3,504 research outputs found
Fast and Deterministic Approximations for k-Cut
In an undirected graph, a k-cut is a set of edges whose removal breaks the graph into at least k connected components. The minimum weight k-cut can be computed in n^O(k) time, but when k is treated as part of the input, computing the minimum weight k-cut is NP-Hard [Goldschmidt and Hochbaum, 1994]. For poly(m,n,k)-time algorithms, the best possible approximation factor is essentially 2 under the small set expansion hypothesis [Manurangsi, 2017]. Saran and Vazirani [1995] showed that a (2 - 2/k)-approximately minimum weight k-cut can be computed via O(k) minimum cuts, which implies a O~(km) randomized running time via the nearly linear time randomized min-cut algorithm of Karger [2000]. Nagamochi and Kamidoi [2007] showed that a (2 - 2/k)-approximately minimum weight k-cut can be computed deterministically in O(mn + n^2 log n) time. These results prompt two basic questions. The first concerns the role of randomization. Is there a deterministic algorithm for 2-approximate k-cuts matching the randomized running time of O~(km)? The second question qualitatively compares minimum cut to 2-approximate minimum k-cut. Can 2-approximate k-cuts be computed as fast as the minimum cut - in O~(m) randomized time?
We give a deterministic approximation algorithm that computes (2 + eps)-minimum k-cuts in O(m log^3 n / eps^2) time, via a (1 + eps)-approximation for an LP relaxation of k-cut
An Investigation Report on Auction Mechanism Design
Auctions are markets with strict regulations governing the information
available to traders in the market and the possible actions they can take.
Since well designed auctions achieve desirable economic outcomes, they have
been widely used in solving real-world optimization problems, and in
structuring stock or futures exchanges. Auctions also provide a very valuable
testing-ground for economic theory, and they play an important role in
computer-based control systems.
Auction mechanism design aims to manipulate the rules of an auction in order
to achieve specific goals. Economists traditionally use mathematical methods,
mainly game theory, to analyze auctions and design new auction forms. However,
due to the high complexity of auctions, the mathematical models are typically
simplified to obtain results, and this makes it difficult to apply results
derived from such models to market environments in the real world. As a result,
researchers are turning to empirical approaches.
This report aims to survey the theoretical and empirical approaches to
designing auction mechanisms and trading strategies with more weights on
empirical ones, and build the foundation for further research in the field
On Conceptually Simple Algorithms for Variants of Online Bipartite Matching
We present a series of results regarding conceptually simple algorithms for
bipartite matching in various online and related models. We first consider a
deterministic adversarial model. The best approximation ratio possible for a
one-pass deterministic online algorithm is , which is achieved by any
greedy algorithm. D\"urr et al. recently presented a -pass algorithm called
Category-Advice that achieves approximation ratio . We extend their
algorithm to multiple passes. We prove the exact approximation ratio for the
-pass Category-Advice algorithm for all , and show that the
approximation ratio converges to the inverse of the golden ratio
as goes to infinity. The convergence is
extremely fast --- the -pass Category-Advice algorithm is already within
of the inverse of the golden ratio.
We then consider a natural greedy algorithm in the online stochastic IID
model---MinDegree. This algorithm is an online version of a well-known and
extensively studied offline algorithm MinGreedy. We show that MinDegree cannot
achieve an approximation ratio better than , which is guaranteed by any
consistent greedy algorithm in the known IID model.
Finally, following the work in Besser and Poloczek, we depart from an
adversarial or stochastic ordering and investigate a natural randomized
algorithm (MinRanking) in the priority model. Although the priority model
allows the algorithm to choose the input ordering in a general but well defined
way, this natural algorithm cannot obtain the approximation of the Ranking
algorithm in the ROM model
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