Lifting Linear Extension Complexity Bounds to the Mixed-Integer Setting

Mixed-integer mathematical programs are among the most commonly used models for a wide set of problems in Operations Research and related fields. However, there is still very little known about what can be expressed by small mixed-integer programs. In particular, prior to this work, it was open whether some classical problems, like the minimum odd-cut problem, can be expressed by a compact mixed-integer program with few (even constantly many) integer variables. This is in stark contrast to linear formulations, where recent breakthroughs in the field of extended formulations have shown that many polytopes associated to classical combinatorial optimization problems do not even admit approximate extended formulations of sub-exponential size. We provide a general framework for lifting inapproximability results of extended formulations to the setting of mixed-integer extended formulations, and obtain almost tight lower bounds on the number of integer variables needed to describe a variety of classical combinatorial optimization problems. Among the implications we obtain, we show that any mixed-integer extended formulation of sub-exponential size for the matching polytope, cut polytope, traveling salesman polytope or dominant of the odd-cut polytope, needs $\Omega(n/\log n)$ many integer variables, where $n$ is the number of vertices of the underlying graph. Conversely, the above-mentioned polyhedra admit polynomial-size mixed-integer formulations with only $O(n)$ or $O(n \log n)$ (for the traveling salesman polytope) many integer variables. Our results build upon a new decomposition technique that, for any convex set $C$, allows for approximating any mixed-integer description of $C$ by the intersection of $C$ with the union of a small number of affine subspaces.Comment: A conference version of this paper will be presented at SODA 201

Skewincidence

We introduce a new class of problems lying halfway between questions about graph capacity and intersection. We say that two binary sequences x and y of the same length have a skewincidence if there is a coordinate i for which x_i=y_{i+1}=1 or vice versa. We give rather sharp bounds on the maximum number of binary sequences of length n any pair of which has a skewincidence

On the number of types in sparse graphs

We prove that for every class of graphs $\mathcal{C}$ which is nowhere dense, as defined by Nesetril and Ossona de Mendez, and for every first order formula $\phi(\bar x,\bar y)$, whenever one draws a graph $G\in \mathcal{C}$ and a subset of its nodes $A$, the number of subsets of $A^{|\bar y|}$ which are of the form $\{\bar v\in A^{|\bar y|}\, \colon\, G\models\phi(\bar u,\bar v)\}$ for some valuation $\bar u$ of $\bar x$ in $G$ is bounded by $\mathcal{O}(|A|^{|\bar x|+\epsilon})$, for every $\epsilon>0$. This provides optimal bounds on the VC-density of first-order definable set systems in nowhere dense graph classes. We also give two new proofs of upper bounds on quantities in nowhere dense classes which are relevant for their logical treatment. Firstly, we provide a new proof of the fact that nowhere dense classes are uniformly quasi-wide, implying explicit, polynomial upper bounds on the functions relating the two notions. Secondly, we give a new combinatorial proof of the result of Adler and Adler stating that every nowhere dense class of graphs is stable. In contrast to the previous proofs of the above results, our proofs are completely finitistic and constructive, and yield explicit and computable upper bounds on quantities related to uniform quasi-wideness (margins) and stability (ladder indices)

On largest volume simplices and sub-determinants

We show that the problem of finding the simplex of largest volume in the convex hull of $n$ points in $\mathbb{Q}^d$ can be approximated with a factor of $O(\log d)^{d/2}$ in polynomial time. This improves upon the previously best known approximation guarantee of $d^{(d-1)/2}$ by Khachiyan. On the other hand, we show that there exists a constant $c>1$ such that this problem cannot be approximated with a factor of $c^d$, unless $P=NP$. % This improves over the $1.09$ inapproximability that was previously known. Our hardness result holds even if $n = O(d)$, in which case there exists a \bar c\,^{d}-approximation algorithm that relies on recent sampling techniques, where $\bar c$ is again a constant. We show that similar results hold for the problem of finding the largest absolute value of a subdeterminant of a $d\times n$ matrix

A new graph parameter related to bounded rank positive semidefinite matrix completions

The Gram dimension \gd(G) of a graph $G$ is the smallest integer $k\ge 1$ such that any partial real symmetric matrix, whose entries are specified on the diagonal and at the off-diagonal positions corresponding to edges of $G$, can be completed to a positive semidefinite matrix of rank at most $k$ (assuming a positive semidefinite completion exists). For any fixed $k$ the class of graphs satisfying \gd(G) \le k is minor closed, hence it can characterized by a finite list of forbidden minors. We show that the only minimal forbidden minor is $K_{k+1}$ for $k\le 3$ and that there are two minimal forbidden minors: $K_5$ and $K_{2,2,2}$ for $k=4$. We also show some close connections to Euclidean realizations of graphs and to the graph parameter $\nu^=(G)$ of \cite{H03}. In particular, our characterization of the graphs with \gd(G)\le 4 implies the forbidden minor characterization of the 3-realizable graphs of Belk and Connelly \cite{Belk,BC} and of the graphs with $\nu^=(G) \le 4$ of van der Holst \cite{H03}.Comment: 31 pages, 6 Figures. arXiv admin note: substantial text overlap with arXiv:1112.596