Some diffusive and recurrence properties of Lorentz Lattice Gas Cellular Automata (LLGCA) have been expensively studied in terms of the densities of some of the left/right static/flipping mirrors/rotators. In this paper, for any combination S of these well known scatters, we study the computational complexity of the following problem which we call PERIODICITY on the S-model: given a finite configuration that distributes only those scatters in S, whether a particle visits the starting position periodically or not. Previously, the flipping mirror model and the occupied flipping rotator model have been shown unbounded, i.e. the process is always diffusive [17]. On the other hand, PERIODICITY is shown PSPACE-complete in the unoccupied flipping rotator model [21]. In this paper, we show that PERIODICITY is PSPACE-compete in any S-model that is neither occupied, unbounded, nor static. Particularly, we prove that PERIODICITY in any unoccupied and bounded model containing flipping mirror is PSPACE-complete.

In the theory of average-case NP-completeness, Levin introduced the polynomial domination and Gurevich did the average polynomial domination. Ben-David et al. proved that if P-samp (the class of polynomial-time samplable distributions) is polynomially dominated by P-comp (the class of polynomial-time computable distributions) then there exists no strong one-way function. This result will be strengthened by relaxing the assumption from the polynomial domination to the average polynomial domination. Our results also include incompleteness for average polynomial-time one-one reducibility from (NP,P-samp) to (NP,P-comp). To derive these and other related results, a prefix-search algorithm presented by Ben-David et al. will be modified to survive the average polynomial domination.

H-coloring problem is a coloring problem with restrictions such that some pairs of colors cannot be used for adjacent vertices, where H is a graph representing the restrictions of colors. We deal with the case that H is the complement graph of a cycle of odd order 2p + 1. This paper presents the following results: (1) chordal graphs and internally maximal planar graphs are -colorable if and only if they are p-colorable (p 2), (2) -coloring problem on planar graphs is NP-complete, and (3) there exists a class that includes infinitely many -colorable but non-3-colorable planar graphs.

We define two restricted classes of Boolean circuits by assuming the following conditions on underlying graphs of circuits, and prove, for each class, nonlinear lower bounds on size of circuits computing cyclic shifts: for any two paths from the same input to the same output node, the sequences of depths of nodes along these paths are the same. A circuit is partitioned into subcircuits such that each subcircuit has at most o(log n) output gates and the multivalued circuit obtained from the partition is directed tree. These two conditions are driven from different points of view, and lower bounds are established for each one of them.

Chen and Kanj considered the VERTEX COVER problem for graphs with perfect matchings (VC-PM). They showed that: (i) There is a reduction from general VERTEX COVER to VC-PM, which guarantees that if one can achieve an approximation factor of less than two for VC-PM, then one can do so for general VERTEX COVER as well. (ii) There is an algorithm for VC-PM whose approximation factor is given as 1.069+0.069 where is the average degree of the given graph. In this paper we improve (ii). Namely we give a new VC-PM algorithm which greatly outperforms the above one and its approximation factor is roughly . Our algorithm also works for graphs with "large" matchings, although its approximation factor is degenerated.

It is a well-known and useful problem to find a matching in a given graph G whose size is at most a given parameter k and whose weight is maximized (over all matchings of size at most k in G). In this paper, we consider two natural extensions of this problem. One is to find t disjoint matchings in a given graph G whose total size is at most a given parameter k and whose total weight is maximized, where t is a (small) constant integer. Previously, only the special case where t=2 was known to be fixed-parameter tractable. In this paper, we show that the problem is fixed-parameter tractable for any constant t. When t=2, the time complexity of the new algorithm is significantly better than that of the previously known algorithm. The other is to find a set of vertex-disjoint paths each of length 1 or 2 in a given graph whose total length is at most a given parameter k and whose total weight is maximized. As interesting applications, we further use the algorithms to speed up several known approximation algorithms (for related NP-hard problems) whose approximation ratio depends on a fixed parameter 0<ε<1 and whose running time is dominated by the time needed for exactly solving the problems on graphs in which each connected component has at most ε-1 vertices.