arXiv subjects

Tony Huynh

Publications and source records attributed to Tony Huynh.

61 records · Page 4Linked to original sources

arXiv2015

Transfinite Ford-Fulkerson on a Finite Network

It is well-known that the Ford-Fulkerson algorithm for finding a maximum flow in a network need not terminate if we allow the arc capacities to take irrational values. Every non-terminating example converges to a limit flow, but this limit flow need not be a maximum flow. Hence, one may pass to the limit and begin the algorithm again. In this way, we may view the Ford-Fulkerson algorithm as a transfinite algorithm. We analyze the transfinite running-time of the Ford-Fulkerson algorithm using ordinal numbers, and prove that the worst case running-time is $ω^{Θ(|E|)}$. For the lower bound, we show that we can model the Euclidean algorithm via Ford-Fulkerson on an auxiliary network. By running this example on a pair of incommensurable numbers, we obtain a new robust non-terminating example. We then describe how to glue $k$ copies of our Euclidean example in parallel to obtain running-time $ω^k$. An upper bound of $ω^{|E|}$ is established via induction on $|E|$. We conclude by illustrating a close connection to transfinite chip-firing as previously investigated by the first author.

math.CO↗

arXiv2015

Space proof complexity for random 3-CNFs

We investigate the space complexity of refuting $3$-CNFs in Resolution and algebraic systems. We prove that every Polynomial Calculus with Resolution refutation of a random $3$-CNF $ϕ$ in $n$ variables requires, with high probability, $Ω(n)$ distinct monomials to be kept simultaneously in memory. The same construction also proves that every Resolution refutation $ϕ$ requires, with high probability, $Ω(n)$ clauses each of width $Ω(n)$ to be kept at the same time in memory. This gives a $Ω(n^2)$ lower bound for the total space needed in Resolution to refute $ϕ$. These results are best possible (up to a constant factor). The main technical innovation is a variant of Hall's Lemma. We show that in bipartite graphs $G$ with bipartition $(L,R)$ and left-degree at most 3, $L$ can be covered by certain families of disjoint paths, called VW-matchings, provided that $L$ expands in $R$ by a factor of $(2-ε)$, for $ε< 1/23$.

cs.CC↗

arXiv2014

Space proof complexity for random $3$-CNFs via a $(2-ε)$-Hall's Theorem

We investigate the space complexity of refuting $3$-CNFs in Resolution and algebraic systems. No lower bound for refuting any family of $3$-CNFs was previously known for the total space in resolution or for the monomial space in algebraic systems. We prove that every Polynomial Calculus with Resolution refutation of a random $3$-CNF $ϕ$ in $n$ variables requires, with high probability, $Ω(n/\log n)$ distinct monomials to be kept simultaneously in memory. The same construction also proves that every Resolution refutation $ϕ$ requires, with high probability, $Ω(n/\log n)$ clauses each of width $Ω(n/\log n)$ to be kept at the same time in memory. This gives a $Ω(n^2/\log^2 n)$ lower bound for the total space needed in Resolution to refute $ϕ$. The main technical innovation is a variant of Hall's theorem. We show that in bipartite graphs $G$ with bipartition $(L,R)$ and left-degree at most 3, $L$ can be covered by certain families of disjoint paths, called $(2,4)$-matchings, provided that $L$ expands in $R$ by a factor of $(2-ε)$, for $ε< \frac{1}{23}$.

cs.CC↗

arXiv2014

Extremal Problems for Subset Divisors

Let $A$ be a set of $n$ positive integers. We say that a subset $B$ of $A$ is a divisor of $A$, if the sum of the elements in $B$ divides the sum of the elements in $A$. We are interested in the following extremal problem. For each $n$, what is the maximum number of divisors a set of $n$ positive integers can have? We determine this function exactly for all values of $n$. Moreover, for each $n$ we characterize all sets that achieve the maximum. We also prove results for the $k$-subset analogue of our problem. For this variant, we determine the function exactly in the special case that $n=2k$. We also characterize all sets that achieve this bound when $n=2k$.

math.CO↗

arXiv2014

On Hilbert bases of cuts

A Hilbert basis is a set of vectors X such that the integer cone (semigroup) generated by X is the intersection of the lattice generated by X with the cone generated by X. Define a graph to be (cut) Hilbert if its set of cuts forms a Hilbert basis. We show that the Hilbert property is not closed under edge deletions, subdivisions, nor 2-sums. Furthermore, no graph having K_6-e as a minor is Hilbert. This corrects an error in [M. Laurent. Hilbert bases of cuts. Discrete Math., 150(1-3):257-279 (1996)]. For positive results, we give conditions under which the 2-sum of two graphs produces a Hilbert graph. Using these conditions we show that all H-minor-free graphs are Hilbert , where H is the unique 3-connected graph obtained by uncontracting an edge of K_5. We also establish a relationship between edge deletion and subdivision. Namely, if G' is obtained from a Hilbert graph G by subdividing an edge e two or more times, then G-e is Hilbert if and only if G' is Hilbert.

math.CO↗

arXiv2014

Ontogeny of aerial righting and wing flapping in juvenile birds

Mechanisms of aerial righting in juvenile Chukar Partridge (Alectoris chukar) were studied from hatching through 14 days post hatching (dph). Asymmetric movements of the wings were used from 1 to 8 dph to effect progressively more successful righting behaviour via body roll. Following 8 dph, wing motions transitioned to bilaterally symmetric flapping that yielded aerial righting via nose down pitch, along with substantial increases in vertical force production during descent. Ontogenetically, the use of such wing motions to effect aerial righting precedes both symmetric flapping and a previously documented behaviour in chukar (i.e., wing assisted incline running) hypothesized to be relevant to incipient flight evolution in birds. These findings highlight the importance of asymmetric wing activation and controlled aerial manoeuvres during bird development, and are potentially relevant to understanding the origins of avian flight.

physics.bio-ph↗

arXiv2014

Shifts in stability and control effectiveness during evolution of Paraves support aerial maneuvering hypotheses for flight origins

The capacity for aerial maneuvering shaped the evolution of flying animals. Here we evaluate consequences of aviaian morphology for aerial performance (1,2) by quantifying static stability and control effectiveness of physical models (3) for numerous taxa sampled from within the lineage leading to birds (Paraves, 4). Results of aerodynamic testing are mapped phylogenetically (5-9) to examine how maneuvering characteristics correlate with tail shortening, fore- and hindwing elaboration, and other morphological features (10). In the evolution of the Avialae we observe shifts from static stability to inherently unstable aerial planforms; control effectiveness also migrated from tails to the forewings. These shifts suggest that some degree of aerodynamic control and and capacity for maneuvering preceded the evolution of strong power stroke. The timing of shifts also suggests some features normally considered in light of development of a power stroke may play important roles in control.

q-bio.PE↗