591 research outputs found
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
Maximization of Non-Monotone Submodular Functions
A litany of questions from a wide variety of scientific disciplines can be cast as non-monotone submodular maximization problems. Since this class of problems includes max-cut, it is NP-hard. Thus, general purpose algorithms for the class tend to be approximation algorithms. For unconstrained problem instances, one recent innovation in this vein includes an algorithm of Buchbinder et al. (2012) that guarantees a ½ - approximation to the maximum. Building on this, for problems subject to cardinality constraints, Buchbinderet al. (2014) o_er guarantees in the range [0:356; ½ + o(1)]. Earlier work has the best approximation factors for more complex constraints and settings. For constraints that can be characterized as a solvable polytope, Chekuri et al. (2011) provide guarantees. For the online secretary setting, Gupta et al. (2010) provide guarantees. In sum, the current body of work on non-monotone submodular maximization lays strong foundations. However, there remains ample room for future algorithm development
A Randomness Threshold for Online Bipartite Matching, via Lossless Online Rounding
Over three decades ago, Karp, Vazirani and Vazirani (STOC'90) introduced the
online bipartite matching problem. They observed that deterministic algorithms'
competitive ratio for this problem is no greater than , and proved that
randomized algorithms can do better. A natural question thus arises: \emph{how
random is random}? i.e., how much randomness is needed to outperform
deterministic algorithms? The \textsc{ranking} algorithm of Karp et
al.~requires random bits, which, ignoring polylog terms,
remained unimproved. On the other hand, Pena and Borodin (TCS'19) established a
lower bound of random bits for any
competitive ratio.
We close this doubly-exponential gap, proving that, surprisingly, the lower
bound is tight. In fact, we prove a \emph{sharp threshold} of random bits for the randomness necessary and sufficient to
outperform deterministic algorithms for this problem, as well as its
vertex-weighted generalization. This implies the same threshold for the advice
complexity (nondeterminism) of these problems.
Similar to recent breakthroughs in the online matching literature, for
edge-weighted matching (Fahrbach et al.~FOCS'20) and adwords (Huang et
al.~FOCS'20), our algorithms break the barrier of by randomizing matching
choices over two neighbors. Unlike these works, our approach does not rely on
the recently-introduced OCS machinery, nor the more established randomized
primal-dual method. Instead, our work revisits a highly-successful online
design technique, which was nonetheless under-utilized in the area of online
matching, namely (lossless) online rounding of fractional algorithms. While
this technique is known to be hopeless for online matching in general, we show
that it is nonetheless applicable to carefully designed fractional algorithms
with additional (non-convex) constraints
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