Many problems in interprocedural program analysis can be modeled as the context-free language (CFL) reachability problem on graphs and can be solved in cubic time. Despite years of efforts, there are no known truly sub-cubic algorithms for this problem. We study the related certification task: given an instance of CFL reachability, are there small and efficiently checkable certificates for the existence and for the non-existence of a path? We show that, in both scenarios, there exist succinct certificates ( O ( n 2 ) in the size of the problem) and these certificates can be checked in subcubic (matrix multiplication) time. The certificates are based on grammar-based compression of paths (for reachability) and on invariants represented as matrix inequalities (for non-reachability). Thus, CFL reachability lies in nondeterministic and co-nondeterministic subcubic time. A natural question is whether faster algorithms for CFL reachability will lead to faster algorithms for combinatorial problems such as Boolean satisfiability (SAT). As a consequence of our certification results, we show that there cannot be a fine-grained reduction from SAT to CFL reachability for a conditional lower bound stronger than n ω , unless the nondeterministic strong exponential time hypothesis (NSETH) fails. In a nutshell, reductions from SAT are unlikely to explain the cubic bottleneck for CFL reachability. Our results extend to related subcubic equivalent problems: pushdown reachability and 2NPDA recognition; as well as to all-pairs CFL reachability. For example, we describe succinct certificates for pushdown non-reachability (inductive invariants) and observe that they can be checked in matrix multiplication time. We also extract a new hardest 2NPDA language, capturing the “hard core” of all these problems.
In the general AntiFactor problem, a graph G is given with a set Xv ⊆ N of forbidden degrees for every vertex v and the task is to find a set S of edges such that the degree of v in S is not in the set Xv. Standard techniques (dynamic programming + fast convolution) can be used to show that if M is the largest forbidden degree, then the problem can be solved in time (M + 2) tw • n O(1) if a tree decomposition of width tw is given. However, significantly faster algorithms are possible if the sets Xv are sparse: our main algorithmic result shows that if every vertex has at most x forbidden degrees (we call this special case AntiFactorx), then the problem can be solved in time (x + 1) O(tw) • n O(1) . That is, the AntiFactorx is fixed-parameter tractable parameterized by treewidth tw and the maximum number x of excluded degrees.Our algorithm uses the technique of representative sets, which can be generalized to the optimization version, but (as expected) not to the counting version of the problem. In fact, we show that #AntiFactor1 is already #W[1]-hard parameterized by the width of the given decomposition. Moreover, we show that, unlike for the decision version, the standard dynamic programming algorithm is essentially optimal for the counting version. Formally, for a fixed nonempty set X, we denote by X-AntiFactor the special case where every vertex v has the same set Xv = X of forbidden degrees. We show the following lower bound for every fixed set X: if there is an > 0 such that #X-AntiFactor can be solved in time (max X + 2 − ) tw • n O(1) on a tree decomposition of width tw, then the Counting Strong Exponential-Time Hypothesis (#SETH) fails.
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