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On the Complexity of Problems on Tree-structured Graphs

“…In this paper, we introduce a new class of parameterized complexity, which appears to be the natural home of several 'tree structured' parameterized problems. This class, which we call XALP, can be seen as the parameterized version of a class known in classic complexity theory as NAuxPDA[poly, log] (see [3]), or ASPSZ(log n, n O (1) ) [23].…”

“…This is defined by taking the treewidth of an associated graph, usually a bipartite graph on the variables on one side and the clauses on the other, where there is an edge if the variable appears in the clause. Alekhnovitch and Razborov [2] raised the question of whether satisfiability of formulas of small treewidth can be checked in polynomial space, which was positively answered by Allender et al [3]. However, the running time of the algorithm is 3 T W(φ) log |φ| rather than 2 O(T W(φ)) |φ| O (1) , where |φ| = n + m for n the number of variables and m the number of clauses.…”

“…To support this conjecture, Allender et al [3] show that Satisfiability where the treewidth of the associated graph is O(log n) is complete for a class of problems called SAC 1 : these are the problems that can be recognized by 'uniform' circuits with semi-unbounded fanin of depth O(log n) and polynomial size. This class has also been shown to be equivalent to classes of problems that are defined using Alternating Turing Machines and non-deterministic Turing machines with access to an auxiliary stack [23,25].…”

“…This class has also been shown to be equivalent to classes of problems that are defined using Alternating Turing Machines and non-deterministic Turing machines with access to an auxiliary stack [23,25]. In fact, Allender et al [3] give a more general result for treewidth O(log k n) for k ≥ 1.…”

Beating Brute Force for (Quantified) Satisfiability of Circuits of Bounded Treewidth

“…It should be noted that Alekhnovich and Razborov [1] and Allender et al [2] have previously investigated the time and space complexity of CNF-SAT from the perspective of the treewidth of the underlying graph. For a CNF φ with n input variables and m clauses, they define the treewidth ω of φ as the treewidth of the clause-variable graph of φ. Alekhnovich and Razborov [1] showed that the satisfiability of such CNF can be solved in time .…”