This paper introduces a special family of randomized algorithms for Max DICUT that we call oblivious algorithms. Let the bias of a vertex be the ratio between the total weight of its outgoing edges and the total weight of all its edges. An oblivious algorithm selects at random in which side of the cut to place a vertex v, with probability that only depends on the bias of v, independently of other vertices. The reader may observe that the algorithm that ignores the bias and chooses each side with probability 1/2 has an approximation ratio of 1/4, whereas no oblivious algorithm can have an approximation ratio better than 1/2 (with an even directed cycle serving as a negative example). We attempt to characterize the best approximation ratio achievable by oblivious algorithms, and present results that are nearly tight. The paper also discusses natural extensions of the notion of oblivious algorithms, and extensions to the more general problem of Max 2-AND.
The factor graph of an instance of a symmetric constraint satisfaction problem on n Boolean variables and m constraints (CSPs such as k-SAT, k-AND, k-LIN) is a bipartite graph describing which variables appear in which constraints. The factor graph describes the instance up to the polarity of the variables, and hence there are up to 2 km instances of the CSP that share the same factor graph. It is well known that factor graphs with certain structural properties make the underlying CSP easier to either solve exactly (e.g., for tree structures) or approximately (e.g., for planar structures). We are interested in the following question: is there a factor graph for which if one can solve every instance of the CSP with this particular factor graph, then one can solve every instance of the CSP regardless of the factor graph (and similarly, for approximation)? We call such a factor graph universal. As one needs different factor graphs for different values of n and m, this gives rise to the notion of a family of universal factor graphs. We initiate a systematic study of universal factor graphs, and present some results for max-kSAT. Our work has connections with the notion of preprocessing as previously studied for closest codeword and closest lattice-vector problems, with proofs for the PCP theorem, and with tests for the long code. Many questions remain open.
We show (under standard assumptions) that there are NP optimization problems for which estimation is easier than approximation. Namely, one can estimate the value of the optimal solution within a ratio of ρ, but it is difficult to find a solution whose value is within ρ of optimal. As an important special case, we show that there are linear programming relaxations for which no polynomial time rounding technique matches the integrality gap of the linear program.
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