The legal aspects of funding public education through real property taxation : 1971 Serrano to the present

All states use the tax on real property to some degree in the financing of public elementary and secondary education. The problem of this usage stems largely from the local control of the collection and disbursement of the monies by the individual school districts. This method is claimed by some to create inequities whereby the quality of available educational opportunity rests on the wealth of the school district in which the student resides. Reformers have sought relief from these alleged inequities in both the United States Constitution's Fourteenth Amendment, and also in the equal protection provisions of the various state constitutions. The data for this study are contained primarily in significant court cases from 1968 to the present. Additional data have been collected through a review of the literature, which intensified in quantity primarily during the interim between Serrano v. Priest in 1971 and Rodriguez v. San Antonio Independent School District in 1973

The structure of optimal solutions to the submodular

function minimization proble

Independence and port oracles for matroids, with an application to computational learning theory

Given a matroid M with distinguished element e, a port oracle with respect to e reports whether or not a given subset contains a circuit that contains e. The first main result of this paper is an algorithm for computing an e-based ear decomposition (that is, an ear decomposition every circuit of which contains element e) of a matroid using only a polynomial number of elementary operations and port oracle calls. In the case that M is binary, the incidence vectors of the circuits in the ear decomposition form a matrix representation for M. Thus, this algorithm solves a problem in computational learning theory; it learns the class of binary matroid port (BMP) functions with membership queries in polynomial time. In this context, the algorithm generalizes results of Angluin, Hellerstein, and Karpinski [1], and Raghavan and Schach [17], who showed that certain subclasses of the BMP functions are learnable in polynomial time using membership queries. The second main result of this paper is an algorithm for testing independence of a given input set of the matroid M. This algorithm, which uses the ear decomposition algorithm as a subroutine, uses only a polynomial number of elementary operations and port oracle calls. The algorithm proves a constructive version of an early theorem of Lehman [13], which states that the port of a connected matroid uniquely determines the matroid

On cycle cones and polyhedra

AbstractGiven an undirected graph G and a cost associated with each edge, the weighted girth problem is to find a simple cycle of G having minimum total cost. We consider several variants of the weighted girth problem, some of which are NP-hard and some of which are solvable in polynomial time. We also consider the polyhedra associated with each of these problems. Two of these polyhedra are the cycle cone of G, which is the cone generated by the incidence vectors of cycles of G, and the cycle polytope of G, which is the convex hull of the incidence vectors of cycles of G. First we give a short proof of Seymour's characterization of the cycle cone of G. Next we give a polyhedral composition result for the cycle polytope of G. In particular, we prove that if G decomposes via a 3-edge cut into graphs G1 and G2, say, then defining linear systems for the cycle polytopes of G1 and G2 can be combined in a certain way to obtain a defining linear system for the cycle polytope of G. We also describe a polynomial decomposition-based algorithm for the weighted girth problem on Halin graphs, and we give a complete linear description for the cycle polytope of G, in the case G is a Halin graph

On chains of 3-connected matroids

AbstractA sequence of k-connected matroids N0, N1,…,Nm is called a k-chain, from N0 to Nm, if Ni−1 is a minor of Ni(i=1,…,m); this chain is said to have gap t = max {|E(Ni)| − |E(Ni−1)|: i = 1,…,m}. Chains of gap 1 are said to be dense.If M and N are 3-connected matroids, M is not a wheel or whirl, |E(N)| ⩾ 4 and N is a minor of M, then there is a dense 3-chain from N′ to M where N′ is isomorphic to N. For graphs this is a theorem of Negami, and for matroids a theorem of Seymour and Tan. Truemper has proved that for M and N 3-connected and N a minor of M, if we do not allow the insertion of an isomorphic copy for N, then there is always a 3-chain from N to M of gap at most 3. We investigate the structure of these chains, and show that if N has no circuits or cocircuits with 3 or fewer elements, then there is a 3-chain of gap at most 2

Extensions of Tutte's wheels-and-whirls theorem

AbstractTutte's wheels-and-whirls theorem states that if M is a 3-connected matroid and, for every element e, both the deletion and the contraction of e destroy 3-connectivity, then M is a wheel or a whirl. We prove some extensions of this theorem, one of which states that if M is 3-connected and has both a wheel and a whirl minor, then either M has only seven elements or there is some element the deletion or contraction of which maintains 3-connectivity and leaves a matroid with both a wheel and a whirl minor