skip to main content


Title: Improved Bounds in Stochastic Matching and Optimization
Real-world problems often have parameters that are uncertain during the optimization phase; stochastic optimization or stochastic programming is a key approach introduced by Beale and by Dantzig in the 1950s to address such uncertainty. Matching is a classical problem in combinatorial optimization. Modern stochastic versions of this problem model problems in kidney exchange, for instance. We improve upon the current-best approximation bound of 3.709 for stochastic matching due to Adamczyk et al. (in: Algorithms-ESA 2015, Springer, Berlin, 2015) to 3.224; we also present improvements on Bansal et al. (Algorithmica 63(4):733–762, 2012) for hypergraph matching and for relaxed versions of the problem. These results are obtained by improved analyses and/or algorithms for rounding linear-programming relaxations of these problems.  more » « less
Award ID(s):
1749864
NSF-PAR ID:
10111320
Author(s) / Creator(s):
; ; ; ;
Date Published:
Journal Name:
Algorithmica
Volume:
80
ISSN:
0178-4617
Page Range / eLocation ID:
3225-3252
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Chiappa, Silvia ; Calandra, Roberto (Ed.)
    The article considers smooth optimization of functions on Lie groups. By generalizing NAG variational principle in vector space (Wibisono et al., 2016) to general Lie groups, continuous Lie-NAG dynamics which are guaranteed to converge to local optimum are obtained. They correspond to momentum versions of gradient flow on Lie groups. A particular case of SO(𝑛) is then studied in details, with objective functions corresponding to leading Generalized EigenValue problems: the Lie-NAG dynamics are first made explicit in coordinates, and then discretized in structure preserving fashions, resulting in optimization algorithms with faithful energy behavior (due to conformal symplecticity) and exactly remaining on the Lie group. Stochastic gradient versions are also investigated. Numerical experiments on both synthetic data and practical problem (LDA for MNIST) demonstrate the effectiveness of the proposed methods as optimization algorithms (\emph{not} as a classification method). 
    more » « less
  2. Abstract

    Quantum computing is a rapidly growing field with the potential to change how we solve previously intractable problems. Emerging hardware is approaching a complexity that requires increasingly sophisticated programming and control. Scaffold is an older quantum programming language that was originally designed for resource estimation for far-future, large quantum machines, and ScaffCC is the corresponding LLVM-based compiler. For the first time, we provide a full and complete overview of the language itself, the compiler as well as its pass structure. While previous works Abhariet al(2015Parallel Comput.452–17), Abhariet al(2012 Scaffold: quantum programming languagehttps://cs.princeton.edu/research/techreps/TR-934-12), have piecemeal descriptions of different portions of this toolchain, we provide a more full and complete description in this paper. We also introduce updates to ScaffCC including conditional measurement and multidimensional qubit arrays designed to keep in step with modern quantum assembly languages, as well as an alternate toolchain targeted at maintaining correctness and low resource count for noisy-intermediate scale quantum (NISQ) machines, and compatibility with current versions of LLVM and Clang. Our goal is to provide the research community with a functional LLVM framework for quantum program analysis, optimization, and generation of executable code.

     
    more » « less
  3. Column-sparse packing problems arise in several contexts in both deterministic and stochastic discrete optimization. We present two unifying ideas, (non-uniform) attenuation and multiple-chance algorithms, to obtain improved approximation algorithms for some well-known families of such problems. As three main examples, we attain the integrality gap, up to lower-order terms, for known LP relaxations for k-column sparse packing integer programs (Bansal et al., Theory of Computing, 2012) and stochastic k-set packing (Bansal et al., Algorithmica, 2012), and go “half the remaining distance” to optimal for a major integrality-gap conjecture of Furedi, Kahn and Seymour on hypergraph matching (Combinatorica, 1993). 
    more » « less
  4. Given a graph and an integer k, Densest k-Subgraph is the algorithmic task of finding the subgraph on k vertices with the maximum number of edges. This is a fundamental problem that has been subject to intense study for decades, with applications spanning a wide variety of fields. The state-of-the-art algorithm is an O(n^{1/4+ϵ})-factor approximation (for any ϵ>0) due to Bhaskara et al. [STOC '10]. Moreover, the so-called log-density framework predicts that this is optimal, i.e. it is impossible for an efficient algorithm to achieve an O(n^{1/4−ϵ})-factor approximation. In the average case, Densest k-Subgraph is a prototypical noisy inference task which is conjectured to exhibit a statistical-computational gap. In this work, we provide the strongest evidence yet of hardness for Densest k-Subgraph by showing matching lower bounds against the powerful Sum-of-Squares (SoS) algorithm, a meta-algorithm based on convex programming that achieves state-of-art algorithmic guarantees for many optimization and inference problems. For k ≤ n^1/2, we obtain a degree n^δ SoS lower bound for the hard regime as predicted by the log-density framework. To show this, we utilize the modern framework for proving SoS lower bounds on average-case problems pioneered by Barak et al. [FOCS '16]. A key issue is that small denser-than-average subgraphs in the input will greatly affect the value of the candidate pseudo-expectation operator around the subgraph. To handle this challenge, we devise a novel matrix factorization scheme based on the positive minimum vertex separator. We then prove an intersection tradeoff lemma to show that the error terms when using this separator are indeed small. 
    more » « less
  5. From optimal transport to robust dimensionality reduction, a plethora of machine learning applications can be cast into the min-max optimization problems over Riemannian manifolds. Though many min-max algorithms have been analyzed in the Euclidean setting, it has proved elusive to translate these results to the Riemannian case. Zhang et al. [2022] have recently shown that geodesic convex concave Riemannian problems always admit saddle-point solutions. Inspired by this result, we study whether a performance gap between Riemannian and optimal Euclidean space convex-concave algorithms is necessary. We answer this question in the negative—we prove that the Riemannian corrected extragradient (RCEG) method achieves last-iterate convergence at a linear rate in the geodesically strongly-convex-concave case, matching the Euclidean result. Our results also extend to the stochastic or non-smooth case where RCEG and Riemanian gradient ascent descent (RGDA) achieve near-optimal convergence rates up to factors depending on curvature of the manifold. 
    more » « less