60,778 research outputs found
Sublinear Algorithms for -Approximate Matching
We study sublinear time algorithms for estimating the size of maximum
matching. After a long line of research, the problem was finally settled by
Behnezhad [FOCS'22], in the regime where one is willing to pay an approximation
factor of . Very recently, Behnezhad et al.[SODA'23] improved the
approximation factor to using
time. This improvement over the factor is, however, minuscule and they
asked if even -approximation is possible in time. We
give a strong affirmative answer to this open problem by showing
-approximation algorithms that run in
time. Our approach is conceptually simple and
diverges from all previous sublinear-time matching algorithms: we show a
sublinear time algorithm for computing a variant of the edge-degree constrained
subgraph (EDCS), a concept that has previously been exploited in dynamic
[Bernstein Stein ICALP'15, SODA'16], distributed [Assadi et al. SODA'19] and
streaming [Bernstein ICALP'20] settings, but never before in the sublinear
setting. Independent work: Behnezhad, Roghani and Rubinstein [BRR'23]
independently showed sublinear algorithms similar to our Theorem 1.2 in both
adjacency list and matrix models. Furthermore, in [BRR'23], they show
additional results on strictly better-than-1.5 approximate matching algorithms
in both upper and lower bound sides
Minimizing Flow Time in the Wireless Gathering Problem
We address the problem of efficient data gathering in a wireless network
through multi-hop communication. We focus on the objective of minimizing the
maximum flow time of a data packet. We prove that no polynomial time algorithm
for this problem can have approximation ratio less than \Omega(m^{1/3) when
packets have to be transmitted, unless . We then use resource
augmentation to assess the performance of a FIFO-like strategy. We prove that
this strategy is 5-speed optimal, i.e., its cost remains within the optimal
cost if we allow the algorithm to transmit data at a speed 5 times higher than
that of the optimal solution we compare to
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