Vectors in a Box

For an integer d>=1, let tau(d) be the smallest integer with the following property: If v1,v2,...,vt is a sequence of t>=2 vectors in [-1,1]^d with v1+v2+...+vt in [-1,1]^d, then there is a subset S of {1,2,...,t} of indices, 2<=|S|<=tau(d), such that \sum_{i\in S} vi is in [-1,1]^d. The quantity tau(d) was introduced by Dash, Fukasawa, and G\"unl\"uk, who showed that tau(2)=2, tau(3)=4, and tau(d)=Omega(2^d), and asked whether tau(d) is finite for all d. Using the Steinitz lemma, in a quantitative version due to Grinberg and Sevastyanov, we prove an upper bound of tau(d) <= d^{d+o(d)}, and based on a construction of Alon and Vu, whose main idea goes back to Hastad, we obtain a lower bound of tau(d)>= d^{d/2-o(d)}. These results contribute to understanding the master equality polyhedron with multiple rows defined by Dash et al., which is a "universal" polyhedron encoding valid cutting planes for integer programs (this line of research was started by Gomory in the late 1960s). In particular, the upper bound on tau(d) implies a pseudo-polynomial running time for an algorithm of Dash et al. for integer programming with a fixed number of constraints. The algorithm consists in solving a linear program, and it provides an alternative to a 1981 dynamic programming algorithm of Papadimitriou.Comment: 12 pages, 1 figur

Oriented Open-Closed String Theory Revisited

String theory on D-brane backgrounds is open-closed string theory. Given the relevance of this fact, we give details and elaborate upon our earlier construction of oriented open-closed string field theory. In order to incorporate explicitly closed strings, the classical sector of this theory is open strings with a homotopy associative A_\infty algebraic structure. We build a suitable Batalin-Vilkovisky algebra on moduli spaces of bordered Riemann surfaces, the construction of which involves a few subtleties arising from the open string punctures and cyclicity conditions. All vertices coupling open and closed strings through disks are described explicitly. Subalgebras of the algebra of surfaces with boundaries are used to discuss symmetries of classical open string theory induced by the closed string sector, and to write classical open string field theory on general closed string backgrounds. We give a preliminary analysis of the ghost-dilaton theorem.Comment: 61 pages, 19 eps figures, phyzzx, BoxedEP

Fibrator Properties of PL Manifolds

In the early 90s, R.Daverman defined the concept of the PL fibrator ([12]). PL fibrators, by definition, provide detection of PL approximate fibrations. Daver- man defines a closed, connected, orientable PL n-manifold to be a codimension-k PL orientable fibrator if for all closed, connected, orientable PL (n + k)-manifolds M and PL maps p : M → B, where B is a polyhedron, such that each fiber collapses to an n-complex homotopy equivalent to Nn, p is always an approximate fibration. If N is a codimension-k PL orientable fibrator for all k \u3e 0, N is called a PL orientable fibrator. Until now only a few classes of manifolds are known not to be PL fibrators. Following this concept of Daverman, in this dissertation we attempt to find to what extent such results can be obtained for PL maps p : Mn+k → B between manifolds, such that each fiber has the homotopy type (or more generally the shape) of N, but does not necessarily collapse to an n-complex, which is a severe restriction. Here we use the following slightly changed PL setting: M is a closed, connected, orientable PL (n + k)-manifold, B is a simplicial triangulated manifold (not necessarily PL), p : Mn+k → B a PL proper, surjective map, and N a fixed closed, connected, orientable PL n-manifold. We call N a codimension-k shape msimplo-fibrator if for all orientable, PL (n + k)-manifolds Mn+k and PL maps p : Mn+k → B, such that each fiber is homotopy equivalent to Nn, p is always an approximate fibration. If N is a codimension-k shape msimplo-fibrator for all k \u3e 0, N is a shape msimplo-fibrator. We are interested in PL manifolds N with π1(N ) /= 1, that force every map f : N → N , with 1 /= f# (π1 (N )) π1 (N ), to be a homotopy equivalence. We call PL manifolds N with this property special manifolds. There is a similar group theoretic term: a group G is super Hopfian if every homomorphism φ : G → Gwith 1 /= φ(G) In the first part of the dissertation we study which groups posses this property of being super Hopfian. We find that every non-abelian group of the order pq where p, q are distinct primes is super Hopfian. Also, a free product of non-trivial, finitely generated, residually finite groups at least one of which is not Z2 is super Hopfian. Then we give an example of special manifolds to which we apply our main results in the second part of this dissertation. First we prove that all orientable, special manifolds N with non-cyclic fun- damental groups are codimension-2 shape msimpl o-fibrators. Then we find which 3- manifolds have this property. Next we prove which manifolds are codimension-4 shape msimpl o-fibrators. Our main result gives that an orientable, special PL n-manifold N with a non-trivial first homology group is a shape msimplo-fibrators if N is a codimension-2 shape msimplo-fibrator. The condition of N being a codimension-2 PL shape orientable fibrator can be replaced with N having a non-cyclic fundamental group. In the last section we list some open questions

Deformations of bordered Riemann surfaces and associahedral polytopes

We consider the moduli space of bordered Riemann surfaces with boundary and marked points. Such spaces appear in open-closed string theory, particularly with respect to holomorphic curves with Lagrangian submanifolds. We consider a combinatorial framework to view the compactification of this space based on the pair-of-pants decomposition of the surface, relating it to the well-known phenomenon of bubbling. Our main result classifies all such spaces that can be realized as convex polytopes. A new polytope is introduced based on truncations of cubes, and its combinatorial and algebraic structures are related to generalizations of associahedra and multiplihedra.Comment: 25 pages, 31 figure

Reformulation and decomposition of integer programs

In this survey we examine ways to reformulate integer and mixed integer programs. Typically, but not exclusively, one reformulates so as to obtain stronger linear programming relaxations, and hence better bounds for use in a branch-and-bound based algorithm. First we cover in detail reformulations based on decomposition, such as Lagrangean relaxation, Dantzig-Wolfe column generation and the resulting branch-and-price algorithms. This is followed by an examination of Benders’ type algorithms based on projection. Finally we discuss in detail extended formulations involving additional variables that are based on problem structure. These can often be used to provide strengthened a priori formulations. Reformulations obtained by adding cutting planes in the original variables are not treated here.Integer program, Lagrangean relaxation, column generation, branch-and-price, extended formulation, Benders' algorithm