2,145 research outputs found
Finding Even Cycles Faster via Capped k-Walks
In this paper, we consider the problem of finding a cycle of length (a
) in an undirected graph with nodes and edges for constant
. A classic result by Bondy and Simonovits [J.Comb.Th.'74] implies that
if , then contains a , further implying that
one needs to consider only graphs with .
Previously the best known algorithms were an algorithm due to Yuster
and Zwick [J.Disc.Math'97] as well as a algorithm by Alon et al. [Algorithmica'97].
We present an algorithm that uses time and finds a
if one exists. This bound is exactly when . For
-cycles our new bound coincides with Alon et al., while for every our
bound yields a polynomial improvement in .
Yuster and Zwick noted that it is "plausible to conjecture that is
the best possible bound in terms of ". We show "conditional optimality": if
this hypothesis holds then our algorithm is tight as well.
Furthermore, a folklore reduction implies that no combinatorial algorithm can
determine if a graph contains a -cycle in time for any
under the widely believed combinatorial BMM conjecture. Coupled
with our main result, this gives tight bounds for finding -cycles
combinatorially and also separates the complexity of finding - and
-cycles giving evidence that the exponent of in the running time should
indeed increase with .
The key ingredient in our algorithm is a new notion of capped -walks,
which are walks of length that visit only nodes according to a fixed
ordering. Our main technical contribution is an involved analysis proving
several properties of such walks which may be of independent interest.Comment: To appear at STOC'1
On the existence and strength of stable membrane protrusions
We present a mathematical model for the protrusion of lamellipodia in motile cells. The model lamellipodium consists of a viscoelastic actin gel in the bulk and a dynamic boundary layer of newly polymerized filaments at the leading edge called the semiflexible region (SR). The density of filaments in the SR can increase due to nucleation of new filaments and decrease due to capping and severing of existing filaments. Following on from previous publications, we present important approximations that make the model feasible and accessible to fast computational analysis. It reveals that there are three qualitatively different parameter regimes: a stable, stationarily protruding lamellipodium; a stable lamellipodium showing oscillatory motion of the leading edge; and zero filament density and no stable lamellipodium. Hence, the model defines criteria for the existence of lamellipodia and the ability of cells to move effectively, and we discuss which parameter changes can induce transitions between the different states. Furthermore, stable lamellipodia have to be able to exert and withstand substantial forces. We can fit the experimentally measured dynamic force–velocity relation that describes how cells can adapt to increasing external forces when encountering an obstacle in their environment during motion. Moreover, we predict a different stationary force–velocity relation that should apply if cells experience a constant force, e.g. exerted by the surrounding tissue
Blazing a Trail via Matrix Multiplications: A Faster Algorithm for Non-Shortest Induced Paths
For vertices and of an -vertex graph , a -trail of is
an induced -path of that is not a shortest -path of . Berger,
Seymour, and Spirkl [Discrete Mathematics 2021] gave the previously only known
polynomial-time algorithm, running in time, to either output a
-trail of or ensure that admits no -trail. We reduce the
complexity to the time required to perform a poly-logarithmic number of
multiplications of Boolean matrices, leading to a largely
improved -time algorithm.Comment: 18 pages, 6 figures, a preliminary version appeared in STACS 202
The Strongish Planted Clique Hypothesis and Its Consequences
We formulate a new hardness assumption, the Strongish Planted Clique Hypothesis (SPCH), which postulates that any algorithm for planted clique must run in time n^?(log n) (so that the state-of-the-art running time of n^O(log n) is optimal up to a constant in the exponent).
We provide two sets of applications of the new hypothesis. First, we show that SPCH implies (nearly) tight inapproximability results for the following well-studied problems in terms of the parameter k: Densest k-Subgraph, Smallest k-Edge Subgraph, Densest k-Subhypergraph, Steiner k-Forest, and Directed Steiner Network with k terminal pairs. For example, we show, under SPCH, that no polynomial time algorithm achieves o(k)-approximation for Densest k-Subgraph. This inapproximability ratio improves upon the previous best k^o(1) factor from (Chalermsook et al., FOCS 2017). Furthermore, our lower bounds hold even against fixed-parameter tractable algorithms with parameter k.
Our second application focuses on the complexity of graph pattern detection. For both induced and non-induced graph pattern detection, we prove hardness results under SPCH, improving the running time lower bounds obtained by (Dalirrooyfard et al., STOC 2019) under the Exponential Time Hypothesis
Counting 4-Patterns in Permutations Is Equivalent to Counting 4-Cycles in Graphs
Permutation ? appears in permutation ? if there exists a subsequence of ? that is order-isomorphic to ?. The natural algorithmic question is to check if ? appears in ?, and if so count the number of occurrences. Only since very recently we know that for any fixed length k, we can check if a given pattern of length k appears in a permutation of length n in time linear in n, but being able to count all such occurrences in f(k)? n^o(k/log k) time would refute the exponential time hypothesis (ETH). Together with practical applications in statistics, this motivates a systematic study of the complexity of counting occurrences for different patterns of fixed small length k. We investigate this question for k = 4. Very recently, Even-Zohar and Leng [arXiv 2019] identified two types of 4-patterns. For the first type they designed an ??(n) time algorithm, while for the second they were able to provide an ??(n^1.5) time algorithm. This brings up the question whether the permutations of the second type are inherently harder than the first type.
We establish a connection between counting 4-patterns of the second type and counting 4-cycles (not necessarily induced) in a sparse undirected graph. By designing two-way reductions we show that the complexities of both problems are the same, up to polylogarithmic factors. This allows us to leverage the work done on the latter to provide a reasonable argument for why there is a difference in the complexities for counting 4-patterns of the first and the second type. In particular, even for the seemingly simpler problem of detecting a 4-cycle in a graph on m edges, the best known algorithm works in ?(m^{4/3}) time. Our reductions imply that an ?(n^{4/3-?}) time algorithm for counting occurrences of any 4-pattern of the second type in a permutation of length n would imply an exciting breakthrough for counting (and hence also detecting) 4-cycles. In the other direction, by plugging in the fastest known algorithm for counting 4-cycles, we obtain an algorithm for counting occurrences of any 4-pattern of the second type in ?(n^1.48) time
Improved Algorithms for Recognizing Perfect Graphs and Finding Shortest Odd and Even Holes
Various classes of induced subgraphs are involved in the deepest results of
graph theory and graph algorithms. A prominent example concerns the {\em
perfection} of that the chromatic number of each induced subgraph of
equals the clique number of . The seminal Strong Perfect Graph Theorem
confirms that the perfection of can be determined by detecting odd holes in
and its complement. Chudnovsky et al. show in 2005 an algorithm
for recognizing perfect graphs, which can be implemented to run in
time for the exponent of square-matrix
multiplication. We show the following improved algorithms.
1. The tractability of detecting odd holes was open for decades until the
major breakthrough of Chudnovsky et al. in 2020. Their algorithm is
later implemented by Lai et al. to run in time, leading to the best
formerly known algorithm for recognizing perfect graphs. Our first result is an
algorithm for detecting odd holes, implying an algorithm for
recognizing perfect graphs.
2. Chudnovsky et al. extend in 2021 the algorithms for detecting odd
holes (2020) and recognizing perfect graphs (2005) into the first polynomial
algorithm for obtaining a shortest odd hole, which runs in time. We
reduce the time for finding a shortest odd hole to .
3. Conforti et al. show in 1997 the first polynomial algorithm for detecting
even holes, running in about time. It then takes a line of
intensive efforts in the literature to bring down the complexity to
, , , and finally . On the other hand,
the tractability of finding a shortest even hole has been open for 16 years
until the very recent algorithm of Cheong and Lu in 2022. We
improve the time of finding a shortest even hole to .Comment: 29 pages, 5 figure
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