arXiv subjects

Konstantinos Panagiotou

Publications and source records attributed to Konstantinos Panagiotou.

46 records · Page 3Linked to original sources

arXiv2013

The Multiple-orientability Thresholds for Random Hypergraphs

A $k$-uniform hypergraph $H = (V, E)$ is called $\ell$-orientable, if there is an assignment of each edge $e\in E$ to one of its vertices $v\in e$ such that no vertex is assigned more than $\ell$ edges. Let $H_{n,m,k}$ be a hypergraph, drawn uniformly at random from the set of all $k$-uniform hypergraphs with $n$ vertices and $m$ edges. In this paper we establish the threshold for the $\ell$-orientability of $H_{n,m,k}$ for all $k\ge 3$ and $\ell \ge 2$, i.e., we determine a critical quantity $c_{k, \ell}^*$ such that with probability $1-o(1)$ the graph $H_{n,cn,k}$ has an $\ell$-orientation if $c < c_{k, \ell}^*$, but fails doing so if $c > c_{k, \ell}^*$. Our result has various applications including sharp load thresholds for cuckoo hashing, load balancing with guaranteed maximum load, and massive parallel access to hard disk arrays.

cs.DM↗

arXiv2012

Going after the k-SAT Threshold

Random $k$-SAT is the single most intensely studied example of a random constraint satisfaction problem. But despite substantial progress over the past decade, the threshold for the existence of satisfying assignments is not known precisely for any $k\geq3$. The best current results, based on the second moment method, yield upper and lower bounds that differ by an additive $k\cdot \frac{\ln2}2$, a term that is unbounded in $k$ (Achlioptas, Peres: STOC 2003). The basic reason for this gap is the inherent asymmetry of the Boolean value `true' and `false' in contrast to the perfect symmetry, e.g., among the various colors in a graph coloring problem. Here we develop a new asymmetric second moment method that allows us to tackle this issue head on for the first time in the theory of random CSPs. This technique enables us to compute the $k$-SAT threshold up to an additive $\ln2-\frac12+O(1/k)\approx 0.19$. Independently of the rigorous work, physicists have developed a sophisticated but non-rigorous technique called the "cavity method" for the study of random CSPs (Mézard, Parisi, Zecchina: Science 2002). Our result matches the best bound that can be obtained from the so-called "replica symmetric" version of the cavity method, and indeed our proof directly harnesses parts of the physics calculations.

math.CO↗

arXiv2012

Random Hyperbolic Graphs: Degree Sequence and Clustering

In the last decades, the study of models for large real-world networks has been a very popular and active area of research. A reasonable model should not only replicate all the structural properties that are observed in real world networks (for example, heavy tailed degree distributions, high clustering and small diameter), but it should also be amenable to mathematical analysis. There are plenty of models that succeed in the first task but are hard to analyze rigorously. On the other hand, a multitude of proposed models, like classical random graphs, can be studied mathematically, but fail in creating certain aspects that are observed in real-world networks. Recently, Papadopoulos, Krioukov, Boguna and Vahdat [INFOCOM'10] introduced a random geometric graph model that is based on hyperbolic geometry. The authors argued empirically and by some preliminary mathematical analysis that the resulting graphs have many of the desired properties. Moreover, by computing explicitly a maximum likelihood fit of the Internet graph, they demonstrated impressively that this model is adequate for reproducing the structure of real graphs with high accuracy. In this work we initiate the rigorous study of random hyperbolic graphs. We compute exact asymptotic expressions for the expected number of vertices of degree k for all k up to the maximum degree and provide small probabilities for large deviations. We also prove a constant lower bound for the clustering coefficient. In particular, our findings confirm rigorously that the degree sequence follows a power-law distribution with controllable exponent and that the clustering is nonvanishing.

math.CO↗

arXiv2011

Catching the k-NAESAT Threshold

The best current estimates of the thresholds for the existence of solutions in random constraint satisfaction problems ('CSPs') mostly derive from the first and the second moment method. Yet apart from a very few exceptional cases these methods do not quite yield matching upper and lower bounds. According to deep but non-rigorous arguments from statistical mechanics, this discrepancy is due to a change in the geometry of the set of solutions called condensation that occurs shortly before the actual threshold for the existence of solutions (Krzakala, Montanari, Ricci-Tersenghi, Semerjian, Zdeborova: PNAS 2007). To cope with condensation, physicists have developed a sophisticated but non-rigorous formalism called Survey Propagation (Mezard, Parisi, Zecchina: Science 2002). This formalism yields precise conjectures on the threshold values of many random CSPs. Here we develop a new Survey Propagation inspired second moment method for the random k-NAESAT problem, which is one of the standard benchmark problems in the theory of random CSPs. This new technique allows us to overcome the barrier posed by condensation rigorously. We prove that the threshold for the existence of solutions in random $k$-NAESAT is $2^{k-1}\ln2-(\frac{\ln2}2+\frac14)+\eps_k$, where $|\eps_k| \le 2^{-(1-o_k(1))k}$, thereby verifying the statistical mechanics conjecture for this problem.

cs.DM↗

arXiv2011

Explosive Percolation in Erdös-Rényi-Like Random Graph Processes

The evolution of the largest component has been studied intensely in a variety of random graph processes, starting in 1960 with the Erdös-Rényi process. It is well known that this process undergoes a phase transition at n/2 edges when, asymptotically almost surely, a linear-sized component appears. Moreover, this phase transition is continuous, i.e., in the limit the function f(c) denoting the fraction of vertices in the largest component in the process after cn edge insertions is continuous. A variation of the Erdös-Rényi process are the so-called Achlioptas processes in which in every step a random pair of edges is drawn, and a fixed edge-selection rule selects one of them to be included in the graph while the other is put back. Recently, Achlioptas, D'Souza and Spencer (2009) gave strong numerical evidence that a variety of edge-selection rules exhibit a discontinuous phase transition. However, Riordan and Warnke (2011) very recently showed that all Achlioptas processes have a continuous phase transition. In this work we prove discontinuous phase transitions for a class of Erdös-Rényi-like processes in which in every step we connect two vertices, one chosen randomly from all vertices, and one chosen randomly from a restricted set of vertices.

cs.DM↗

arXiv2010

Rumor Spreading on Random Regular Graphs and Expanders

Broadcasting algorithms are important building blocks of distributed systems. In this work we investigate the typical performance of the classical and well-studied push model. Assume that initially one node in a given network holds some piece of information. In each round, every one of the informed nodes chooses independently a neighbor uniformly at random and transmits the message to it. In this paper we consider random networks where each vertex has degree d, which is at least 3, i.e., the underlying graph is drawn uniformly at random from the set of all d-regular graphs with n vertices. We show that with probability 1 - o(1) the push model broadcasts the message to all nodes within (1 + o(1))C_d ln n rounds, where C_d = 1/ ln(2(1-1/d)) - 1/(d ln(1 - 1/d)). In particular, we can characterize precisely the effect of the node degree to the typical broadcast time of the push model. Moreover, we consider pseudo-random regular networks, where we assume that the degree of each node is very large. There we show that the broadcast time is (1+o(1))C ln n with probability 1 - o(1), where C= 1/ ln 2 + 1, is the limit of C_d as d grows.

math.CO↗

arXiv2009

Sharp Load Thresholds for Cuckoo Hashing

The paradigm of many choices has influenced significantly the design of efficient data structures and, most notably, hash tables. Cuckoo hashing is a technique that extends this concept. There,we are given a table with $n$ locations, and we assume that each location can hold one item. Each item to be inserted chooses randomly k>1 locations and has to be placed in any one of them. How much load can cuckoo hashing handle before collisions prevent the successful assignment of the available items to the chosen locations? Practical evaluations of this method have shown that one can allocate a number of elements that is a large proportion of the size of the table, being very close to 1 even for small values of k such as 4 or 5. In this paper we show that there is a critical value for this proportion: with high probability, when the amount of available items is below this value, then these can be allocated successfully, but when it exceeds this value, the allocation becomes impossible. We give explicitly for each k>1 this critical value. This answers an open question posed by Mitzenmacher (ESA '09) and underpins theoretically the experimental results. Our proofs are based on the translation of the question into a hypergraph setting, and the study of the related typical properties of random k-uniform hypergraphs.

cs.DS↗

arXiv2009

Extremal Subgraphs of Random Graphs: an Extended Version

We prove that there is a constant $c >0$, such that whenever $p \ge n^{-c}$, with probability tending to 1 when $n$ goes to infinity, every maximum triangle-free subgraph of the random graph $G_{n,p}$ is bipartite. This answers a question of Babai, Simonovits and Spencer (Journal of Graph Theory, 1990). The proof is based on a tool of independent interest: we show, for instance, that the maximum cut of almost all graphs with $M$ edges, where $M >> n$, is ``nearly unique''. More precisely, given a maximum cut $C$ of $G_{n,M}$, we can obtain all maximum cuts by moving at most $O(\sqrt{n^3/M})$ vertices between the parts of $C$.

math.PR↗

arXiv2009

3-Connected Cores In Random Planar Graphs

The study of the structural properties of large random planar graphs has become in recent years a field of intense research in computer science and discrete mathematics. Nowadays, a random planar graph is an important and challenging model for evaluating methods that are developed to study properties of random graphs from classes with structural side constraints. In this paper we focus on the structure of random biconnected planar graphs regarding the sizes of their 3-connected building blocks, which we call cores. In fact, we prove a general theorem regarding random biconnected graphs. If B_n is a graph drawn uniformly at random from a class B of labeled biconnected graphs, then we show that with probability 1-o(1) B_n belongs to exactly one of the following categories: (i) Either there is a unique giant core in B_n, that is, there is a 0 < c < 1 such that the largest core contains ~ cn vertices, and every other core contains at most n^a vertices, where 0 < a < 1; (ii) or all cores of B_n contain O(log n) vertices. Moreover, we find the critical condition that determines the category to which B_n belongs, and also provide sharp concentration results for the counts of cores of all sizes between 1 and n. As a corollary, we obtain that a random biconnected planar graph belongs to category (i), where in particular c = 0.765... and a = 2/3.

math.CO↗

arXiv2009

The Speed of Broadcasting in Random Networks: Density Does Not Matter

Broadcasting algorithms are of fundamental importance for distributed systems engineering. In this paper we revisit the classical and well-studied push protocol for message broadcasting. Assuming that initially only one node has some piece of information, at each stage every one of the informed nodes chooses randomly and independently one of its neighbors and passes the message to it. The performance of the push protocol on a fully connected network, where each node is joined by a link to every other node, is very well understood. In particular, Frieze and Grimmett proved that with probability 1-o(1) the push protocol completes the broadcasting of the message within (1 +/- ε) (log_2 n + ln n) stages, where n is the number of nodes of the network. However, there are no tight bounds for the broadcast time on networks that are significantly sparser than the complete graph. In this work we consider random networks on n nodes, where every edge is present with probability p, independently of every other edge. We show that if p > f(n)ln n/ n, where f(n) is any function that tends to infinity as n grows, then the push protocol broadcasts the message within (1 +/- ε) (log_2 n + ln n) stages with probability 1-o(1). In other words, in almost every network of density d such that d > f(n)ln n, the push protocol broadcasts a message as fast as in a fully connected network. This is quite surprising in the sense that the time needed remains essentially unaffected by the fact that most of the links are missing.

math.CO↗