Partitionable starters for twin prime power type

AbstractSkew starters, balanced starters, partitionable starters are used in the construction of various combinatorial designs and configurations such as Room squares, Howell designs and Howell rotations. In this paper, we construct partitionable starters of order n when n is a product of two prime powers differing by 2. These partitionable starters are shown to be skew for n ⩾ 143. The results imply the existence of certain balanced Howell rotations. Moreover, we show the existence of partionable balanced starters of order n = 2m −1

On inefficient special cases of NP-complete problems

AbstractEvery intractable set A has a polynomial complexity core, a set H such that for any P-subset S of A or of Ā, S∩H is finite. A complexity core H of A is proper if H⊆A. It is shown here that if P≠NP, then every currently known (i.e., either invertibly paddable or k-creative) NP-complete set A and its complement Ā have proper polynomial complexity cores that are nonsparse and are accepted by deterministic machines in time 2cn for some constant c. Turning to the intractable class DEXT=∪c>0DTIME(2cn), it is shown that every set that is ⩽pm-complete for DEXT has an infinite proper polynomial complexity core that is nonsparse and recursive

Disproofs of Generalized Gilbert–Pollak Conjecture on the Steiner Ratio in Three or More Dimensions

AbstractThe Gilbert–Pollak conjecture, posed in 1968, was the most important conjecture in the area of “Steiner trees.” The “Steiner minimal tree” (SMT) of a point setPis the shortest network of “wires” which will suffice to “electrically” interconnectP. The “minimum spanning tree” (MST) is the shortest such network when onlyintersite line segmentsare permitted. The generalized GP conjecture stated thatρd=infP⊂Rd(lSMT(P)/lMST(P)) was achieved whenPwas the vertices of a regulard-simplex. It was showed previously that the conjecture is true ford=2 and false for 3⩽d⩽9. We settle remaining cases completely in this paper. Indeed, we show that any point set achievingρdmust have cardinality growing at least exponentially withd. The real question now is: What are the true minimal-ρpoint sets? This paper introduces the “d-dimensional sausage” point sets, which may have a lit to do with the answer

A (log23 + 12) competitive algorithm for the counterfeit coin problem

AbstractConsider a set of coins where each coin is either of the heavy type or the light type. The problem is to identify the type of each coin with minimal number of weighings on a balanced scale. The case that only one coin, called a counterfeit, has a different weight from others, is a classic mathematical puzzle. Later works study the case of more than one counterfeit, but the number of counterfeits is always assumed known. Recently, Hu and Hwang gave an algorithm which does not depend on the knowledge of the number of counterfeits, and yet perform uniformly good whatever that number turns out to be in the sample considered. Such an algorithm is known as a competitive algorithm and the uniform guarantee is measured by its competitive constant. Their algorithm has competitive ratio 2log2 3. In this paper, we give a new competitive algorithm with competitive ratio log2 3 + 12