Valid Orderings of Real Hyperplane Arrangements

Given a real finite hyperplane arrangement A and a point p not on any of the hyperplanes, we define an arrangement vo(A,p), called the *valid order arrangement*, whose regions correspond to the different orders in which a line through p can cross the hyperplanes in A. If A is the set of affine spans of the facets of a convex polytope P and p lies in the interior of P, then the valid orderings with respect to p are just the line shellings of p where the shelling line contains p. When p is sufficiently generic, the intersection lattice of vo(A,p) is the *Dilworth truncation* of the semicone of A. Various applications and examples are given. For instance, we determine the maximum number of line shellings of a d-polytope with m facets when the shelling line contains a fixed point p. If P is the order polytope of a poset, then the sets of facets visible from a point involve a generalization of chromatic polynomials related to list colorings.Comment: 15 pages, 2 figure

Valid Orderings of Real Hyperplane Arrangements

Given a real finite hyperplane arrangement A and a point p not on any of the hyperplanes, we define an arrangement vo(A,p), called the valid order arrangement, whose regions correspond to the different orders in which a line through p can cross the hyperplanes in A. If A is the set of affine spans of the facets of a convex polytope P and p lies in the interior of P, then the valid orderings with respect to p are just the line shellings of P where the shelling line contains p. When p is sufficiently generic, the intersection lattice of vo(A,p) is the Dilworth truncation of the semicone of A. Various applications and examples are given. For instance, we determine the maximum number of line shellings of a d-polytope with m facets when the shelling line contains a fixed point p. If P is the order polytope of a poset, then the sets of facets visible from a point involve a generalization of chromatic polynomials related to list colorings.National Science Foundation (U.S.) (Grant DMS-1068625

Supersolvability and Freeness for ψ-Graphical Arrangements

Let G be a simple graph on the vertex set {v[subscript 1],…,v[subscript n]} with edge set E. Let K be a field. The graphical arrangement A[subscript G] in K[superscript n] is the arrangement x[subscript i]−x[subscript j]=0,v[subscript i]v[subscript j] ∈ E. An arrangement A is supersolvable if the intersection lattice L(c(A)) of the cone c(A) contains a maximal chain of modular elements. The second author has shown that a graphical arrangement A[subscript G] is supersolvable if and only if G is a chordal graph. He later considered a generalization of graphical arrangements which are called ψ-graphical arrangements. He conjectured a characterization of the supersolvability and freeness (in the sense of Terao) of a ψ-graphical arrangement. We provide a proof of the first conjecture and state some conditions on free ψ-graphical arrangements.China Scholarship CouncilNational Science Foundation (U.S.) (Grant DMS-1068625

Matroids arising from electrical networks

This paper introduces Dirichlet matroids, a generalization of graphic matroids arising from electrical networks. We present four main results. First, we exhibit a matroid quotient formed by the dual of a network embedded in a surface with boundary and the dual of the associated Dirichlet matroid. This generalizes an analogous result for graphic matroids of cellularly embedded graphs. Second, we characterize the Bergman fans of Dirichlet matroids as explicit subfans of graphic Bergman fans. In doing so, we generalize the connection between Bergman fans of complete graphs and phylogenetic trees. Third, we use the half-plane property of Dirichlet matroids to prove an interlacing result on the real zeros and poles of the trace of the response matrix. And fourth, we bound the coefficients of the precoloring polynomial of a network by the coefficients of the chromatic polynomial of the underlying graph.Comment: 27 pages, 14 figure

Fair allocation of a multiset of indivisible items

We study the problem of fairly allocating a multiset $M$ of $m$ indivisible items among $n$ agents with additive valuations. Specifically, we introduce a parameter $t$ for the number of distinct types of items and study fair allocations of multisets that contain only items of these $t$ types, under two standard notions of fairness: 1. Envy-freeness (EF): For arbitrary $n$, $t$, we show that a complete EF allocation exists when at least one agent has a unique valuation and the number of items of each type exceeds a particular finite threshold. We give explicit upper and lower bounds on this threshold in some special cases. 2. Envy-freeness up to any good (EFX): For arbitrary $n$, $m$, and for $t\le 2$, we show that a complete EFX allocation always exists. We give two different proofs of this result. One proof is constructive and runs in polynomial time; the other is geometrically inspired.Comment: 34 pages, 6 figures, 1 table, 1 algorith

Valid Orderings of Real Hyperplane Arrangements

Given a real finite hyperplane arrangement $$\mathcal {A}$$ A and a point $$p$$ p not on any of the hyperplanes, we define an arrangement $$\mathrm {vo}(\mathcal {A},p)$$ vo ( A , p ) , called the valid order arrangement, whose regions correspond to the different orders in which a line through $$p$$ p can cross the hyperplanes in $$\mathcal {A}$$ A . If $$\mathcal {A}$$ A is the set of affine spans of the facets of a convex polytope $$\mathcal {P}$$ P and $$p$$ p lies in the interior of $$\mathcal {P}$$ P , then the valid orderings with respect to $$p$$ p are just the line shellings of $$\mathcal {P}$$ P where the shelling line contains $$p$$ p . When $$p$$ p is sufficiently generic, the intersection lattice of $$\mathrm {vo}(\mathcal {A},p)$$ vo ( A , p ) is the Dilworth truncation of the semicone of $$\mathcal {A}$$ A . Various applications and examples are given. For instance, we determine the maximum number of line shellings of a $$d$$ d -polytope with $$m$$ m facets when the shelling line contains a fixed point $$p$$ p . If $$\mathcal {P}$$ P is the order polytope of a poset, then the sets of facets visible from a point involve a generalization of chromatic polynomials related to list colorings