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22
Poisson cloning model for random graphs
 International Congress of Mathematicians (ICM), 2006 (preprint in
, 2004
"... Abstract. In the random graph G(n, p) with pn bounded, the degrees of the vertices are almost i.i.d Poisson random variables with mean λ: = p(n − 1). Motivated by this fact, we introduce the Poisson cloning model GPC(n, p) for random graphs in which the degrees are i.i.d Poisson random variables wit ..."
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Abstract. In the random graph G(n, p) with pn bounded, the degrees of the vertices are almost i.i.d Poisson random variables with mean λ: = p(n − 1). Motivated by this fact, we introduce the Poisson cloning model GPC(n, p) for random graphs in which the degrees are i.i.d Poisson random variables with mean λ. Then, we first establish a theorem that shows the new model is equivalent to the classical model G(n, p) in an asymptotic sense. Next, we introduce a useful algorithm, called the cutoff line algorithm, to generate the random graph GPC(n, p). The Poisson cloning model GPC(n, p) equipped with the cutoff line algorithm enables us to very precisely analyze the sizes of the largest component and the tcore of G(n, p). This new approach to the problems yields not only elegant proofs but also improved bounds that are essentially best possible. We also consider the Poisson cloning models for random hypergraphs and random kSAT problems. Then, the tcore problem for random hypergraphs and the pure literal algorithm for random kSAT problems are analyzed. 1
Rigorous results for random (2 + p)SAT
, 2001
"... In recent years there has been significant interest in the study of random kSAT formulae. For a given set of n Boolean variables, let Bk denote the set of all possible disjunctions of k distinct, noncomplementary literals from its variables (kclauses). A random kSAT formula Fk(n; m) is formed by ..."
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Cited by 19 (4 self)
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In recent years there has been significant interest in the study of random kSAT formulae. For a given set of n Boolean variables, let Bk denote the set of all possible disjunctions of k distinct, noncomplementary literals from its variables (kclauses). A random kSAT formula Fk(n; m) is formed by selecting uniformly and independently m clauses from Bk and taking their conjunction. Motivated by insights from statistical mechanics that suggest a possible relationship between the “order” of phase transitions and computational complexity, Monasson and Zecchina (Phys. Rev. E 56(2) (1997) 1357) proposed the random (2+p)SAT model: for a given p ∈ [0; 1], a random (2 + p)SAT formula, F2+p(n; m), has m randomly chosen clauses over n variables, where pm clauses are chosen from B3 and (1 − p)m from B2. Using the heuristic “replica method” of statistical mechanics, Monasson and Zecchina gave a number of nonrigorous predictions on the behavior of random (2 + p)SAT formulae. In this paper we give the first rigorous results for random (2 + p)SAT, including the following surprising fact: for p 6 2=5, with probability 1 − o(1), a random (2 + p)SAT formula is satis able i its 2SAT subformula is satisfiable. That is, for p 6 2=5, random (2 + p)SAT behaves like random 2SAT.
A 1.875–Approximation Algorithm for the Stable Marriage Problem
, 2007
"... We consider the problem of finding a stable matching of maximum size when both ties and unacceptable partners are allowed in preference lists. This problem is known to be APXhard, and the current best known approximation algorithm achieves the approximation ratio 2 − c 1 √ N,where c is some positiv ..."
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Cited by 14 (2 self)
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We consider the problem of finding a stable matching of maximum size when both ties and unacceptable partners are allowed in preference lists. This problem is known to be APXhard, and the current best known approximation algorithm achieves the approximation ratio 2 − c 1 √ N,where c is some positive constant. In this paper, we give a 1.875– approximation algorithm, which is the first result on the approximation ratio better than two.
Greedy Algorithms for Minimisation Problems in Random Regular Graphs
, 2001
"... . In this paper we introduce a general strategy for approximating the solution to minimisation problems in random regular graphs. We describe how the approach can be applied to the minimum vertex cover (MVC), minimum independent dominating set (MIDS) and minimum edge dominating set (MEDS) proble ..."
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Cited by 13 (4 self)
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. In this paper we introduce a general strategy for approximating the solution to minimisation problems in random regular graphs. We describe how the approach can be applied to the minimum vertex cover (MVC), minimum independent dominating set (MIDS) and minimum edge dominating set (MEDS) problems. In almost all cases we are able to improve the best known results for these problems. Results for the MVC problem translate immediately to results for the maximum independent set problem. We also derive lower bounds on the size of an optimal MIDS. 1
Vertex and Edge Covers with Clustering Properties: Complexity and Algorithms
, 2006
"... We consider the concepts of a ttotal vertex cover and a ttotal edge cover (t 1), which generalize the notions of a vertex cover and an edge cover, respectively. A ttotal vertex (respectively edge) cover of a connected graph G is a vertex (edge) cover S of G such that each connected component of ..."
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Cited by 9 (2 self)
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We consider the concepts of a ttotal vertex cover and a ttotal edge cover (t 1), which generalize the notions of a vertex cover and an edge cover, respectively. A ttotal vertex (respectively edge) cover of a connected graph G is a vertex (edge) cover S of G such that each connected component of the subgraph of G induced by S has least t vertices (edges). These denitions are motivated by combining the concepts of clustering and covering in graphs. Moreover they yield a spectrum of parameters that essentially range from a vertex cover to a connected vertex cover (in the vertex case) and from an edge cover to a spanning tree (in the edge case). For various values of t, we present NPcompleteness and approximability results (both upper and lower bounds) and FPT algorithms for problems concerned with nding the minimum size of a ttotal vertex cover, ttotal edge cover and connected vertex cover, in particular improving on a previous FPT algorithm for the latter problem.
Algorithmics of Twosided Matching Problems
, 2003
"... In this thesis, we study several types of twosided matching problems. Such a problem involves two disjoint sets of participants, say U and W, each of whom ranks a subset of the other set of participants in order of preference. A matching in this context is a pairing of members of U with members of ..."
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Cited by 9 (1 self)
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In this thesis, we study several types of twosided matching problems. Such a problem involves two disjoint sets of participants, say U and W, each of whom ranks a subset of the other set of participants in order of preference. A matching in this context is a pairing of members of U with members of W that satises certain problemspecific cardinality and ranking constraints. Chapter 1 contains a brief introduction to twosided matching problems, and provides the necessary background for the remaining thesis. In Chapter 2, we introduce the studentproject allocation problem (SPA), which generalizes the classical hospitals/residents problem (HR). An instance of SPA consists of two sets of participants, namely students and projects, where each project is oered by a unique lecturer. Each student ranks a subset of the projects in order of preference, and similarly, each lecturer ranks a subset of the students in order of preference. We present two optimal lineartime algorithms for nding a stable matching of students to
Coupon Collectors, qBinomial Coefficients and the Unsatisfiability Threshold
"... The problem of determining the unsatisfiability threshold for random 3SAT formulas consists in determining the clause to variable ratio that marks the (experimentally observed) abrupt change from almost surely satisfiable formulas to almost surely unsatisfiable. Up to now, there have been rigorousl ..."
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Cited by 6 (1 self)
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The problem of determining the unsatisfiability threshold for random 3SAT formulas consists in determining the clause to variable ratio that marks the (experimentally observed) abrupt change from almost surely satisfiable formulas to almost surely unsatisfiable. Up to now, there have been rigorously established increasingly better lower and upper bounds to the actual threshold value. An upper bound of 4.506 was announced by Dubois et al. in 1999 but, to the best of our knowledge, no complete proof has been made available from the authors yet. We consider the problem of bounding the threshold value from above using methods that, we believe, are of interest on their own right. More specifically, we explain how the method of local maximum satisfying truth assignments can be combined with results for coupon collector's probabilities in order to achieve an upper bound for the unsatisfiability threshold less than 4.571. Thus, we improve over the best, with an available complete proof, previous upper bound, which was 4.596. In order to obtain this value, we also establish a bound on the qbinomial coefficients (a generalization of the binomial coefficients) which may be of independent interest.
Small Maximal Matchings in Random Graphs
 IN PROC. LATIN 2000, PP 1827. LNCS 1776
, 2000
"... We look at the minimal size of a maximal matching in general, bipartite and dregular random graphs. We prove that with high probability the ratio between the sizes of any two maximal matchings approaches one in dense random graphs and random bipartite graphs. Weaker bounds hold for sparse random g ..."
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Cited by 6 (1 self)
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We look at the minimal size of a maximal matching in general, bipartite and dregular random graphs. We prove that with high probability the ratio between the sizes of any two maximal matchings approaches one in dense random graphs and random bipartite graphs. Weaker bounds hold for sparse random graphs and random dregular graphs. We also describe an algorithm that with high probability finds a matching of size strictly less than n/2 in a cubic graph. The result is based on approximating the algorithm dynamics by a number of systems of linear differential equations.
Maximum Induced Matchings of Random Cubic Graphs
 JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
"... We present a heuristic for finding a large induced matching of cubic graphs. We analyse the performance of this heuristic, which is a random greedy algorithm, on random cubic graphs using dierential equations and obtain a lower bound on the expected size of the induced matching, M, returned by ..."
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Cited by 5 (1 self)
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We present a heuristic for finding a large induced matching of cubic graphs. We analyse the performance of this heuristic, which is a random greedy algorithm, on random cubic graphs using dierential equations and obtain a lower bound on the expected size of the induced matching, M, returned by the algorithm. A corresponding upper bound is derived by means of a direct expectation argument. We prove that M asymptotically almost surely satises 0:270413n M 0:282069n.
On operators satisfying T
 T 2 T ≥ T ∗ T 2 T ∗ , Linear Alg. Appl
"... Abstract. For the random 2SAT formula F(n,p), let FC(n,p) be the formula left after the pure literal algorithm applied to F(n,p) stops. Using the recently developed Poisson cloning model together with the cutoff line algorithm (COLA), we completely analyze the structure of FC(n,p). In particular, ..."
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Cited by 5 (0 self)
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Abstract. For the random 2SAT formula F(n,p), let FC(n,p) be the formula left after the pure literal algorithm applied to F(n,p) stops. Using the recently developed Poisson cloning model together with the cutoff line algorithm (COLA), we completely analyze the structure of FC(n,p). In particular, it is shown that, for λ: = p(2n − 1) = 1 + σ with σ ≫ n −1/3, the core of F(n,p) has θ 2 λ n + O((θ λ n)1/2) variables and θ 2 λ λn + O((θ λ n))1/2 clauses, with high probability, where θ λ is the larger solution of the equation θ − (1 − e −θ λ λ) = 0. We also estimate the probability of F(n,p) being satisfiable to obtain λ Pr[F2(n,