Misc. background material.
Excerpts
Probabilistic method
A brief introduction to the probabilistic method, followed by some basic bounds and a few examples.
Alon, Spencer, and Erdős (1992) describe the method as follows:
In order to prove the existence of a combinatorial structure with certain properties, we construct an appropriate probability space and show that a randomly chosen element in the space has the desired properties with positive probability. [1]
The method has come into wide use since about 1950, even in areas of mathematics that, apriori, have nothing to do with probability. This is roughly because probability provides a useful highlevel conceptual framework for understanding (formalizing, organizing, communicating) many widely applicable counting and averaging arguments.
basic bounds
Existence proofs, the naive union bound, linearity of expectation, Markov bound.
examples (Max Cut, Turán’s theorem)
Examples: 2approximation for Max Cut; Turán’s theorem
method of conditional probabilities (Max Cut)
The method of conditional probabilities converts a probabilistic existence proof into a deterministic algorithm.
The method of conditional probabilities is a systematic method for converting nonconstructive probabilistic existence proofs into efficient deterministic algorithms that explicitly construct the desired object.
pessimistic estimators (Turáns theorem)
Pessimistic estimators in the method of conditional probabilities.
In applying the method of conditional probabilities, exact conditional probabilities (or expectations) are sometimes hard to compute. Pessimistic estimators can be used instead. We illustrate the idea by example, using Turán’s theorem.
Chernoff bound
A brief introduction to Chernoff bounds.
If you’re already familiar with Chernoff bounds, you may prefer to skip directly to the statement and proof of a typical Chernoff bound.
Chernoff bounds (a.k.a. tail bounds, Hoeffding/Azuma/Talagrand inequalities, the method of bounded differences, etc. [1, 2]) are used to bound the probability that some function (typically a sum) of many “small” random variables falls in the tail of its distribution (far from its expectation).
Chernoff bound proof
The statement and proof of a typical Chernoff bound.
Wald’s equation
Wald’s equation, a form of linearity of expectation for sums with randomly many terms.
Consider a sum \sum _{t=1}^ T x_ t of random variables, where the number of terms T is itself a random variable. If each term x_ t has expectation at most (or at least) \mu , then the expectation of the sum is at most (or at least) \mu \, \textrm{E}[T] (the bound on the expectation of each term, times the expected number of terms).
Wald’s equation
Wald’s equation, a form of linearity of expectation for sums with randomly many terms.
Consider a sum \sum _{t=1}^ T x_ t of random variables, where the number of terms T is itself a random variable. If each term x_ t has expectation at most (or at least) \mu , then the expectation of the sum is at most (or at least) \mu \, \textrm{E}[T] (the bound on the expectation of each term, times the expected number of terms). This holds provided the random variables in the sum are bounded above or below and T is a stopping time with finite expectation.
Wald’s equation for dependent increments
A variant of Wald’s equation to use when the expected increment depends on the sum so far.
Wald’s equation applies to any sequence that, with each step, increases (or decreases) in expectation by a constant additive amount. What about sequences where the expected change with each step depends on the current value? For example, suppose Alice starts with n coins. In each round t=1,2,\ldots ,T, she flips each remaining coin and discards those that come up tails. She stops once all coins are discarded. What is the expected number of rounds?
Markov bound for supermartingales
For any nonnegative supermartingale, the probability that it ever exceeds its initial value by a factor of c is at most 1/c.
Alice goes to the casino with $1. At the casino, she plays the following game repeatedly: she bets half her current balance on a fair coin flip. (For example, on the first flip, she bets 50 cents, so she wins 50 cents with probability 1/2 and loses 50 cents with probability 1/2.) Will Alice’s winnings ever reach $10 or more? The bound here says this happens with probability at most 1/10.
“Stoppingtime” Chernoff bounds
Extending the Chernoff bound to handle sums with randomly many terms.
Alice goes to the casino and plays bets on a sequence of fair coin flips. On the tth bet, Alice chooses an amount a_ t\in [0,1] to bet: she wins a_ t if this flip is heads, otherwise she loses a_ t. Since it’s Vegas, she never stops playing. Fix any \varepsilon \gt 0. Let W_ t be the sum of bets won after the tth bet. Let L_ t be the sum of bets lost after the tth bet.
Expected maximum (or minimum) of many sums
Bounds on the expected maximum (or minimum) among a collection of sums.
It can be technically convenient to work with expectations directly, instead of working with probabilities. Here, given a collection of sums of 0/1 random variables, we bound the expected maximum (or minimum) sum in the collection.
For example, suppose Alice throws balls randomly into n bins just until the first bin has n balls. The bound says that the expected maximum number of balls in any bin will be at most n+2\sqrt {n\ln n}+\ln n. Similarly, the expected minimum number of balls in any bin will be at least n2\sqrt {n\ln n}.
Expected deviation of a sum
Bounds on the expected deviation of a sum from a threshold.
Here are bounds on the expected deviation of a sum of 0/1random variables above or below some threshold (typically near its mean).
For example, suppose Alice flips a fair coin n times. She pays Bob $1 for each head after the first (1+\varepsilon )n/2 heads (if any). What is her expected payment to Bob? The bounds here say: at most \varepsilon ^{1} \exp (\varepsilon ^2 n/6). For example, if \varepsilon =\Omega (1/\sqrt n), the expected payment is O(\sqrt n).
Linear programming
Notes on Linear Programming and Integer Linear Programming.
modeling Set Cover and Multicommodity Flow
Set Cover and Multicommodity flow as (integer) linear programs.
Modeling a problem as a linear program or integer linear program is a basic skill. Here are two examples.
rounding an LP relaxation
A simple example of computing an approximate solution by rounding the solution to a linearprogram relaxation.
The basic paradigm:

Model your problem as an integer linear program.

Solve its linear program relaxation.

Somehow round the solution x^* of the relaxed problem to get a solution \tilde x of the original problem.
The rounding step is typically most easily done with a socalled randomized rounding scheme.
Rounding a relaxation is one of two standard ways to use linearprogramming relaxations to design approximation algorithms. (The other way is the primaldual method.)
Randomized rounding
Randomly rounding a fractional LP solution to an integer solution.
The idea, introduced by Raghavan and Thompson in 1987, is to use the probabilistic method to round the solution of a linear program, converting it into an approximately optimal integer solution [3]. It’s a broadly useful technique. For many problems, randomized rounding yields algorithms with optimal approximation ratios (assuming P\neq NP). This note describes randomized rounding and gives a few examples.
Set Cover / greedy algorithm
A brief review of the greedy algorithm for the SetCover problem.
Set Cover / Wolsey’s generalization
Wolsey’s generalization of the greedy SetCover algorithm to a large class of problems.
It is natural to ask what general classes of problems the greedy SetCover algorithm generalizes to. Here we describe one such class, due to Wolsey (1982), that captures many, but not all, such problems.
Lagrangian relaxation / example
A simple example of a Lagrangianrelaxation algorithm.
The algorithm is for Maximum Multicommodity Flow. It illustrates some prototypical aspects of Lagrangian relaxation.
Lagrangian relaxation / discussion
A brief yet longwinded discussion of Lagrangianrelaxation algorithms.