Sparse instances of hard problems

In this thesis, we use and refine methods of computational complexity theory to analyze the complexity of sparse instances, such as graphs with few edges or formulas with few constraints of bounded width. Two natural questions arise in this context: • Is there an efficient algorithm that reduces arbitrary instances of an NP-hard problem to equivalent, sparse instances? • Is there an algorithm that solves sparse instances of an NP-hard problem significantly faster than general instances can be solved? We formalize these questions for different problems and show that positive answers for these formalizations would lead to consequences in complexity theory that are considered unlikely. The first question is modeled by the following two-player communication process to decide a language L: The first player holds the entire input x but is polynomially bounded; the second player is computationally unbounded but does not know any part of x; their goal is to decide cooperatively whether x belongs to L at small cost, where the cost measure is the number of bits of communication from the first player to the second player. For any integer d ≥ 3 and positive real we show that if satisfiability for n-variable d-CNF formulas has a protocol of cost O(nd− ) then coNP is in NP/poly, which implies that the polynomial-time hierarchy collapses to its third level. We obtain similar results for various NP-complete covering and packing problems in graphs and hypergraphs. The results even hold when the first player is conondeterministic, and are tight as there exists a trivial protocol for = 0. Under the hypothesis that coNP is not in NP/poly, our results imply surprisingly tight lower bounds for parameters of interest in several areas, namely sparsification, kernelization in parameterized complexity, lossy compression, and probabilistically checkable proofs. We study the second question from above for counting problems in the exponential time setting. The Exponential Time Hypothesis (ETH) is the complexity assumption that the satisfiability of n-variable 3-CNF formulas cannot be decided in time exp(o(n)). Assuming (variants of) ETH, we obtain asymptotically tight, exponential lower bounds for well-studied #P-hard problems: • Computing the number of satisfying assignments of a 2-CNF formula, • Computing the number of all independent sets in a graph, • Computing the permanent of a matrix with entries 0 and 1, • Evaluating the Tutte polynomial of multigraphs at fixed evaluation points. We also obtain results for the Tutte polynomial of simple graphs, where our lower bounds are asymptotically tight up to polylogarithmic factors in the exponent of the running time.