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This note describes six apparently new regular compounds of polytopes in \(\mathbb {E}^4\), that were missed by Coxeter in his systematic faceting procedure. In fact, three have the same symbols as ones in Coxeter’s list (and two are nearly the same); however, their symmetry groups are much smaller. The treatment relies heavily on the use of quaternions.

The “tensorization trick” consists in proving some geometric result for a set of vectors \(\left\{ v_i\right\} \) in some vector space V and then applying the same result to the tensor powers \(\left\{ v_i^{\otimes k}\right\} \) in \(V^{\otimes k}\), which in turn produces a considerably stronger version of the original result for vectors \(\left\{ v_i\right\} \). Our main examples concern packing vectors in the sphere, approximation of convex bodies by algebraic hypersurfaces and approximation of convex bodies by polytopes. We also discuss applications of a closely related polynomial method to constructing neighborly polytopes, bounding the Grothendieck constant, proving the polynomial ham sandwich theorem, bounding the number of equiangular lines in \({\mathbb R}^d\) and to constructing a counterexample to Borsuk’s conjecture.

A class of differential inclusion systems – the linear polytope systems – is discussed in this chapter. This kind of system can be viewed as another extension of the linear control systems to the set-valued mappings. This chapter contains four sections. In the first section, we present the definition of the linear polytope system and the motivation of investigation. Section 4.2 deals with the convex hull Lyapunov function which is the main tool in this chapter. Then Sect. 4.3 considers the control of the linear polytope system. We apply the conclusions of Sect. 4.3 to deal with saturated control at the last section of this chapter.

In many games, players' decisions consist of multiple sub-decisions, and hence can give rise to an exponential number of pure strategies. However, this set of pure strategies is often structured, allowing it to be represented compactly, as in network congestion games, security games, and extensive form games. Reduction to the standard normal form generally introduces exponential blow-up in the strategy space and therefore are inefficient for computation purposes. Although individual classes of such games have been studied, there currently exists no general purpose algorithms for computing solutions. equilibrium. To address this, we define multilinear games generalizing all. Informally, a game is multilinear if its utility functions are linear in each player's strategy, while fixing other players' strategies. Thus, if pure strategies, even if they are exponentially many, are vectors in polynomial dimension, then we show that mixed-strategies have an equivalent representation in terms of marginals forming a polytope in polynomial dimension. The canonical representation for multilinear games can still be exponential in the number of players, a typical obstacle in multi-player games. Therefore, it is necessary to assume additional structure that allows computation of certain sub-problems in polynomial time. Towards this, we identify two key subproblems: computation of utility gradients, and optimizing linear functions over strategy polytope. Given a multilinear game, with polynomial time subroutines for these two tasks, we show the following: a We can construct a polynomially computable and continuous fixed-point formulation, and show that its approximate fixed-points are approximate NE. This gives containment of approximate NE computation in PPAD, and settles its complexity to PPAD-complete. b Even though a coarse correlated equilibrium can potentially have exponential representation , through LP duality and a carefully designed separation oracle, we provide a polynomial-time algorithm to compute one with polynomial representation. c We show existence of an approximate NE with support-size logarithmic in the strategy polytope dimensions.

We discuss six topics related to inscribable polytopes, both in dimension 3 (where the topic was started with a problem posed by Steiner in 1832) and in higher dimensions.

In the former chapter, control problems of linear polytope systems are considered by using the convex hull Lyapunov function. The linear polytope system is a convex combination of a finite set of finite linear systems. Involved set-valued mapping in the differential inclusion is the convex combination; hence, convex theory can be applied to deal with these control problems. In this chapter, the Lure differential inclusion system and its relative control problems are considered. This kind of differential inclusions is different from linear convex hulls; the set-valued mapping satisfies so-called sector condition, i.e., the image of the set-valued mapping is in a cone. Hence, it is a naturally nonlinear mapping.

Let G be a finite abelian group. This paper is concerned with nonnegative functions on G that are sparse with respect to the Fourier basis. We establish combinatorial conditions on subsets S and T of Fourier basis elements under which nonnegative functions with Fourier support S are sums of squares of functions with Fourier support T. Our combinatorial condition involves constructing a chordal cover of a graph related to G and S (the Cayley graph Cay(G^,S)) with maximal cliques related to T. Our result relies on two main ingredients: the decomposition of sparse positive semidefinite matrices with a chordal sparsity pattern, as well as a simple but key observation exploiting the structure of the Fourier basis elements of G (the characters of G). We apply our general result to two examples. First, in the case where G=Zn2, by constructing a particular chordal cover of the half-cube graph, we prove that any nonnegative quadratic form in n binary variables is a sum of squares of functions of degree at most ⌈n/2⌉, establishing a conjecture of Laurent. Second, we consider nonnegative functions of degree d on ZN (when d divides N). By constructing a particular chordal cover of the dth power of the N-cycle, we prove that any such function is a sum of squares of functions with at most 3dlog(N/d) nonzero Fourier coefficients. Dually this shows that a certain cyclic polytope in R2d with N vertices can be expressed as a projection of a section of the cone of positive semidefinite matrices of size 3dlog(N/d). Putting N=d2 gives a family of polytopes in R2d with linear programming extension complexity Ω(d2) and semidefinite programming extension complexity O(dlog(d)). To the best of our knowledge, this is the first explicit family of polytopes (Pd) in increasing dimensions where xcPSD(Pd)=o(xcLP(Pd)), where xcPSD and xcLP are respectively the SDP and LP extension complexity., United States. Air Force Office of Scientific Research (AFOSR FA9550-11-1-0305), United States. Air Force Office of Scientific Research (AFOSR FA9550-12-1-0287)

Symmetric submodular functions are an important family of submodular functions capturing many interesting cases including cut functions of graphs and hypergraphs. Maximization of such functions subject to various constraints receives little attention by current research, unlike similar minimization problems which have been widely studied. In this work, we identify a few submodular maximization problems for which one can get a better approximation for symmetric objectives than the state of the art approximation for general submodular functions. We first consider the problem of maximizing a non-negative symmetric submodular function $f\colon 2^\mathcal{N} \to \mathbb{R}^+$ subject to a down-monotone solvable polytope $\mathcal{P} \subseteq [0, 1]^\mathcal{N}$. For this problem we describe an algorithm producing a fractional solution of value at least $0.432 \cdot f(OPT)$, where $OPT$ is the optimal integral solution. Our second result considers the problem $\max \{f(S) : |S| = k\}$ for a non-negative symmetric submodular function $f\colon 2^\mathcal{N} \to \mathbb{R}^+$. For this problem, we give an approximation ratio that depends on the value $k / |\mathcal{N}|$ and is always at least $0.432$. Our method can also be applied to non-negative non-symmetric submodular functions, in which case it produces $1/e - o(1)$ approximation, improving over the best known result for this problem. For unconstrained maximization of a non-negative symmetric submodular function we describe a deterministic linear-time $1/2$-approximation algorithm. Finally, we give a $[1 - (1 - 1/k)^{k - 1}]$-approximation algorithm for Submodular Welfare with $k$ players having identical non-negative submodular utility functions, and show that this is the best possible approximation ratio for the problem., 31 pages, an extended abstract appeared in ESA 2015

We obtain upper bounds for the number of distinct solutions generated by the simplex method for linear programming (LP). One of the upper bounds is polynomial in the number of variables, the number of constraints, and the ratio of the maximum to the minimum positive components in all the basic feasible solutions. We show that they are good upper bounds for some special LP problems including those on 0-1 polytopes, those with totally unimodular matrices, and the Markov decision problems. We also show that the upper bounds are almost tight by using an LP instance on a 0-1 polytope and a simple variant of the Klee-Minty example.

This chapter deals with \(H_\infty \) filtering of both discrete-time systems and continuous-time systems with polytopic uncertainties. The uncertain parameters are supposed to reside in a polytope. By using the parameter-dependent Lyapunov function approach and introducing some auxiliary matrix variables, sufficient conditions for the \(H_\infty \) filter design are presented in terms of solutions to a set of linear matrix inequalities (LMIs). In contrast to the existing results of \(H_\infty \) filter design, the main advantage of the proposed design methods is the reduced conservativeness. In addition, the LMI decoupling approach is also considered for designing \(H_\infty \) filters. An example is provided to demonstrate the effectiveness of the proposed methods.

Darwinian evolution can be illustrated as an uphill walk in a landscape, where the surface consists of genotypes, the height coordinates represent fitness, and each step corresponds to a point mutation. Epistasis, roughly defined as the dependence between the fitness effects of mutations, is a key concept in the theory of adaptation. Important recent approaches depend on graphs and polytopes. Fitness graphs are useful for describing coarse properties of a landscape, such as mutational trajectories and the number of peaks. The graphs have been used for relating global and local properties of fitness landscapes. The geometric theory of gene interaction, or the shape theory, is the most fine-scaled approach to epistasis. Shapes, defined as triangulations of polytopes for any number of loci, replace the well established concepts of positive and negative epistasis for two mutations. From the shape one can identify the fittest populations, i.e., populations where allele shuffling (recombination) will not increase the mean fitness. Shapes and graphs provide complementary information. The approaches make no structural assumptions about the underlying fitness landscapes, which make them well suited for empirical work.

There are many examples of optimization problems whose associated polyhedra can be described much nicer, and with way less inequalities, by projections of higher dimensional polyhedra than this would be possible in the original space. However, currently not many general tools to construct such extended formulations are available. In this paper, we develop a framework of polyhedral relations that generalizes inductive constructions of extended formulations via projections, and we particularly elaborate on the special case of reflection relations. The latter ones provide polynomial size extended formulations for several polytopes that can be constructed as convex hulls of the unions of (exponentially) many copies of an input polytope obtained via sequences of reflections at hyperplanes. We demonstrate the use of the framework by deriving small extended formulations for the G-permutahedra of all finite reflection groups G (generalizing both Goeman's [6] extended formulation of the permutahedron of size O(n log n) and Ben-Tal and Nemirovski's [2] extended formulation with O(k) inequalities for the regular 2k-gon) and for Huffman-polytopes (the convex hulls of the weight-vectors of Huffman codes).

Rothvoss [1] showed that there exists a 0/1 polytope (a polytope whose vertices are in {0,1} n ) such that any higher-dimensional polytope projecting to it must have 2Ω(n) facets, i.e., its linear extension complexity is exponential. The question whether there exists a 0/1 polytope with high PSD extension complexity was left open. We answer this question in the affirmative by showing that there is a 0/1 polytope such that any spectrahedron projecting to it must be the intersection of a semidefinite cone of dimension 2Ω(n) and an affine space. Our proof relies on a new technique to rescale semidefinite factorizations.

We discuss several methods of determining all facets of “small” polytopes and polyhedra and give several criteria for classifying the facets into different facet types, so as to bring order into the multitude of facets as, e.g., produced by the application of the double description algorithm (DDA). Among the forms that we consider are the normal, irreducible and minimum support representations of facets. We study symmetries of vertex and edge figures under permissible permutations that leave the underlying polyhedron unchanged with the aim of reducing the numerical effort to find all facets efficiently. Then we introduce a new notion of the rank of facets and integer polyhedra. In the last section, we present old and new results on the facets of the symmetric traveling salesman polytope \(\mathcal{Q}_{T}^{n}\) with up to n=10 cities based on our computer calculations and state a conjecture that, in the notion of rank ρ(P) introduced here, asserts \(\rho(\mathcal{Q}_{T}^{n}) = n - 5\) for all n≥5. This conjecture is supported by our calculations up to n=9 and, possibly, n=10.

The Maximum (Node-) Weight Connected Subgraph Problem (MWCS) searches for a connected subgraph with maximum total weight in a node-weighted (di)graph. In this work we introduce a new integer linear programming formulation built on node variables only, which uses new constraints based on node-separators. We theoretically compare its strength to previously used MIP models in the literature and study the connected subgraph polytope associated with our new formulation. In our computational study we compare branch-and-cut implementations of the new model with two models recently proposed in the literature: one of them using the transformation into the Prize-Collecting Steiner Tree problem, and the other one working on the space of node variables only. The obtained results indicate that the new formulation outperforms the previous ones in terms of the running time and in terms of the stability with respect to variations of node weights.

We characterize the self-parallel (mitered) offsets of a general nonconvex polytope \(\mathcal{Q}\) in 3-space and give a canonical algorithm that constructs a straight skeleton for \(\mathcal{Q}\).

We consider a problem where one desires to intercept a vehicle before it reaches a specific (target) location. The vehicle travels on a road network. There are unattended ground sensors (UGSs) on the road network. The data obtained by a UGS is uploaded to an unmanned aerial vehicle (UAV) when the UAV overflies the UGS. There is an interceptor vehicle which may travel the road network. The interceptor uses intruder state estimates based on the data uploaded from the UGSs to the UAV. There is a negative payoff if the intruder is intercepted prior to reaching the target location, and a positive payoff if the intruder reaches the target location without having been previously intercepted. The intruder position is modeled as a Markov chain. The observations are corrupted by random noise. A dynamic programming approach is taken where the value is propagated using a min-plus curseof- dimensionality-free algorithm. The double-description method for convex polytopes is used to reduce computational loads.

In this paper we extend recent results of Fiorini et al. on the extension complexity of the cut polytope and related polyhedra. We first describe a lifting argument to show exponential extension complexity for a number of NP-complete problems including subset-sum and three dimensional matching. We then obtain a relationship between the extension complexity of the cut polytope of a graph and that of its graph minors. Using this we are able to show exponential extension complexity for the cut polytope of a large number of graphs, including those used in quantum information and suspensions of cubic planar graphs.

In many questions, the geometric approach gives an intuitive visualization that leads to a better understanding of a problem and sometimes even to its solution. This chapter is entirely dedicated to notions, definitions, and basic properties of integer geometry. We start with general definitions of integer geometry, and in particular, define integer lengths, distances, areas of triangles, and indexes of angles. Further we extend the notion of integer area to the case of arbitrary polygons whose vertices have integer coordinates. Then we formulate and prove the famous Pick’s formula that shows how to find areas of polytopes simply by counting points with integer coordinates contained in them. Finally we formulate one theorem in the spirit of Pick’s theorem: it is the so-called twelve-point theorem.