1,700 research outputs found
Incremental (1-?)-Approximate Dynamic Matching in O(poly(1/?)) Update Time
In the dynamic approximate maximum bipartite matching problem we are given bipartite graph G undergoing updates and our goal is to maintain a matching of G which is large compared the maximum matching size ?(G). We define a dynamic matching algorithm to be ? (respectively (?, ?))-approximate if it maintains matching M such that at all times |M | ? ?(G) ? ? (respectively |M| ? ?(G) ? ? - ?).
We present the first deterministic (1-?)-approximate dynamic matching algorithm with O(poly(?^{-1})) amortized update time for graphs undergoing edge insertions. Previous solutions either required super-constant [Gupta FSTTCS\u2714, Bhattacharya-Kiss-Saranurak SODA\u2723] or exponential in 1/? [Grandoni-Leonardi-Sankowski-Schwiegelshohn-Solomon SODA\u2719] update time. Our implementation is arguably simpler than the mentioned algorithms and its description is self contained. Moreover, we show that if we allow for additive (1, ??n)-approximation our algorithm seamlessly extends to also handle vertex deletions, on top of edge insertions. This makes our algorithm one of the few small update time algorithms for (1-?)-approximate dynamic matching allowing for updates both increasing and decreasing the maximum matching size of G in a fully dynamic manner.
Our algorithm relies on the weighted variant of the celebrated Edge-Degree-Constrained-Subgraph (EDCS) datastructure introduced by [Bernstein-Stein ICALP\u2715]. As far as we are aware we introduce the first application of the weighted-EDCS for arbitrarily dense graphs. We also present a significantly simplified proof for the approximation ratio of weighed-EDCS as a matching sparsifier compared to [Bernstein-Stein], as well as simple descriptions of a fractional matching and fractional vertex cover defined on top of the EDCS. Considering the wide range of applications EDCS has found in settings such as streaming, sub-linear, stochastic and more we hope our techniques will be of independent research interest outside of the dynamic setting
Incremental -approximate dynamic matching in update time
In the dynamic approximate maximum bipartite matching problem we are given
bipartite graph undergoing updates and our goal is to maintain a matching
of which is large compared the maximum matching size . We define a
dynamic matching algorithm to be (respectively )-approximate if it maintains matching such that at all times (respectively ).
We present the first deterministic -approximate dynamic
matching algorithm with amortized update time for
graphs undergoing edge insertions. Previous solutions either required
super-constant [Gupta FSTTCS'14, Bhattacharya-Kiss-Saranurak SODA'23] or
exponential in
[Grandoni-Leonardi-Sankowski-Schwiegelshohn-Solomon SODA'19] update time. Our
implementation is arguably simpler than the mentioned algorithms and its
description is self contained. Moreover, we show that if we allow for additive
-approximation our algorithm seamlessly extends to also
handle vertex deletions, on top of edge insertions. This makes our algorithm
one of the few small update time algorithms for -approximate
dynamic matching allowing for updates both increasing and decreasing the
maximum matching size of in a fully dynamic manner
Approximate Counting of k-Paths: Deterministic and in Polynomial Space
A few years ago, Alon et al. [ISMB 2008] gave a simple randomized O((2e)^km epsilon^{-2})-time exponential-space algorithm to approximately compute the number of paths on k vertices in a graph G up to a multiplicative error of 1 +/- epsilon. Shortly afterwards, Alon and Gutner [IWPEC 2009, TALG 2010] gave a deterministic exponential-space algorithm with running time (2e)^{k+O(log^3k)}m log n whenever epsilon^{-1}=k^{O(1)}. Recently, Brand et al. [STOC 2018] provided a speed-up at the cost of reintroducing randomization. Specifically, they gave a randomized O(4^km epsilon^{-2})-time exponential-space algorithm. In this article, we revisit the algorithm by Alon and Gutner. We modify the foundation of their work, and with a novel twist, obtain the following results.
- We present a deterministic 4^{k+O(sqrt{k}(log^2k+log^2 epsilon^{-1}))}m log n-time polynomial-space algorithm. This matches the running time of the best known deterministic polynomial-space algorithm for deciding whether a given graph G has a path on k vertices.
- Additionally, we present a randomized 4^{k+O(log k(log k + log epsilon^{-1}))}m log n-time polynomial-space algorithm. While Brand et al. make non-trivial use of exterior algebra, our algorithm is very simple; we only make elementary use of the probabilistic method.
Thus, the algorithm by Brand et al. runs in time 4^{k+o(k)}m whenever epsilon^{-1}=2^{o(k)}, while our deterministic and randomized algorithms run in time 4^{k+o(k)}m log n whenever epsilon^{-1}=2^{o(k^{1/4})} and epsilon^{-1}=2^{o(k/(log k))}, respectively. Prior to our work, no 2^{O(k)}n^{O(1)}-time polynomial-space algorithm was known. Additionally, our approach is embeddable in the classic framework of divide-and-color, hence it immediately extends to approximate counting of graphs of bounded treewidth; in comparison, Brand et al. note that their approach is limited to graphs of bounded pathwidth
Constant-Factor FPT Approximation for Capacitated k-Median
Capacitated k-median is one of the few outstanding optimization problems for which the existence of a polynomial time constant factor approximation algorithm remains an open problem. In a series of recent papers algorithms producing solutions violating either the number of facilities or the capacity by a multiplicative factor were obtained. However, to produce solutions without violations appears to be hard and potentially requires different algorithmic techniques. Notably, if parameterized by the number of facilities k, the problem is also W[2] hard, making the existence of an exact FPT algorithm unlikely. In this work we provide an FPT-time constant factor approximation algorithm preserving both cardinality and capacity of the facilities. The algorithm runs in time 2^O(k log k) n^O(1) and achieves an approximation ratio of 7+epsilon
Simple and Faster Algorithms for Knapsack
In this paper, we obtain a number of new simple pseudo-polynomial time
algorithms on the well-known knapsack problem, focusing on the running time
dependency on the number of items , the maximum item weight
, and the maximum item profit . Our results
include:
- An -time
randomized algorithm for 0-1 knapsack, improving the previous
[Bringmann and Cassis, ESA'23] for the small case.
- An -time
randomized algorithm for bounded knapsack, improving the previous
[Polak, Rohwedder and
Wegrzyck, ICALP'21]
Two new results about quantum exact learning
We present two new results about exact learning by quantum computers. First,
we show how to exactly learn a -Fourier-sparse -bit Boolean function from
uniform quantum examples for that function. This
improves over the bound of uniformly random classical
examples (Haviv and Regev, CCC'15). Our main tool is an improvement of Chang's
lemma for the special case of sparse functions. Second, we show that if a
concept class can be exactly learned using quantum membership
queries, then it can also be learned using classical membership queries. This improves the
previous-best simulation result (Servedio and Gortler, SICOMP'04) by a -factor.Comment: v3: 21 pages. Small corrections and clarification
On the central levels problem
The \emph{central levels problem} asserts that the subgraph of the -dimensional hypercube induced by all bitstrings with at least many 1s and at most many 1s, i.e., the vertices in the middle levels, has a Hamilton cycle for any and .
This problem was raised independently by Buck and Wiedemann, Savage, Gregor and {\v{S}}krekovski, and by Shen and Williams, and it is a common generalization of the well-known \emph{middle levels problem}, namely the case , and classical binary Gray codes, namely the case .
In this paper we present a general constructive solution of the central levels problem.
Our results also imply the existence of optimal cycles through any sequence of consecutive levels in the -dimensional hypercube for any and .
Moreover, extending an earlier construction by Streib and Trotter, we construct a Hamilton cycle through the -dimensional hypercube, , that contains the symmetric chain decomposition constructed by Greene and Kleitman in the 1970s, and we provide a loopless algorithm for computing the corresponding Gray code
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