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The circuits of a polyhedron are a superset of its edge directions. Circuit walks, a sequence of steps along circuits, generalize edge walks and are "short" if they have few steps or small total length. Both interpretations of short are relevant to the theory and application of linear programming. We study the hardness of several problems relating to the construction of short circuit walks. We establish that for a pair of vertices of a $0/1$-network-flow polytope, it is NP-complete to determine the length of a shortest circuit walk, even if we add the requirement that the walk must be sign-compatible. Our results also imply that determining the minimal number of circuits needed for a sign-compatible decomposition is NP-complete. Further, we show that it is NP-complete to determine the smallest total length (for $p$-norms $\lVert \cdot \rVert_p$, $1 < p \leq \infty$) of a circuit walk between a pair of vertices. One method to construct a short circuit walk is to pick up a correct facet at each step, which generalizes a non-revisiting walk. We prove that it is NP-complete to determine if there is a circuit direction that picks up a correct facet; in contrast, this problem can be solved in polynomial time for TU polyhedra.

The simplex algorithm is one of the most popular algorithms to solve linear programs (LPs). Starting at an extreme point solution of an LP, it performs a sequence of basis exchanges (called pivots) that allows one to move to a better extreme point along an improving edge-direction of the underlying polyhedron. A key issue in the simplex algorithm's performance is degeneracy, which may lead to a (potentially long) sequence of basis exchanges which do not change the current extreme point solution. In this paper, we prove that it is always possible to limit the number of consecutive degenerate pivots that the simplex algorithm performs to $n-m-1$, where $n$ is the number of variables and $m$ is the number of equality constraints of a given LP in standard equality form., Comment: 18 pages, 2 figures

We present new pivot rules for the Simplex method for LPs over 0/1 polytopes. We show that the number of non-degenerate steps taken using these rules is strongly polynomial and even linear in the dimension or in the number of variables. Our bounds on the number of steps are asymptotically optimal on several well-known combinatorial polytopes. Our analysis is based on the geometry of 0/1 polytopes and novel modifications to the classical Steepest-Edge and Shadow-Vertex pivot rules. We draw interesting connections between our pivot rules and other well-known algorithms in combinatorial optimization., Comment: 32 Pages, 3 figures

The stable marriage problem with ties is a well-studied and interesting problem in game theory. We are given a set of men and a set of women. Each individual has a preference ordering on the opposite group, which can possibly contain ties. A stable marriage is given by a matching between men and women for which there is no blocking pair, i.e., a men and a women who strictly prefer each other to their current partner in the matching. In this paper, we study the diameter of the polytope given by the convex hull of characteristic vectors of stable marriages, in the setting with ties. We prove an upper bound of $\lfloor \frac{n}{3}\rfloor$ on the diameter, where $n$ is the total number of men and women, and give a family of instances for which the bound holds tight. Our result generalizes the bound on the diameter of the standard stable marriage polytope (i.e., the well-known polytope that describes the setting without ties), developed previously in the literature.

Circuit-augmentation algorithms are generalizations of the Simplex method, where in each step one is allowed to move along a fixed set of directions, called circuits, that is a superset of the edges of a polytope. We show that in the circuit-augmentation framework the greatest-improvement and Dantzig pivot rules are NP-hard, already for 0/1-LPs. Differently, the steepest-descent pivot rule can be carried out in polynomial time in the 0/1 setting, and the number of circuit augmentations required to reach an optimal solution according to this rule is strongly-polynomial for 0/1-LPs. The number of circuit augmentations has been of interest as a proxy for the number of steps in the Simplex method, and the circuit-diameter of polyhedra has been studied as a lower bound to the combinatorial diameter of polyhedra. Extending prior results, we show that for any polyhedron $P$ the circuit-diameter is bounded by a polynomial in the input bit-size of $P$. This is in contrast with the best bounds for the combinatorial diameter of polyhedra. Interestingly, we show that the circuit-augmentation framework can be exploited to make novel conclusions about the classical Simplex method itself: In particular, as a byproduct of our circuit results, we prove that (i) computing the shortest (monotone) path to an optimal solution on the 1-skeleton of a polytope is NP-hard, and hard to approximate within a factor better than 2, and (ii) for $0/1$ polytopes, a monotone path of strongly-polynomial length can be constructed using steepest improving edges.

The (combinatorial) diameter of a polytope $P \subseteq \mathbb R^d$ is the maximum value of a shortest path between a pair of vertices on the 1-skeleton of $P$, that is the graph where the nodes are given by the $0$-dimensional faces of $P$, and the edges are given the 1-dimensional faces of $P$. The diameter of a polytope has been studied from many different perspectives, including a computational complexity point of view. In particular, [Frieze and Teng, 1994] showed that computing the diameter of a polytope is (weakly) NP-hard. In this paper, we show that the problem of computing the diameter is strongly NP-hard even for a polytope with a very simple structure: namely, the \emph{fractional matching} polytope. We also show that computing a pair of vertices at maximum shortest path distance on the 1-skeleton of this polytope is an APX-hard problem. We prove these results by giving an \emph{exact characterization} of the diameter of the fractional matching polytope, that is of independent interest.

The combinatorial diameter of a polytope $P$ is the maximum value of a shortest path between two vertices of $P$, where the path uses the edges of $P$ only. In contrast to the combinatorial diameter, the circuit diameter of $P$ is defined as the maximum value of a shortest path between two vertices of $P$, where the path uses potential edge directions of $P$ i.e., all edge directions that can arise by translating some of the facets of $P$. In this paper, we study the circuit diameter of polytopes corresponding to classical combinatorial optimization problems, such as the Matching polytope, the Traveling Salesman polytope and the Fractional Stable Set polytope.

In a standard $f$-connectivity network design problem, we are given an undirected graph $G=(V,E)$, a cut-requirement function $f:2^V \rightarrow {\mathbb{N}}$, and non-negative costs $c(e)$ for all $e \in E$. We are then asked to find a minimum-cost vector $x \in {\mathbb{N}}^E$ such that $x(\delta(S)) \geq f(S)$ for all $S \subseteq V$. We focus on the class of such problems where $f$ is a proper function. This encodes many well-studied NP-hard problems such as the generalized survivable network design problem. In this paper we present the first strongly polynomial time FPTAS for solving the LP relaxation of the standard IP formulation of the $f$-connectivity problem with general proper functions $f$. Implementing Jain's algorithm, this yields a strongly polynomial time $(2+\epsilon)$-approximation for the generalized survivable network design problem (where we consider rounding up of rationals an arithmetic operation).

We show that there exists a family of Knapsack polytopes such that, for each polytope P from this family and each {\epsilon} > 0, any {\epsilon}-approximated formulation of P in the original space R^n requires a number of inequalities that is super-polynomial in n. This answers a question by Bienstock and McClosky (2012). We also prove that, for any down-monotone polytope, an {\epsilon}-approximated formulation in the original space can be obtained with inequalities using at most O(min{log(n/{\epsilon}),n}/{\epsilon}) different coefficients.

The symmetric Virtual Private Network Design (VPND) problem is concerned with buying capacity on links (edges) in a communication network such that certain traffic demands can be met. We investigate a natural generalization of VPND where the cost per unit of capacity may decrease if a larger amount of capacity is reserved (economies of scale principle). The growth of the cost of capacity is modelled by a non-decreasing concave function $f$. We call the problem the concave symmetric Virtual Private Network Design (cVPND) problem. After showing that a generalization of the so-called Pyramidal Routing problem and hence also the cVPND have the so-called tree routing property, we study approximation algorithms for cVPND. For general $f$, using known results on the so-called Single Source Buy at Bulk problem by Grandoni and Italiano, we give a randomized 24.92-approximation algorithm., Comment: Oberwolfach abstract

The VPN Tree Routing Conjecture is a conjecture about the Virtual Private Network Design problem. It states that the symmetric version of the problem always has an optimum solution which has a tree-like structure. In recent work, Hurkens, Keijsper and Stougie (Proc. IPCO XI, 2005; SIAM J. Discrete Math., 2007) have shown that the conjecture holds when the network is a ring. A shorter proof of the VPN Conjecture for rings was found a few months ago by Grandoni, Kaibel, Oriolo and Skutella (to appear in Oper. Res. Lett., 2008). In their paper, Grandoni et al. introduce another conjecture, called the Pyramidal Routing Conjecture (or simply PR Conjecture), which implies the VPN Conjecture. Here we consider a strengthened version of the PR Conjecture. First we establish several general tools which can be applied in arbitrary networks. Then we use them to prove that outerplanar networks satisfy the PR Conjecture.