We present a new method for proving rank lower bounds for the cutting planes procedures of Gomory and Chvátal (GC) and Lovász and Schrijver (LS), when viewed as proof systems for unsatisfiability. We apply this method to obtain the following new results: First, we prove near-optimal rank bounds for GC and LS proofs for several prominent unsatisfiable CNF examples, including random kCNF formulas and the Tseitin graph formulas. It follows from these lower bounds that a linear number of rounds of GC or LS procedures when applied to the standard MAXSAT linear relaxation does not reduce the integrality gap. Second, we give unsatisfiable examples that have constant rank GC and LS proofs but that require linear rank Resolution proofs. Third, we give examples where the GC rank is O(log n) but the LS rank is linear. Finally, we address the question of size

We consider a model (BT) for backtracking algorithms. Our model generalizes both the priority model of Borodin, Nielson and Rackoff, 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 satisfiability.

We propose a model called priority branching trees (pBT ) for backtracking and dynamic programming algorithms. Our model generalizes both the priority model of Borodin, Nielson and Rackoff, 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 Satisfiability.

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.

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