 
Summary: Toward a Model for Backtracking and Dynamic Programming
Michael Alekhnovich Allan Borodin y Joshua BureshOppenheim z
Russell Impagliazzo zx{ Avner Magen y Toniann Pitassi y{
January 7, 2005
Abstract
We consider a model (BT) for backtracking algorithms. Our model generalizes both the
priority model of Borodin, Nielson and Racko, as well as a simple dynamic programming
model due to Woeginger, and hence spans a wide spectrum of algorithms. After witnessing the
strength of the model, we then show its limitations by providing lower bounds for algorithms in
this model for several classical problems such as interval scheduling, knapsack and satisability.
1 Introduction
Proving unconditional lower bounds for computing explicit functions remains one of the most
challenging problems in computational complexity. Since 1949, when Shannon showed that a
random function has large circuit complexity [25], little progress has been made toward proving
lower bounds for the size of unrestricted Boolean circuits that compute explicit functions. One
explanation for this phenomenon was given by the Natural Proofs approach of Razborov and
Rudich [24] who showed that most of the existing lower bound techniques are incapable of proving
such lower bounds. One way to investigate the complexity of explicit functions in spite of these
diĘculties is to study reductions between problems, e.g. to identify a canonical problem (like an
NPcomplete problem) and show that a given computational task is \not easier" than solving this
