Kazuhisa Seto

A Moderately Exponential Time Algorithm for k-IBDD Satisfiability

A k-indexed Binary Decision Diagram (k-IBDD) is a branching program with k-layers and each layer consists of an Ordered Binary Decision Diagram (OBDD). This paper studies the satisfiability of k-IBDD (k-IBDD SAT). A k-IBDD SAT is, given a k-IBDD, to ask whether there exists a consistent path from the root to the 1-sink. We propose a moderately exponential time algorithm using exponential space for k-IBDD SAT of n variables and cn size. Our algorithm runs in time \(O\left( 2^{(1-\mu (c))n}\right) \), where \(\mu (c)=\Omega \left( \frac{1}{(\log {c})^{2^{k-1}-1}}\right) \). As a corollary, we obtain a polynomial space and deterministic algorithm, which solves k-IBDD SAT of size polynomial in n and runs in \(O\left( 2^{ n - n^{ 1/2^{k-1} }}\right) \) time.

Bounded Depth Circuits with Weighted Symmetric Gates: Satisfiability, Lower Bounds and Compression

A Boolean function f:{0,1}^n -> {0,1} is weighted symmetric if there exist a function g: Z -> {0,1} and integers w_0, w_1, ..., w_n such that f(x_1, ...,x_n) = g(w_0+sum_{i=1}^n w_i x_i) holds. In this paper, we present algorithms for the circuit satisfiability problem of bounded depth circuits with AND, OR, NOT gates and a limited number of weighted symmetric gates. Our algorithms run in time super-polynomially faster than 2^n even when the number of gates is super-polynomial and the maximum weight of symmetric gates is nearly exponential. With an additional trick, we give an algorithm for the maximum satisfiability problem that runs in time poly(n^t)*2^{n-n^{1/O(t)}} for instances with n variables, O(n^t) clauses and arbitrary weights. To the best of our knowledge, this is the first moderately exponential time algorithm even for Max 2SAT instances with arbitrary weights. Through the analysis of our algorithms, we obtain average-case lower bounds and compression algorithms for such circuits and worst-case lower bounds for majority votes of such circuits, where all the lower bounds are against the generalized Andreev function. Our average-case lower bounds might be of independent interest in the sense that previous ones for similar circuits with arbitrary symmetric gates rely on communication complexity lower bounds while ours are based on the restriction method.

Improved Exact Algorithms for Mildly Sparse Instances of Max SAT

We present improved exponential time exact algorithms for Max SAT. Our algorithms run in time of the form O(2^{(1-mu(c))n}) for instances with n variables and m=cn clauses. In this setting, there are three incomparable currently best algorithms: a deterministic exponential space algorithm with mu(c)=1/O(c * log(c)) due to Dantsin and Wolpert [SAT 2006], a randomized polynomial space algorithm with mu(c)=1/O(c * log^3(c)) and a deterministic polynomial space algorithm with mu(c)=1/O(c^2 * log^2(c)) due to Sakai, Seto and Tamaki [Theory Comput. Syst., 2015]. Our first result is a deterministic polynomial space algorithm with mu(c)=1/O(c * log(c)) that achieves the previous best time complexity without exponential space or randomization. Furthermore, this algorithm can handle instances with exponentially large weights and hard constraints. The previous algorithms and our deterministic polynomial space algorithm run super-polynomially faster than 2^n only if m=O(n^2). Our second results are deterministic exponential space algorithms for Max SAT with mu(c)=1/O((c * log(c))^{2/3}) and for Max 3-SAT with mu(c)=1/O(c^{1/2}) that run super-polynomially faster than 2^n when m=o(n^{5/2}/log^{5/2}(n)) and m=o(n^3/log^2(n)) respectively.

Solving Sparse Instances of Max SAT via Width Reduction and Greedy Restriction

We present a moderately exponential time polynomial space algorithm for sparse instances of Max SAT. Our algorithms run in time of the form O(2(1 − μ(c))n ) for instances with n variables and cn clauses. Our deterministic and randomized algorithm achieve \(\mu(c) = \Omega(\frac{1}{c^2\log^2 c})\) and \(\mu(c) = \Omega(\frac{1}{c \log^3 c})\) respectively. Previously, an exponential space deterministic algorithm with \(\mu(c) = \Omega(\frac{1}{c\log c})\) was shown by Dantsin and Wolpert [SAT 2006] and a polynomial space deterministic algorithm with \(\mu(c) = \Omega(\frac{1}{2^{O(c)}})\) was shown by Kulikov and Kutzkov [CSR 2007].

Efficient Algorithms for Sorting k-Sets in Bins

We give efficient algorithms for Sorting k-Sets in Bins. The Sorting k-Sets in Bins problem can be described as follows: We are given numbered n bins with k balls in each bin. Balls in the i-th bin are numbered n − i + 1. We can only swap balls between adjacent bins. How many swaps are needed to move all balls to the same numbered bins. For this problem, we design an efficient greedy algorithm with \(\frac{k+1}{4}n^2+O(kn)\) swaps. As k and n increase, this approaches the lower bound of \(\lceil \binom{kn}{2}/(2k-1) \rceil\). In addition, we design a more efficient recursive algorithm using \(\frac{15}{16}n^2+O(n)\) swaps for the k = 3 case.

A Moderately Exponential Time Algorithm for k -IBDD Satisfiability

We present a satisfiability algorithm for k-indexed binary decision diagrams (k-IBDDs). The proposed exponential space and deterministic algorithm solves the satisfiability of k-IBDDs, i.e., k-IBDD SAT, for instances with n variables and cn nodes in

k

O\left( 2^{(1-\mu _k(c))n}\right)

Improved exact algorithms for mildly sparse instances of Max SAT

O2(1-μk(c))n time, where