Counting Unique-Sink Orientations

Unique-sink orientations (USOs) are an abstract class of orientations of the n-cube graph. We consider some classes of USOs that are of interest in connection with the linear complementarity problem. We summarise old and show new lower and upper bounds on the sizes of some such classes. Furthermore, we provide a characterisation of K-matrices in terms of their corresponding USOs.Comment: 13 pages; v2: proof of main theorem expanded, plus various other corrections. Now 16 pages; v3: minor correction

The Niceness of Unique Sink Orientations

Random Edge is the most natural randomized pivot rule for the simplex algorithm. Considerable progress has been made recently towards fully understanding its behavior. Back in 2001, Welzl introduced the concepts of \emph{reachmaps} and \emph{niceness} of Unique Sink Orientations (USO), in an effort to better understand the behavior of Random Edge. In this paper, we initiate the systematic study of these concepts. We settle the questions that were asked by Welzl about the niceness of (acyclic) USO. Niceness implies natural upper bounds for Random Edge and we provide evidence that these are tight or almost tight in many interesting cases. Moreover, we show that Random Edge is polynomial on at least $n^{\Omega(2^n)}$ many (possibly cyclic) USO. As a bonus, we describe a derandomization of Random Edge which achieves the same asymptotic upper bounds with respect to niceness and discuss some algorithmic properties of the reachmap.Comment: An extended abstract appears in the proceedings of Approx/Random 201

Enumeration of PLCP-orientations of the 4-cube

The linear complementarity problem (LCP) provides a unified approach to many problems such as linear programs, convex quadratic programs, and bimatrix games. The general LCP is known to be NP-hard, but there are some promising results that suggest the possibility that the LCP with a P-matrix (PLCP) may be polynomial-time solvable. However, no polynomial-time algorithm for the PLCP has been found yet and the computational complexity of the PLCP remains open. Simple principal pivoting (SPP) algorithms, also known as Bard-type algorithms, are candidates for polynomial-time algorithms for the PLCP. In 1978, Stickney and Watson interpreted SPP algorithms as a family of algorithms that seek the sink of unique-sink orientations of $n$-cubes. They performed the enumeration of the arising orientations of the $3$-cube, hereafter called PLCP-orientations. In this paper, we present the enumeration of PLCP-orientations of the $4$-cube.The enumeration is done via construction of oriented matroids generalizing P-matrices and realizability classification of oriented matroids.Some insights obtained in the computational experiments are presented as well

Generic method for bijections between blossoming trees and planar maps

This article presents a unified bijective scheme between planar maps and blossoming trees, where a blossoming tree is defined as a spanning tree of the map decorated with some dangling half-edges that enable to reconstruct its faces. Our method generalizes a previous construction of Bernardi by loosening its conditions of applications so as to include annular maps, that is maps embedded in the plane with a root face different from the outer face. The bijective construction presented here relies deeply on the theory of \alpha-orientations introduced by Felsner, and in particular on the existence of minimal and accessible orientations. Since most of the families of maps can be characterized by such orientations, our generic bijective method is proved to capture as special cases all previously known bijections involving blossoming trees: for example Eulerian maps, m-Eulerian maps, non separable maps and simple triangulations and quadrangulations of a k-gon. Moreover, it also permits to obtain new bijective constructions for bipolar orientations and d-angulations of girth d of a k-gon. As for applications, each specialization of the construction translates into enumerative by-products, either via a closed formula or via a recursive computational scheme. Besides, for every family of maps described in the paper, the construction can be implemented in linear time. It yields thus an effective way to encode and generate planar maps. In a recent work, Bernardi and Fusy introduced another unified bijective scheme, we adopt here a different strategy which allows us to capture different bijections. These two approaches should be seen as two complementary ways of unifying bijections between planar maps and decorated trees.Comment: 45 pages, comments welcom

Approximating Holant problems by winding

We give an FPRAS for Holant problems with parity constraints and not-all-equal constraints, a generalisation of the problem of counting sink-free-orientations. The approach combines a sampler for near-assignments of "windable" functions -- using the cycle-unwinding canonical paths technique of Jerrum and Sinclair -- with a bound on the weight of near-assignments. The proof generalises to a larger class of Holant problems; we characterise this class and show that it cannot be extended by expressibility reductions. We then ask whether windability is equivalent to expressibility by matchings circuits (an analogue of matchgates), and give a positive answer for functions of arity three

Sinks in Acyclic Orientations of Graphs

Greene and Zaslavsky proved that the number of acyclic orientations of a graph with a unique sink is, up to sign, the linear coefficient of the chromatic polynomial. We give three new proofs of this result using pure induction, noncommutative symmetric functions, and an algorithmic bijection.Comment: 17 pages, 1 figur