and thus have implications on the power of the multiplicative bidding language itself. Inspired by empirical studies of search engine price data, we then codify the relevant restrictions of the problem, and give further algorithmic and hardness results. Our main technical contribution is an O(log n)-approximation for the case of multiplicative prices and monotone values. We also provide empirical validations of our problem restrictions, and test our algorithms on real data against natural benchmarks. Our experiments show that they perform favorably compare with the baseline.

Subquadratic submodular function minimization

Submodular function minimization (SFM) is a fundamental discrete optimization problem which generalizes many well known problems, has applications in various fields, and can be solved in polynomial time. Owing to applications in computer vision and machine learning, fast SFM algorithms are highly desirable. The current fastest algorithms [Lee, Sidford, Wong, 2015] run in O(n2lognM· EO + n3logO(1)nM) time and O(n3log2n· EO +n4logO(1)n)time respectively, where M is the largest absolute value of the function (assuming the range is integers) and is the time taken to evaluate the function on any set. Although the best known lower bound on the query complexity is only Ω(n) [Harvey, 2008], the current shortest non-deterministic proof [Cunningham, 1985] certifying the optimum value of a function requires Ω(n2) function evaluations. The main contribution of this paper are subquadratic SFM algorithms. For integer-valued submodular functions, we give an SFM algorithm which runs in O(nM3logn· EO) time giving the first nearly linear time algorithm in any known regime. For real-valued submodular functions with range in [-1,1], we give an algorithm which in Õ(n5/3· EO/ε2) time returns an ε-additive approximate solution. At the heart of it, our algorithms are projected stochastic subgradient descent methods on the Lovasz extension of submodular functions where we crucially exploit submodularity and data structures to obtain fast, i.e. sublinear time, subgradient updates. The latter is crucial for beating the n2 bound - we show that algorithms which access only subgradients of the Lovasz extension, and these include the empirically fast Fujishige-Wolfe heuristic [Fujishige, 1980; Wolfe, 1976]

Matroid Online Bipartite Matching and Vertex Cover

The Adwords and Online Bipartite Matching problems have enjoyed a renewed attention over the past decade due to their connection to Internet advertising. Our community has contributed, among other things, new models (notably stochastic) and extensions to the classical formulations to address the issues that arise from practical needs. In this paper, we propose a new generalization based on matroids and show that many of the previous results extend to this more general setting. Because of the rich structures and expressive power of matroids, our new setting is potentially of interest both in theory and in practice. In the classical version of the problem, the offline side of a bipartite graph is known initially while vertices from the online side arrive one at a time along with their incident edges. The objective is to maintain a decent approximate matching from which no edge can be removed. Our generalization, called Matroid Online Bipartite Matching, additionally requires that the set of matched offline vertices be independent in a given matroid. In particular, the case of partition matroids corresponds to the natural scenario where each advertiser manages multiple ads with a fixed total budget. Our algorithms attain the same performance as the classical version of the problems considered, which are often provably the best possible. We present

Magnetic resonance imaging assisted cryosurgery

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Tight Bounds for Approximate Carathéodory and Beyond

-competitive algorithms for Matroid Online Bipartite Matching under the small bid assumption, as well as a

Computing walrasian equilibria: fast algorithms and structural properties

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Computing Walrasian Equilibria: Fast Algorithms and Economic Insights.

-competitive algorithm for Matroid Online Bipartite Matching in the random arrival model. A key technical ingredient of our results is a carefully designed primal-dual waterfilling procedure that accommodates for matroid constraints. This is inspired by the extension of our recent charging scheme for Online Bipartite Vertex Cover. Finally, given that only few online problems were studied in the submodular fashion, the techniques introduced in this paper for tackling submodularity in the online setting may be of independent interest.