25 research outputs found

    No small nondeterministic read-once branching programs for CNFs of bounded treewidth

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    In this paper, given a parameter k, we demonstrate an infinite class of cnfs of treewidth at most k of their primal graphs such that equivalent nondeterministic read-once branching programs (nrobps) are of size at least nck for some universal constant c. Thus we rule out the possibility of fixed-parameter tractable space complexity of nrobps parameterized by the smallest treewidth of equivalent cnfs

    On oblivious branching programs with bounded repetition that cannot efficiently compute CNFs of bounded treewidth

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    In this paper we study complexity of an extension of ordered binary decision diagrams (OBDDs) called c-OBDDs on CNFs of bounded (primal graph) treewidth. In particular, we show that for each k ≥ 3 there is a class of CNFs of treewidth k for which the equivalent c-OBDDs are of size Ω(nk/(8c−4)). Moreover, this lower bound holds if c-OBDDs are non-deterministic and semantic. Our second result uses the above lower bound to separate the above model from sentential decision diagrams (SDDs). In order to obtain the lower bound, we use a structural graph parameter called matching width. Our third result shows that matching width and pathwidth are linearly related

    Parameterized Compilation Lower Bounds for Restricted CNF-formulas

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    We show unconditional parameterized lower bounds in the area of knowledge compilation, more specifically on the size of circuits in decomposable negation normal form (DNNF) that encode CNF-formulas restricted by several graph width measures. In particular, we show that - there are CNF formulas of size nn and modular incidence treewidth kk whose smallest DNNF-encoding has size nΩ(k)n^{\Omega(k)}, and - there are CNF formulas of size nn and incidence neighborhood diversity kk whose smallest DNNF-encoding has size nΩ(k)n^{\Omega(\sqrt{k})}. These results complement recent upper bounds for compiling CNF into DNNF and strengthen---quantitatively and qualitatively---known conditional low\-er bounds for cliquewidth. Moreover, they show that, unlike for many graph problems, the parameters considered here behave significantly differently from treewidth

    Classification of OBDD Size for Monotone 2-CNFs

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    We introduce a new graph parameter called linear upper maximum induced matching width lu-mim width, denoted for a graph G by lu(G). We prove that the smallest size of the obdd for ?, the monotone 2-cnf corresponding to G, is sandwiched between 2^{lu(G)} and n^{O(lu(G))}. The upper bound is based on a combinatorial statement that might be of an independent interest. We show that the bounds in terms of this parameter are best possible. The new parameter is closely related to two existing parameters: linear maximum induced matching width (lmim width) and linear special induced matching width (lsim width). We prove that lu-mim width lies strictly in between these two parameters, being dominated by lsim width and dominating lmim width. We conclude that neither of the two existing parameters can be used instead of lu-mim width to characterize the size of obdds for monotone 2-cnfs and this justifies introduction of the new parameter
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