About the complexity of two-stage stochastic IPs

We consider so called 2-stage stochastic integer programs (IPs) and their generalized form, so called multi-stage stochastic IPs. A 2-stage stochastic IP is an integer program of the form $$\max \{ c^T x \mid {\mathcal {A}} x = b, \,l \le x \le u,\, x \in {\mathbb {Z}}^{s + nt} \}$$ max { c T x ∣ A x = b , l ≤ x ≤ u , x ∈ Z s + n t } where the constraint matrix $${\mathcal {A}} \in {\mathbb {Z}}^{r n \times s +nt}$$ A ∈ Z r n × s + n t consists roughly of n repetitions of a matrix $$A \in {\mathbb {Z}}^{r \times s}$$ A ∈ Z r × s on the vertical line and n repetitions of a matrix $$B \in {\mathbb {Z}}^{r \times t}$$ B ∈ Z r × t on the diagonal. In this paper we improve upon an algorithmic result by Hemmecke and Schultz from 2003 [Hemmecke and Schultz, Math. Prog. 2003] to solve 2-stage stochastic IPs. The algorithm is based on the Graver augmentation framework where our main contribution is to give an explicit doubly exponential bound on the size of the augmenting steps. The previous bound for the size of the augmenting steps relied on non-constructive finiteness arguments from commutative algebra and therefore only an implicit bound was known that depends on parameters r, s, t and $$\Delta $$ Δ , where $$\Delta $$ Δ is the largest entry of the constraint matrix. Our new improved bound however is obtained by a novel theorem which argues about intersections of paths in a vector space. As a result of our new bound we obtain an algorithm to solve 2-stage stochastic IPs in time $$f(r,s,\Delta ) \cdot \mathrm {poly}(n,t)$$ f ( r , s , Δ ) · poly ( n , t ) , where f is a doubly exponential function. To complement our result, we also prove a doubly exponential lower bound for the size of the augmenting steps.

Upper Bounds and Explicit Formulas for the Ruin Probability in the Risk Model with Stochastic Premiums and a Multi-Layer Dividend Strategy

This paper is devoted to the investigation of the ruin probability in the risk model with stochastic premiums where dividends are paid according to a multi-layer dividend strategy. We obtain an exponential bound for the ruin probability and investigate conditions, under which it holds for a number of distributions of the premium and claim sizes. Next, we use the exponential bound to construct non-exponential bounds for the ruin probability. We show that the non-exponential bounds turn out to be tighter than the exponential one in some cases. Moreover, we derive explicit formulas for the ruin probability when the premium and claim sizes have either the hyperexponential or the Erlang distributions and apply them to investigate how tight the bounds are. To illustrate and analyze the results obtained, we give numerical examples.

Ruin Probabilities of Continuous-Time Risk Model with Dependent Claim Sizes and Interarrival Times

In this paper we investigate an insurance continuous-time risk model when the claim sizes and inter-arrival times are m-dependent random variables. We provide an upper exponential bound for the ruin probability.

On a Problem of Danzer

Let C be a bounded convex object in ℝd, and let P be a set of n points lying outside C. Further, let cp, cq be two integers with 1 ⩽ cq ⩽ cp ⩽ n - ⌊d/2⌋, such that every cp + ⌊d/2⌋ points of P contain a subset of size cq + ⌊d/2⌋ whose convex hull is disjoint from C. Then our main theorem states the existence of a partition of P into a small number of subsets, each of whose convex hulls are disjoint from C. Our proof is constructive and implies that such a partition can be computed in polynomial time.In particular, our general theorem implies polynomial bounds for Hadwiger--Debrunner (p, q) numbers for balls in ℝd. For example, it follows from our theorem that when p > q = (1+β)⋅d/2 for β > 0, then any set of balls satisfying the (p, q)-property can be hit by O((1+β)2d2p1+1/β logp) points. This is the first improvement over a nearly 60 year-old exponential bound of roughly O(2d).Our results also complement the results obtained in a recent work of Keller, Smorodinsky and Tardos where, apart from improvements to the bound on HD(p, q) for convex sets in ℝd for various ranges of p and q, a polynomial bound is obtained for regions with low union complexity in the plane.

On Regularity Lemmas and their Algorithmic Applications

Szemerédi's regularity lemma and its variants are some of the most powerful tools in combinatorics. In this paper, we establish several results around the regularity lemma. First, we prove that whether or not we include the condition that the desired vertex partition in the regularity lemma is equitable has a minimal effect on the number of parts of the partition. Second, we use an algorithmic version of the (weak) Frieze–Kannan regularity lemma to give a substantially faster deterministic approximation algorithm for counting subgraphs in a graph. Previously, only an exponential dependence for the running time on the error parameter was known, and we improve it to a polynomial dependence. Third, we revisit the problem of finding an algorithmic regularity lemma, giving approximation algorithms for several co-NP-complete problems. We show how to use the weak Frieze–Kannan regularity lemma to approximate the regularity of a pair of vertex subsets. We also show how to quickly find, for each ε′>ε, an ε′-regular partition withkparts if there exists an ε-regular partition withkparts. Finally, we give a simple proof of the permutation regularity lemma which improves the tower-type bound on the number of parts in the previous proofs to a single exponential bound.