Abstract: For many random Constraint Satisfaction Problems, by now, we have asymptotically tight estimates of
the largest constraint density for which they have solutions. At the same time, all known polynomial-time algorithms
for many of these problems already completely fail to find solutions at much smaller densities. For example, it is
well-known that it is easy to color a random graph using twice as many colors as its chromatic number. Indeed, some
of the simplest possible coloring algorithms already achieve this goal. Given the simplicity of those algorithms, one
would expect there is a lot of room for improvement. Yet, to date, no algorithm is known that uses (2 - o)÷ colors,
in spite of efforts by numerous researchers over the years.
In view of the remarkable resilience of this factor of 2 against every algorithm hurled at it, we believe it is natural to
inquire into its origin. We do so by analyzing the evolution of the set of k-colorings of a random graph, viewed as a
subset of {1, . . . , k}n, as edges are added. We prove that the factor of 2 corresponds in a precise mathematical sense
to a phasetransition in the geometry of this set. Roughly, the set of k-colorings looks like a giant ball for k ? 2÷, but
like an error-correcting code for k ? (2 - o)÷. We prove that a completely analogous phasetransition also occurs both in random k-SAT and in random hypergraph 2-coloring. And that for each problem, its location corresponds precisely with the point were all known polynomial-time algorithms fail. To prove our results we develop a general technique that allows us to prove rigorously much of the celebrated 1-step Replica-Symmetry-Breaking hypothesis
of statistical physics for random CSPs.

Abstract: For a number of optimization problems on random graphs
and hypergraphs, e.g., k-colorings, there is a very big gap between the
largest average degree for which known polynomial-time algorithms can
find solutions, and the largest average degree for which solutions provably
exist. We study this phenomenon by examining how sets of solutions
evolve as edges are added.We prove in a precise mathematical sense that,
for each problem studied, the barrier faced by algorithms corresponds
to a phasetransition in the problems solution-space geometry. Roughly
speaking, at some problem-specific critical density, the set of solutions
shatters and goes from being a single giant ball to exponentially many,
well-separated, tiny pieces. All known polynomial-time algorithms work
in the ball regime, but stop as soon as the shattering occurs. Besides
giving a geometric view of the solution space of random instances our
results provide novel constructions of one-way functions.

Abstract: We study the problem of fair resource allocation in a simple cooperative multi-agent setting where we have k agents and a set of n objects to be allocated to those agents. Each object is associated with a weight represented by a positive integer or real number. We would like to allocate all objects to the agents so that each object is allocated to only one agent and the weight is distributed fairly. We adopt the fairness index popularized by the networking community as our measure of fairness, and study centralized algorithms for fair resource allocation. Based on the relationship between our problem and number partitioning, we devise a greedy algorithm for fair resource allocation that runs in polynomial time but is not guaranteed to find the optimal solution, and a complete anytime algorithm that finds the optimal solution but runs in exponential time. Then we study the phasetransition behavior of the complete algorithm. Finally, we demonstrate that the greedy algorithm actually performs very well and returns almost perfectly fair allocations.

Abstract: In recent years there has been signi1cant interest in the study of random k-SAT formulae. For
a given set of n Boolean variables, let Bk denote the set of all possible disjunctions of k distinct,
non-complementary literals from its variables (k-clauses). A random k-SAT formula Fk (n;m) is
formed by selectinguniformly and independently m clauses from Bk and takingtheir conjunction.
Motivated by insights from statistical mechanics that suggest a possible relationship between the
?order? of phasetransitions 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. Usingthe heuristic ?replica method? of
statistical mechanics, Monasson and Zecchina gave a number of non-rigorous predictions on the
behavior of random (2 + p)-SAT formulae. In this paper we give the 1rst rigorous results for
random (2 + p)-SAT, includingthe followingsurprisingfact: for p 6 2=5, with probability
1 − o(1), a random (2 + p)-SAT formula is satis1able i@ its 2-SAT subformula is satis1able.
That is, for p 6 2=5, random (2 + p)-SAT behaves like random 2-SAT.

Abstract: One of the most challenging problems in probability and complexity theory is
to establish and determine the satisfiability threshold, or phasetransition, for
random k-SAT instances: Boolean formulas consisting of clauses with exactly k
literals. As the previous part of the volume has explored, empirical observations
suggest that there exists a critical ratio of the number of clauses to the number
of variables, such that almost all randomly generated formulas with a higher
ratio are unsatisfiable while almost all randomly generated formulas with a lower
ratio are satisfiable. The statement that such a crossover point really exists is
called the satisfiability threshold conjecture. Experiments hint at such a direction,
but as far as theoretical work is concerned, progress has been difficult. In an
important advance, Friedgut [23] showed that the phasetransition is a sharp one,
though without proving that it takes place at a “fixed” ratio for large formulas.
Otherwise, rigorous proofs have focused on providing successively better upper
and lower bounds for the value of the (conjectured) threshold. In this chapter, our