On the Complexity of Local Distributed Graph Problems

This paper is centered on the complexity of graph problems in the well-studied LOCAL model of distributed computing, introduced by Linial [FOCS '87]. It is widely known that for many of the classic distributed graph problems (including maximal independent set (MIS) and $(\Delta+1)$-vertex coloring), the randomized complexity is at most polylogarithmic in the size $n$ of the network, while the best deterministic complexity is typically $2^{O(\sqrt{\log n})}$. Understanding and narrowing down this exponential gap is considered to be one of the central long-standing open questions in the area of distributed graph algorithms. We investigate the problem by introducing a complexity-theoretic framework that allows us to shed some light on the role of randomness in the LOCAL model. We define the SLOCAL model as a sequential version of the LOCAL model. Our framework allows us to prove completeness results with respect to the class of problems which can be solved efficiently in the SLOCAL model, implying that if any of the complete problems can be solved deterministically in $\log^{O(1)} n$ rounds in the LOCAL model, we can deterministically solve all efficient SLOCAL-problems (including MIS and $(\Delta+1)$-coloring) in $\log^{O(1)} n$ rounds in the LOCAL model. We show that a rather rudimentary looking graph coloring problem is complete in the above sense: Color the nodes of a graph with colors red and blue such that each node of sufficiently large polylogarithmic degree has at least one neighbor of each color. The problem admits a trivial zero-round randomized solution. The result can be viewed as showing that the only obstacle to getting efficient determinstic algorithms in the LOCAL model is an efficient algorithm to approximately round fractional values into integer values

Small rainbow cliques in randomly perturbed dense graphs

For two graphs G and H, write G rbw −→ H if G has the property that every proper colouring of its edges yields a rainbow copy of H. We study the thresholds for such so-called anti-Ramsey properties in randomly perturbed dense graphs, which are unions of the form G ∪ G(n, p), where G is an n-vertex graph with edge-density at least d > 0, and d is independent of n. In a companion paper, we proved that the threshold for the property G ∪ G(n, p) rbw −→ K` is n −1/m2(Kd`/2e) , whenever ` ≥ 9. For smaller `, the thresholds behave more erratically, and for 4 ≤ ` ≤ 7 they deviate downwards significantly from the aforementioned aesthetic form capturing the thresholds for large cliques. In particular, we show that the thresholds for ` ∈ {4, 5, 7} are n −5/4 , n −1 , and n −7/15, respectively. For ` ∈ {6, 8} we determine the threshold up to a (1 + o(1))-factor in the exponent: they are n −(2/3+o(1)) and n −(2/5+o(1)), respectively. For ` = 3, the threshold is n −2 ; this follows from a more general result about odd cycles in our companion paper

Strong Ramsey games: Drawing on an infinite board

Consider the following strong Ramsey game. Two players take turns in claiming a previously unclaimed edge of the complete graph on n vertices until all edges have been claimed. The first player to build a copy of K5 is declared the winner of the game. If none of the players win, then the game ends in a draw. A simple strategy stealing argument shows that the second player cannot expect to ever win this game. Moreover, for sufficiently large n, it follows from Ramsey’s Theorem that the game cannot end in a draw and is thus a first player win. A famous question of Beck asks whether the minimum number of moves needed for the first player to win this game on Kn grows with n. This seems unlikely but is still wide open. A striking equivalent formulation of this question is whether the same game played on the infinite complete graph is a first player win or a draw. The target graph of the strong Ramsey game does not have to be K5, it can be any predetermined fixed graph. In fact, it can even be a k-uniform hypergraph (and then the game is played on the infinite k-uniform hypergraph). Since strategy stealing and Ramsey’s Theorem still apply, one can ask the same question: is this game a first player win or a draw? The same intuition which lead most people (including the authors) to believe that the K5 strong Ramsey game on the infinite board is a first player win, would also lead one to believe that the H strong Ramsey game on the infinite board is a first player win for any uniform hypergraph H. However, in this paper we construct a 5-uniform hypergraph for which the corresponding game is a draw

Game saturation of intersecting families

We consider the following combinatorial game: two players, Fast and Slow, claim $k$-element subsets of $[n]=\{1,2,...,n\}$ alternately, one at each turn, such that both players are allowed to pick sets that intersect all previously claimed subsets. The game ends when there does not exist any unclaimed $k$-subset that meets all already claimed sets. The score of the game is the number of sets claimed by the two players, the aim of Fast is to keep the score as low as possible, while the aim of Slow is to postpone the game's end as long as possible. The game saturation number is the score of the game when both players play according to an optimal strategy. To be precise we have to distinguish two cases depending on which player takes the first move. Let $gsat_F(\mathbb{I}_{n,k})$ and $gsat_S(\mathbb{I}_{n,k})$ denote the score of the saturation game when both players play according to an optimal strategy and the game starts with Fast's or Slow's move, respectively. We prove that $\Omega_k(n^{k/3-5}) \le gsat_F(\mathbb{I}_{n,k}),gsat_S(\mathbb{I}_{n,k}) \le O_k(n^{k-\sqrt{k}/2})$ holds

Strong Ramsey games: drawing on an infinite board

We consider the strong Ramsey-type game R(k)(H,ℵ0), played on the edge set of the infinite complete k-uniform hypergraph KkN. Two players, called FP (the first player) and SP (the second player), take turns claiming edges of K^k_N with the goal of building a copy of some finite predetermined k-uniform hypergraph H. The first player to build a copy of H wins. If no player has a strategy to ensure his win in finitely many moves, then the game is declared a draw. In this paper, we construct a 5-uniform hypergraph H such that R(5)(H,ℵ0) is a draw. This is in stark contrast to the corresponding finite game R(5)(H,n), played on the edge set of K5n. Indeed, using a classical game-theoretic argument known as \emph{strategy stealing} and a Ramsey-type argument, one can show that for every k-uniform hypergraph G, there exists an integer n0 such that FP has a winning strategy for R(k)(G,n) for every n≥n0

Is the postpharyngeal gland of a solitary digger wasp homologous to ants? Evidence from chemistry and physiology

The postpharyngeal gland (PPG) was thought to be restricted to ants where it serves a crucial function in the generation of the colony odour. Recently, head glands that closely resemble the PPG of ants were discovered in females of a solitary digger wasp, the European beewolf. The function of this gland necessarily differs from ants: beewolf females apply the secretion of their PPG onto the bodies of paralysed honeybees that serve as larval provisions in order to delay fungus growth. Since ants and digger wasps are not closely related, the occurrence of this gland in these two taxa might either be due to convergent evolution or it is a homologous organ inherited from a common ancestor. Here we test the hypothesis that the PPGs of both taxa are homologous by comparing characteristics of chemical composition and physiology of the PPG of beewolves and ants. Based on reported characteristics of the PPG content of ants, we tested three predictions that were all met. First, the PPG of beewolves contained mainly long-chain hydrocarbons and very few compounds with functional groups. Second, the composition of hydrocarbons in the beewolf PPG was similar to that of the hemolymph. Taking the structure of the gland epithelium and the huge requirements of beewolf females for gland secretion into account this result suggests that the content of the PPG is also sequestered from the hemolymph in beewolves. Third, the chemical composition of the PPG and the cuticle was similar in beewolves since cuticular hydrocarbons derive either from the hemolymph or the PPG. Taking the considerable morphological similarities into account, our results support the hypothesis of a homologous origin of the PPG in beewolves and ants

Detector Description and Performance for the First Coincidence Observations between LIGO and GEO

For 17 days in August and September 2002, the LIGO and GEO interferometer gravitational wave detectors were operated in coincidence to produce their first data for scientific analysis. Although the detectors were still far from their design sensitivity levels, the data can be used to place better upper limits on the flux of gravitational waves incident on the earth than previous direct measurements. This paper describes the instruments and the data in some detail, as a companion to analysis papers based on the first data.Comment: 41 pages, 9 figures 17 Sept 03: author list amended, minor editorial change

Upper limits on the strength of periodic gravitational waves from PSR J1939+2134

The first science run of the LIGO and GEO gravitational wave detectors presented the opportunity to test methods of searching for gravitational waves from known pulsars. Here we present new direct upper limits on the strength of waves from the pulsar PSR J1939+2134 using two independent analysis methods, one in the frequency domain using frequentist statistics and one in the time domain using Bayesian inference. Both methods show that the strain amplitude at Earth from this pulsar is less than a few times $10^{-22}$.Comment: 7 pages, 1 figure, to appear in the Proceedings of the 5th Edoardo Amaldi Conference on Gravitational Waves, Tirrenia, Pisa, Italy, 6-11 July 200

Searching for gravitational waves from known pulsars

We present upper limits on the amplitude of gravitational waves from 28 isolated pulsars using data from the second science run of LIGO. The results are also expressed as a constraint on the pulsars' equatorial ellipticities. We discuss a new way of presenting such ellipticity upper limits that takes account of the uncertainties of the pulsar moment of inertia. We also extend our previous method to search for known pulsars in binary systems, of which there are about 80 in the sensitive frequency range of LIGO and GEO 600.Comment: Accepted by CQG for the proceeding of GWDAW9, 7 pages, 2 figure

Improving the sensitivity to gravitational-wave sources by modifying the input-output optics of advanced interferometers

We study frequency dependent (FD) input-output schemes for signal-recycling interferometers, the baseline design of Advanced LIGO and the current configuration of GEO 600. Complementary to a recent proposal by Harms et al. to use FD input squeezing and ordinary homodyne detection, we explore a scheme which uses ordinary squeezed vacuum, but FD readout. Both schemes, which are sub-optimal among all possible input-output schemes, provide a global noise suppression by the power squeeze factor, while being realizable by using detuned Fabry-Perot cavities as input/output filters. At high frequencies, the two schemes are shown to be equivalent, while at low frequencies our scheme gives better performance than that of Harms et al., and is nearly fully optimal. We then study the sensitivity improvement achievable by these schemes in Advanced LIGO era (with 30-m filter cavities and current estimates of filter-mirror losses and thermal noise), for neutron star binary inspirals, and for narrowband GW sources such as low-mass X-ray binaries and known radio pulsars. Optical losses are shown to be a major obstacle for the actual implementation of these techniques in Advanced LIGO. On time scales of third-generation interferometers, like EURO/LIGO-III (~2012), with kilometer-scale filter cavities, a signal-recycling interferometer with the FD readout scheme explored in this paper can have performances comparable to existing proposals. [abridged]Comment: Figs. 9 and 12 corrected; Appendix added for narrowband data analysi