A branching program is a well-studied model of computation and a representation for Boolean functions. It is a directed acyclic graph with a unique root node, some accepting nodes, and some rejecting nodes. Except for the accepting and rejecting nodes, each node has a label with a variable and each outgoing edge of the node has a label with a 0/1 assignment of the variable. The satisfiability problem for branching programs is, given a branching program with n variables and m nodes, to determine if there exists some assignment that activates a consistent path from the root to an accepting node. The width of a branching program is the maximum number of nodes at any level. The satisfiability problem for width-2 branching programs is known to be NP-complete. In this paper, we present a satisfiability algorithm for width-2 branching programs with n variables and cn nodes, and show that its running time is poly(n)·2(1-µ(c))n, where µ(c)=1/2O(c log c). Our algorithm consists of two phases. First, we transform a given width-2 branching program to a set of some structured formulas that consist of AND and Exclusive-OR gates. Then, we check the satisfiability of these formulas by a greedy restriction method depending on the frequency of the occurrence of variables.

We propose an exact algorithm to determine the satisfiability of oblivious read-twice branching programs. Our algorithm runs in $2^{left(1 - Omega(rac{1}{log c}) ight)n}$ time for instances with n variables and cn nodes.

We propose 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. Our task is to move all of the balls to the same numbered bins. For this problem, we give an efficient greedy algorithm with $rac{k+1}{4}n^2+O(k+n)$ swaps and provide a detailed analysis for k=3. In addition, we give a more efficient recursive algorithm using $rac{15}{16}n^2+O(n)$ swaps for k=3.