Staffing problems and symmetric integer programs

Issued as Final project report, Project no. E-24-63

Complexity of Manipulative Actions When Voting with Ties

Most of the computational study of election problems has assumed that each voter's preferences are, or should be extended to, a total order. However in practice voters may have preferences with ties. We study the complexity of manipulative actions on elections where voters can have ties, extending the definitions of the election systems (when necessary) to handle voters with ties. We show that for natural election systems allowing ties can both increase and decrease the complexity of manipulation and bribery, and we state a general result on the effect of voters with ties on the complexity of control.Comment: A version of this paper will appear in ADT-201

Branch Rings, Thinned Rings, Tree Enveloping Rings

We develop the theory of ``branch algebras'', which are infinite-dimensional associative algebras that are isomorphic, up to taking subrings of finite codimension, to a matrix ring over themselves. The main examples come from groups acting on trees. In particular, for every field k we construct a k-algebra K which (1) is finitely generated and infinite-dimensional, but has only finite-dimensional quotients; (2) has a subalgebra of finite codimension, isomorphic to $M_2(K)$; (3) is prime; (4) has quadratic growth, and therefore Gelfand-Kirillov dimension 2; (5) is recursively presented; (6) satisfies no identity; (7) contains a transcendental, invertible element; (8) is semiprimitive if k has characteristic $\neq2$; (9) is graded if k has characteristic 2; (10) is primitive if k is a non-algebraic extension of GF(2); (11) is graded nil and Jacobson radical if k is an algebraic extension of GF(2).Comment: 35 pages; small changes wrt previous versio

The best shape for a crossdock

The article of record as published may be found at http://dx.doi.org/10.10.1287/trsc.1030.0077Within both retail distribution and less-than-truckload transportation networks crossdocks vary greatly in shape. Docks in the shape of an I, L, or T are most common, but unusual ones may be found, including those in the shape of a U, H, or E. Is there a best shape? We show that the answer depends on the size of the facility and on the pattern of freight flows inside. Our results suggest that many large crossdocks in practice suffer from poor design that increases labor costs on the dock.Office of Naval ResearchNational Science FoundationN00014-95-1-0380 (ONR)DMI-0008313 (NSF)N00014-00-WR-20244 (ONR

Swap Bribery

In voting theory, bribery is a form of manipulative behavior in which an external actor (the briber) offers to pay the voters to change their votes in order to get her preferred candidate elected. We investigate a model of bribery where the price of each vote depends on the amount of change that the voter is asked to implement. Specifically, in our model the briber can change a voter's preference list by paying for a sequence of swaps of consecutive candidates. Each swap may have a different price; the price of a bribery is the sum of the prices of all swaps that it involves. We prove complexity results for this model, which we call swap bribery, for a broad class of election systems, including variants of approval and k-approval, Borda, Copeland, and maximin.Comment: 17 page

Combinatorial Voter Control in Elections

Voter control problems model situations such as an external agent trying to affect the result of an election by adding voters, for example by convincing some voters to vote who would otherwise not attend the election. Traditionally, voters are added one at a time, with the goal of making a distinguished alternative win by adding a minimum number of voters. In this paper, we initiate the study of combinatorial variants of control by adding voters: In our setting, when we choose to add a voter~$v$, we also have to add a whole bundle $\kappa(v)$ of voters associated with $v$. We study the computational complexity of this problem for two of the most basic voting rules, namely the Plurality rule and the Condorcet rule.Comment: An extended abstract appears in MFCS 201

Thurston equivalence of topological polynomials

We answer Hubbard's question on determining the Thurston equivalence class of ``twisted rabbits'', i.e. images of the ``rabbit'' polynomial under n-th powers of the Dehn twists about its ears. The answer is expressed in terms of the 4-adic expansion of n. We also answer the equivalent question for the other two families of degree-2 topological polynomials with three post-critical points. In the process, we rephrase the questions in group-theoretical language, in terms of wreath recursions.Comment: 40 pages, lots of figure

From self-similar groups to self-similar sets and spectra

The survey presents developments in the theory of self-similar groups leading to applications to the study of fractal sets and graphs, and their associated spectra

Representing Terrain With Mathematical Operators

This work describes a mathematical representation of terrain data consisting of a novel operation, the “drill”. It facilitates the representation of legal terrains, capturing the richness of the physics of the terrain’s generation by digging channels in the surface. Given our current reliance on digital map data, hand-held devices, and GPS navigation systems, the accuracy and compactness of terrain data representations are becoming increasingly important. Representing a terrain as a series of operations that can procedurally regenerate the terrains allows for compact representation that retains more information than height fields, TINs, and other popular representations. Our model relies on the hydrography information extracted from the terrain, and so drainage information is retained during encoding. To determine the shape of the drill along each channel in the channel network, a cross section of the channel is extracted, and a quadratic polynomial is fit to it. We extract the drill representation from a mountainous dataset, using a series of parameters (including size and area of influence of the drill, as well as the density of the hydrography data), and present the accuracy calculated using a series of metrics. We demonstrate that the drill operator provides a viable and accurate terrain representation that captures both the terrain shape and the richness of its generation