Matroids are Immune to Braess Paradox

The famous Braess paradox describes the following phenomenon: It might happen that the improvement of resources, like building a new street within a congested network, may in fact lead to larger costs for the players in an equilibrium. In this paper we consider general nonatomic congestion games and give a characterization of the maximal combinatorial property of strategy spaces for which Braess paradox does not occur. In a nutshell, bases of matroids are exactly this maximal structure. We prove our characterization by two novel sensitivity results for convex separable optimization problems over polymatroid base polyhedra which may be of independent interest.Comment: 21 page

Minimum Convex Partitions and Maximum Empty Polytopes

Let $S$ be a set of $n$ points in $\mathbb{R}^d$. A Steiner convex partition is a tiling of ${\rm conv}(S)$ with empty convex bodies. For every integer $d$, we show that $S$ admits a Steiner convex partition with at most $\lceil (n-1)/d\rceil$ tiles. This bound is the best possible for points in general position in the plane, and it is best possible apart from constant factors in every fixed dimension $d\geq 3$. We also give the first constant-factor approximation algorithm for computing a minimum Steiner convex partition of a planar point set in general position. Establishing a tight lower bound for the maximum volume of a tile in a Steiner convex partition of any $n$ points in the unit cube is equivalent to a famous problem of Danzer and Rogers. It is conjectured that the volume of the largest tile is $\omega(1/n)$. Here we give a $(1-\varepsilon)$-approximation algorithm for computing the maximum volume of an empty convex body amidst $n$ given points in the $d$-dimensional unit box $[0,1]^d$.Comment: 16 pages, 4 figures; revised write-up with some running times improve

Convex Integer Optimization by Constantly Many Linear Counterparts

In this article we study convex integer maximization problems with composite objective functions of the form $f(Wx)$, where $f$ is a convex function on $\R^d$ and $W$ is a $d\times n$ matrix with small or binary entries, over finite sets $S\subset \Z^n$ of integer points presented by an oracle or by linear inequalities. Continuing the line of research advanced by Uri Rothblum and his colleagues on edge-directions, we introduce here the notion of {\em edge complexity} of $S$, and use it to establish polynomial and constant upper bounds on the number of vertices of the projection \conv(WS) and on the number of linear optimization counterparts needed to solve the above convex problem. Two typical consequences are the following. First, for any $d$, there is a constant $m(d)$ such that the maximum number of vertices of the projection of any matroid $S\subset\{0,1\}^n$ by any binary $d\times n$ matrix $W$ is $m(d)$ regardless of $n$ and $S$; and the convex matroid problem reduces to $m(d)$ greedily solvable linear counterparts. In particular, $m(2)=8$. Second, for any $d,l,m$, there is a constant $t(d;l,m)$ such that the maximum number of vertices of the projection of any three-index $l\times m\times n$ transportation polytope for any $n$ by any binary $d\times(l\times m\times n)$ matrix $W$ is $t(d;l,m)$; and the convex three-index transportation problem reduces to $t(d;l,m)$ linear counterparts solvable in polynomial time