The existence problems of perfect difference families with block size k, k = 4, 5, and additive sequences of permutations of length n, n = 3, 4, are two outstanding open problems in combinatorial design theory for more than 30 years. In this article, we mainly investigate perfect difference families with block size k = 4 and additive sequences of permutations of length n = 3. The necessary condition for the existence of a perfect difference family with block size 4 and order v, or briefly (v, 4, 1)-PDF, is v ≡ 1(mod 12), and that of an additive sequence of permutations of length 3 and order m, or briefly ASP(3, m), is m ≡ 1(mod 2). So far, (12t +1, 4, 1)-PDFs with t < 50 are known only for t = 1, 4−36, 41, 46 with two definite exceptions of t = 2, 3, and ASP(3, m)'s with odd 3 < m < 200 are known only for m = 5, 7, 13-29, 35, 45, 49, 65, 75, 85, 91, 95, 105, 115, 119, 121, 125, 133, 135, 145, 147, 161, 169, 175, 189, 195 with two definite exceptions of m = 9, 11. In this article, we show that a (12t +1, 4, 1)-PDF exists for any t ≤ 1, 000 except for t = 2, 3, and an ASP(3, m) exists for any odd 3 < m < 200 except for m = 9, 11 and possibly for m = 59. The main idea of this article is to use perfect difference families and additive sequences of permutations with "holes". We first introduce the concepts of an incomplete perfect difference matrix with 415 416 GE, MIAO, AND SUN a regular hole and a perfect difference packing with a regular difference leave, respectively. We show that an additive sequence of permutations is in fact equivalent to a perfect difference matrix, then describe an important recursive construction for perfect difference matrices via perfect difference packings with a regular difference leave. Plenty of perfect difference packings with a desirable difference leave are constructed directly. We also provide a general recursive construction for perfect difference packings, and as its applications, we obtain extensive recursive constructions for perfect difference families, some via incomplete perfect difference matrices with a regular hole. Examples of perfect difference packings directly constructed are used as ingredients in these recursive constructions to produce vast numbers of perfect difference families with block size 4. q 2010 Wiley Periodicals, Inc. J Combin Designs 18: 2010 Keywords: perfect difference family; perfect system of difference sets; perfect difference packing; additive sequence of permutations; perfect difference matrix; semi-perfect group divisible design ‡ Supporting Information may be found in the online version of this article.

One of the most fundamental tasks of wireless sensor networks is to provide coverage of the deployment region. We study the coverage of a line interval with a set of wireless sensors with adjustable coverage ranges. Each coverage range of a sensor is an interval centered at that sensor whose length is decided by the power the sensor chooses. The objective is to find a range assignment with the minimum cost. There are two variants of the optimization problem. In the discrete variant, each sensor can only choose from a finite set of powers, whereas in the continuous variant, each sensor can choose power from a given interval. For the discrete variant of the problem, a polynomial-time exact algorithm is designed. For the continuous variant of the problem, NP-hardness of the problem is proved and followed by an ILP formulation. Then, constant-approximation algorithms are designed when the cost for all sensors is proportional to r κ for some constant κ ≥ 1, where r is the covering radius corresponding to the chosen power. Specifically, if κ = 1, we give a 1.25-approximation algorithm and a fully polynomial-time approximation scheme; if κ > 1, we give a 2-approximation algorithm. We also show that the approximation analyses are tight.

Necessary conditions for the existence of a super‐simple, decomposable, near‐resolvable ( v , 4 , 6 )‐balanced incomplete block design (BIBD) whose 2‐component subdesigns are both near‐resolvable ( v , 4 , 3 )‐BIBDs are v ≡ 1 (mod 4) and v ≥ 17. In this paper, we show that these necessary conditions are sufficient. Using these designs, we also establish that the necessary conditions for the existence of a super‐simple near‐resolvable ( v , 4 , 3 )‐RBIBD, namely v ≡ 1 (mod 4) and v ≥ 9, are sufficient. A few new pairwise balanced designs are also given.

Necessary conditions for the existence of a decomposable super‐simple resolvable (v,4,6)‐BIBD whose two component subdesigns are both resolvable (v,4,3)‐BIBDs are v≡0 (mod 4) and v≥16. In this paper, it is proved that these necessary conditions are sufficient, except possibly for v∈{268,284,292,296}.

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