For graphs F and G an F -matching in G is a subgraph of G consisting of pairwise vertex disjoint ... more For graphs F and G an F -matching in G is a subgraph of G consisting of pairwise vertex disjoint copies of F . The number of F -matchings in G is denoted by s(F, G). We show that for every fixed positive integer m and every fixed tree F , the probability that s(F, Tn) ≡ 0 (mod m), where Tn is a random labeled tree with n vertices, tends to one exponentially fast as n grows to infinity. A similar result is proven for induced F -matchings. This generalizes a recent result of Wagner who showed that the number of independent sets in a random labeled tree is almost surely a zero residue.
We adopt a utilitarian perspective on social choice, assuming that agents have (possibly latent) ... more We adopt a utilitarian perspective on social choice, assuming that agents have (possibly latent) utility functions over some space of alternatives. For many reasons one might consider mechanisms, or social choice functions, that only have access to the ordinal rankings of alternatives by the individual agents rather than their utility functions. In this context, one possible objective for a social choice function is the maximization of (expected) social welfare relative to the information contained in these rankings. We study such optimal social choice functions under three different models, and underscore the important role played by scoring functions. In our worst-case model, no assumptions are made about the underlying distribution and we analyze the worst-case distortion-or degree to which the selected alternative does not maximize social welfare-of optimal social choice functions. In our average-case model, we derive optimal functions under neutral (or impartial culture) distributional models. Finally, a very general learning-theoretic model allows for the computation of optimal social choice functions (i.e., that maximize expected social welfare) under arbitrary, sampleable distributions. In the latter case, we provide both algorithms and sample complexity results for the class of scoring functions, and further validate the approach empirically.
Given a graph G and an integer k, two players take turns coloring the vertices of G one by one us... more Given a graph G and an integer k, two players take turns coloring the vertices of G one by one using k colors so that neighboring vertices get different colors. The first player wins iff at the end of the game all the vertices of G are colored. The game chromatic number χ g (G) is the minimum k for which the first player has a winning strategy. The paper [6] began the analysis of the asymptotic behavior of this parameter for a random graph G n,p . This paper provides some further analysis for graphs with constant average degree i.e. np = O(1) and for random regular graphs.
We describe an algorithm for finding Hamilton cycles in random graphs. Our model is the random gr... more We describe an algorithm for finding Hamilton cycles in random graphs. Our model is the random graph G = G δ≥3 n,m . In this model G is drawn uniformly from graphs with vertex set [n], m edges and minimum degree at least three. We focus on the case where m = cn for constant c. If c is sufficiently large then our algorithm runs in O(n 1+o(1) ) time and succeeds w.h.p. * Research supported in part by NSF Grant CCF2013110 algorithm that reduces this to n 1.5+o(1) for sufficiently large c. The main aim of this paper is to construct an almost linear time algorithm for this model. Theorem 1.1. If c is sufficiently large then our algorithm finds a Hamilton cycle in G δ≥3 n,m , m = cn, and runs in O(n 1+o(1) ) time and succeeds w.h.p.
We regret that the article "On Fractional K-Factors of Random Graphs," which appeared in Random S... more We regret that the article "On Fractional K-Factors of Random Graphs," which appeared in Random Structures and Algorithms (Haber and Krivelevich 30 (2007), 441-463), had an inaccuracy in Lemma 6.1. This in turn led to a misstatement in the description of a family of graphs demonstrating the tightness of the main result. In this short note, we fix this inaccuracy and also take the opportunity, kindly provided by the editors, to close a gap in the proofs of two of the technical lemmas. We like to point the readers to http://www.math.tau.ac.il/ ∼ krivelev/papers.html, where a corrected version of the article appears.
The isoperimetric constant of a graph G on n vertices, i(G), is the minimum of |∂S| |S| , taken o... more The isoperimetric constant of a graph G on n vertices, i(G), is the minimum of |∂S| |S| , taken over all nonempty subsets S ⊂ V (G) of size at most n/2, where ∂S denotes the set of edges with precisely one end in S. A random graph process on n vertices, G(t), is a sequence of n 2 graphs, where G(0) is the edgeless graph on n vertices, and G(t) is the result of adding an edge to G(t − 1), uniformly distributed over all the missing edges. We show that in almost every graph process i( G(t)) equals the minimal degree of G(t) as long as the minimal degree is o(log n). Furthermore, we show that this result is essentially best possible, by demonstrating that along the period in which the minimum degree is typically Θ(log n), the ratio between the isoperimetric constant and the minimum degree falls from 1 to 1 2 , its final value.
For graphs F and G an F -matching in G is a subgraph of G consisting of pairwise vertex disjoint ... more For graphs F and G an F -matching in G is a subgraph of G consisting of pairwise vertex disjoint copies of F . The number of F -matchings in G is denoted by s(F, G). We show that for every fixed positive integer m and every fixed tree F , the probability that s(F, Tn) ≡ 0 (mod m), where Tn is a random labeled tree with n vertices, tends to one exponentially fast as n grows to infinity. A similar result is proven for induced F -matchings. This generalizes a recent result of Wagner who showed that the number of independent sets in a random labeled tree is almost surely a zero residue.
We adopt a utilitarian perspective on social choice, assuming that agents have (possibly latent) ... more We adopt a utilitarian perspective on social choice, assuming that agents have (possibly latent) utility functions over some space of alternatives. For many reasons one might consider mechanisms, or social choice functions, that only have access to the ordinal rankings of alternatives by the individual agents rather than their utility functions. In this context, one possible objective for a social choice function is the maximization of (expected) social welfare relative to the information contained in these rankings. We study such optimal social choice functions under three different models, and underscore the important role played by scoring functions. In our worst-case model, no assumptions are made about the underlying distribution and we analyze the worst-case distortion-or degree to which the selected alternative does not maximize social welfare-of optimal social choice functions. In our average-case model, we derive optimal functions under neutral (or impartial culture) distributional models. Finally, a very general learning-theoretic model allows for the computation of optimal social choice functions (i.e., that maximize expected social welfare) under arbitrary, sampleable distributions. In the latter case, we provide both algorithms and sample complexity results for the class of scoring functions, and further validate the approach empirically.
Given a graph G and an integer k, two players take turns coloring the vertices of G one by one us... more Given a graph G and an integer k, two players take turns coloring the vertices of G one by one using k colors so that neighboring vertices get different colors. The first player wins iff at the end of the game all the vertices of G are colored. The game chromatic number χ g (G) is the minimum k for which the first player has a winning strategy. The paper [6] began the analysis of the asymptotic behavior of this parameter for a random graph G n,p . This paper provides some further analysis for graphs with constant average degree i.e. np = O(1) and for random regular graphs.
We describe an algorithm for finding Hamilton cycles in random graphs. Our model is the random gr... more We describe an algorithm for finding Hamilton cycles in random graphs. Our model is the random graph G = G δ≥3 n,m . In this model G is drawn uniformly from graphs with vertex set [n], m edges and minimum degree at least three. We focus on the case where m = cn for constant c. If c is sufficiently large then our algorithm runs in O(n 1+o(1) ) time and succeeds w.h.p. * Research supported in part by NSF Grant CCF2013110 algorithm that reduces this to n 1.5+o(1) for sufficiently large c. The main aim of this paper is to construct an almost linear time algorithm for this model. Theorem 1.1. If c is sufficiently large then our algorithm finds a Hamilton cycle in G δ≥3 n,m , m = cn, and runs in O(n 1+o(1) ) time and succeeds w.h.p.
We regret that the article "On Fractional K-Factors of Random Graphs," which appeared in Random S... more We regret that the article "On Fractional K-Factors of Random Graphs," which appeared in Random Structures and Algorithms (Haber and Krivelevich 30 (2007), 441-463), had an inaccuracy in Lemma 6.1. This in turn led to a misstatement in the description of a family of graphs demonstrating the tightness of the main result. In this short note, we fix this inaccuracy and also take the opportunity, kindly provided by the editors, to close a gap in the proofs of two of the technical lemmas. We like to point the readers to http://www.math.tau.ac.il/ ∼ krivelev/papers.html, where a corrected version of the article appears.
The isoperimetric constant of a graph G on n vertices, i(G), is the minimum of |∂S| |S| , taken o... more The isoperimetric constant of a graph G on n vertices, i(G), is the minimum of |∂S| |S| , taken over all nonempty subsets S ⊂ V (G) of size at most n/2, where ∂S denotes the set of edges with precisely one end in S. A random graph process on n vertices, G(t), is a sequence of n 2 graphs, where G(0) is the edgeless graph on n vertices, and G(t) is the result of adding an edge to G(t − 1), uniformly distributed over all the missing edges. We show that in almost every graph process i( G(t)) equals the minimal degree of G(t) as long as the minimal degree is o(log n). Furthermore, we show that this result is essentially best possible, by demonstrating that along the period in which the minimum degree is typically Θ(log n), the ratio between the isoperimetric constant and the minimum degree falls from 1 to 1 2 , its final value.
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