arXiv subjects
Publications and source records attributed to Svante Linusson.
We consider the distribution of the value of the optimal k-assignment in an m x n-matrix, where the entries are independent exponential random variables with arbitrary rates. We give closed formulas for both the Laplace transform of this random variable and for its expected value under the condition that there is a zero-cost k-1-assignment.
We prove the main conjecture of the paper ``On the expected value of the minimum assignment'' by Marshall W. Buck, Clara S. Chan, and David P. Robbins (Random Structures & Algorithms 21 (2002), no. 1, 33--58). This is a vast generalization of a formula conjectured by Giorgio Parisi for the $n$ by $n$ random assignment problem. We combine the urn model from the article by Buck, Chan and Robbins, with our proof of Parisi's conjecture in our article posted on the Arxiv in March this year. Our new theorem generalize simultaneously our main theorem from that article and the conjecture by Buck, Chan and Robbins. Using the urn model we avoid several technical difficulties from our previous proof of Parisi's conjecture and get a shorter proof in a conceptually more appealling setting.
An assignment problem is the optimization problem of finding, in an m by n matrix of nonnegative real numbers, k entries, no two in the same row or column, such that their sum is minimal. Such an optimization problem is called a random assignment problem if the matrix entries are random variables. We give a formula for the expected value of the optimal k-assignment in a matrix where some of the entries are zero, and all other entries are independent exponentially distributed random variables with mean 1. Thereby we prove the formula 1+1/4+1/9+...+1/k^2 conjectured by G. Parisi for the case k=m=n, and the generalized conjecture of D. Coppersmith and G. B. Sorkin for arbitrary k, m and n.
We give a conjecture for the expected value of the optimal k-assignment in an m x n-matrix, where the entries are all exp(1)-distributed random variables or zeros. We prove this conjecture in the case there is a zero-cost $k-1$-assignment. Assuming our conjecture, we determine some limits, as $k=m=n\to \infty$, of the expected cost of an optimal n -assignment in an n x n-matrix with zeros in some region. If we take the region outside a quarter-circle inscribed in the square matrix, this limit is thus conjectured to be $π^2/24$. We give a computer-generated verification of a conjecture of Parisi for k=m=n=7 and of a conjecture of Coppersmith and Sorkin for $k\leq 5$. We have used the same computer program to verify this conjecture also for k=6.
Complexes of (not) connected graphs, hypergraphs and their homology appear in the construction of knot invariants given by V. Vassiliev. In this paper we study the complexes of not $i$-connected $k$-hypergraphs on $n$ vertices. We show that the complex of not $2$-connected graphs has the homotopy type of a wedge of $(n-2)!$ spheres of dimension $2n-5$. This answers one of the questions raised by Vassiliev in connection with knot invariants. For this case the $S_n$-action on the homology of the complex is also determined. For complexes of not $2$-connected $k$-hypergraphs we provide a formula for the generating function of the Euler characteristic, and we introduce certain lattices of graphs that encode their topology. We also present partial results for some other cases. In particular, we show that the complex of not $(n-2)$-connected graphs is Alexander dual to the complex of partial matchings of the complete graph. For not $(n-3)$-connected graphs we provide a formula for the generating function of the Euler characteristic.
The Shi arrangement ${\mathcal S}_n$ is the arrangement of affine hyperplanes in ${\mathbb R}^n$ of the form $x_i - x_j = 0$ or $1$, for $1 \leq i < j \leq n$. It dissects ${\mathbb R}^n$ into $(n+1)^{n-1}$ regions, as was first proved by Shi. We give a simple bijective proof of this result. Our bijection generalizes easily to any subarrangement of ${\mathcal S}_n$ containing the hyperplanes $x_i - x_j = 0$ and to the extended Shi arrangements.
Consider the question: Given integers $k N(d,k)$ the answer is yes if and only if $n\equiv 0\quad \pmod {G(d,k)}$. Furthermore, a formula for $G(d,k)$ is given, showing that e.g. $G(d,k)=1$ if $k\ge \left\lfloor\frac{d+1}{2}\right\rfloor$ or if both $d$ and $k$ are even, and also in some other cases (meaning that all numbers beyond $N(d,k)$ occur as the number of $k$-faces of some simple $d$-polytope). This question has previously been studied only for the case of vertices ($k=0$), where Lee \cite{Le} proved the existence of $N(d,0)$ (with $G(d,0)=1$ or $2$ depending on whether $d$ is even or odd), and Prabhu \cite{P2} showed that $N(d,0) \le cd\sqrt {d}$. We show here that asymptotically the true value of Prabhu's constant is $c=\sqrt2$ if $d$ is even, and $c=1$ if $d$ is odd.