My publications:

• Wildstrom, D. Jacob. Cost thresholds for dynamic resource location. Discrete Appl. Math. 156 (2008), no. 10, 1846–1855. (preprint, MR2432946, doi:10.1016/j.dam.2007.09.002)
• Wildstrom, D. Jacob. Resource relocation on asymmetric networks. J. Graph Algorithms Appl. 14 (2010), no. 2, 149–163. (paper)
• Wildstrom, D. Jacob. Dynamic resource location with tropical algebra. Linear Algebra Appl. in press. (doi:10.1016/j.laa.2010.03.028)

I have a number of ongoing research projects. This is a partial list of problems I'm currently engaged in. If you're interested in learning more about any of them, write to me.

Realtime resource relocation

Service providers on communication, transportation, and any other sort of network may relocate in response to changing conditions. For instance, a mobile radio transmitter may be better served by moving physically closer to its clients, or a software construct might provide faster and less expensive service by being reinstalled on a new system as its users' locations shift. A significant question is when such relocations become expedient, given that there are costs associated with both relocation and long-distance service.

The mathematical approach to this problem under consideration is been to consider a finite set of possible client and server locations and costs associated with both relocation and remote service. Then, given an initial server location and a queue of client-requests to serve, there is clearly a selection of relocations to minimize the overall cost of providing service. This can be determined through (highly inefficient) linear programming. Moving beyond the question of perfect optimization with flawless knowledge of the entire request sequence, two fundamental questions (which raise ancillary questions with regard to network architecture and business practice) arise:

• Given imperfect future knowledge, i.e. only the next k requests in an arbitrarily long sequence of requests, is it possible to make a relocation decision which leads to optimal service provision?
• When it is not possible to provide optimal service in the situation described above, what is the ratio between the costs of implementing a well-designed strategy based on limited information and the costs of an omnisciently optimized strategy?

Useful References:

• F. R. K. Chung, R. L. Graham, and M. E. Saks. Dynamic search in graphs. In Discrete algorithms and complexity (Kyoto, 1986), volume 15 of Perspect. Comput., pages 351–387. Academic Press, Boston, MA, 1987.
• F. R. K. Chung, R. L. Graham, and M. E. Saks. A dynamic location problem for graphs. Combinatorica, 9(2):111–131, 1989.
• F. Chung and R. Graham. Dynamic location problems with limited look-ahead. Theoret. Comput. Sci., 261(2):213–226, 2001. Computing and combinatorics (Taipei, 1998).
• R. Cuninghame-Green. Minimax algebra, volume 166 of Lecture Notes in Economics and Mathematical Systems. Springer-Verlag, Berlin, 1979.
• E. Koutsoupias and C. H. Papadimitriou. On the k-server conjecture. J. Assoc. Comput. Mach., 42(5):971–983, 1995.

Permutations of k-tuples with distinct sums

It was shown by Marshall Hall that, given an abelian group of order $n$ and (not necessarily distinct) elements a₁,..., a_n thereof, there is a permutation b₁,..., b_n of elements of the group such that each a_i+b_i is distinct if and only if the a_i add up to zero.

A question which is somewhat similar to this field of inquiry is as follows: if we are given k elements a₁,..., a_k of Z_n, are we guaranteed that there is a permutation σ of 1,...,k so that the sums a_i+σ_i are all distinct modulo n, and if so how many such σ are there? Such a selection is clearly impossible if k>n. Hall's result shows such a selection of σ is not necessarily possible if k=n, but that it always is if k=n-1.

Kézdy and Snevily have conejctured that such a σ always exists if k<n; and furthermore, that the number of such σ is nonstrictly increasing in as a function of n, and nonstrictily increasing as a function of k for k<n.

Useful References:

• N. Alon, Additive Latin transversals. Israel J. Math. 117 (2000), 125–130.
• S. Dasgupta; G. Károlyi; O. Serra; B. Szegedy, Transversals of additive Latin squares, Israel J. Math. 126 (2001), 17–28.
• M. Hall, A combinatorial problem on abelian groups, Proc. Amer. Math. Soc. 3 (1952), 584–587.
• A. Kézdy and H. Snevily, Distinct Sums Modulo n and Tree Embeddings, Combin. Probab. Comput. 11 (2002), 35–42.
• H. Snevily, The Cayley Addition Table of Z_n, Amer. Math Monthly 106 (1999), 584–585.
• Z. Sun, On Snevily's conjecture and restricted sumsets, J. Combin. Theory, Ser. A 103 (2003), 291–304.