A Stronger Model of Dynamic Programming Algorithms

Abstract

We define a formal model of dynamic programming algorithms which we call Prioritized Branching Programs (pBP). Our model is a generalization of the BT model of Alekhnovich et al. (IEEE Conference on Computational Complexity, pp. 308-322, 2005), which is in turn a generalization of the priority algorithms model of Borodin, Nielson and Rackoff. One of the distinguishing features of these models is that they not only capture large classes of algorithms generally considered to be greedy, backtracking or dynamic programming algorithms, but they also allow characterizations of their limitations. Hence they give meaning to the statement that a given problem can or cannot be solved by dynamic programming. After defining the model, we prove three main results: (i) that certain types of natural restrictions of our seemingly more powerful model can be simulated by the BT model; (ii) that in general our model is stronger than the BT model-a fact which is witnessed by the classical shortest paths problem; (iii) that our model has very real limitations, namely that bipartite matching cannot be efficiently computed in it, hence suggesting that there are problems that can be solved efficiently by network flow algorithms and by simple linear programming that cannot be solved by natural dynamic programming approaches.S.

Limitations of Incremental Dynamic Programming

“…We say that the problem has the unique optimum property, if for every feasible solution S ∈ F, there is an input x such that S is a unique optimal solution for x. Note that, if weights 0 and 1 are allowed, then every maximization problem has this property: just set x i = 1 for i ∈ S, and x i = 0 for i ∈ S. The unique optimum property was also used in [3] and [13] to prove lower bounds for prioritized branching trees and branching programs.…”

“…But what about the DP complexity of this problem? It is shown in [13] that this problem requires prioritized branching programs of exponential size. Results of [9] also imply that this problem requires combinatorial dynamic programs of exponential size.…”

“…In this paper we pursue essentially the same goal as in [3] and [13], but with exclusive focus on the dynamic programming paradigm. There is an old field-that of boolean circuit complexity-where the same goal (what small circuits cannot do) was pursuit for now more than 70 years.…”

“…More tractable models of so-called "prioritized branching trees" (pBT) and "prioritized branching programs" (pBP) were recently introduced, respectively, by Alekhnovich et al [3] and Buresh-Oppenheim et al [13]. These models are based on the framework of "priority algorithms" introduced by Borodin et al [11], and subsequently studied and generalized by many authors [4,3,12,14,34,33], just to mention some of them.…”

“…The model of pBP adds to that of pBT an additional feature of "memoization" (or "cashing" or "merging" subtrees), an important aspect of dynamic programming. In [3] it is shown that the Knapsack problem requires pBTs of exponential size, whereas in [13] it is shown that detecting the presence of a perfect matching in bipartite graphs requires (restricted) pBPs of exponential size.…”