We explore the maximum total number of edge crossings and the maximum geometric thickness of the Euclidean minimum-weight (k, ℓ)-tight graph on a planar point set P. In this paper, we show that (10/7－ε)|P| and (11/6－ε)|P| are lower bounds for the maximum total number of edge crossings for any ε ＞ 0 in cases (k,ℓ)=(2,3) and (2,2), respectively. We also show that the lower bound for the maximum geometric thickness is 3 for both cases. In the proofs, we apply the method of arranging isomorphic units regularly. While the method is developed for the proof in case (k,ℓ)=(2,3), it also works for different ℓ.

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.

The planar Hajos calculus (PHC) is the Hajos calculus with the restriction that all the graphs that appear in the construction (including a final graph) must be planar. The degree-d planar Hajos calculus (PHC(dd)) is PHC with the restriction that all the graphs that appear in the construction (including a final graph) must have maximum degree at most d. We prove the followings: (1) If PHC is polynomially bounded, then for any d ≥ 4, PHC(dd+2) can generate any non-3-colorable planar graphs of maximum degree at most d in polynomial steps. (2) If PHC can generate any non-3-colorable planar graphs of maximum degree 4 in polynomial steps, then PHC is polynomially bounded.