Combinatorica 

About: Combinatorica is an academic journal published by Springer Science+Business Media. The journal publishes majorly in the area(s): Conjecture & Upper and lower bounds. It has an ISSN identifier of 0209-9683. Over the lifetime, 1646 publications have been published receiving 78500 citations.

A new polynomial-time algorithm for linear programming

Abstract: We present a new polynomial-time algorithm for linear programming. In the worst case, the algorithm requiresO(n 3.5 L) arithmetic operations onO(L) bit numbers, wheren is the number of variables andL is the number of bits in the input. The running-time of this algorithm is better than the ellipsoid algorithm by a factor ofO(n 2.5). We prove that given a polytopeP and a strictly interior point a eP, there is a projective transformation of the space that mapsP, a toP′, a′ having the following property. The ratio of the radius of the smallest sphere with center a′, containingP′ to the radius of the largest sphere with center a′ contained inP′ isO(n). The algorithm consists of repeated application of such projective transformations each followed by optimization over an inscribed sphere to create a sequence of points which converges to the optimal solution in polynomial time.

The ellipsoid method and its consequences in combinatorial optimization

Abstract: L. G. Khachiyan recently published a polynomial algorithm to check feasibility of a system of linear inequalities. The method is an adaptation of an algorithm proposed by Shor for non-linear optimization problems. In this paper we show that the method also yields interesting results in combinatorial optimization. Thus it yields polynomial algorithms for vertex packing in perfect graphs; for the matching and matroid intersection problems; for optimum covering of directed cuts of a digraph; for the minimum value of a submodular set function; and for other important combinatorial problems. On the negative side, it yields a proof that weighted fractional chromatic number is NP-hard.

The geometry of graphs and some of its algorithmic applications

Abstract: In this paper we explore some implications of viewing graphs asgeometric objects. This approach offers a new perspective on a number of graph-theoretic and algorithmic problems. There are several ways to model graphs geometrically and our main concern here is with geometric representations that respect themetric of the (possibly weighted) graph. Given a graphG we map its vertices to a normed space in an attempt to (i) keep down the dimension of the host space, and (ii) guarantee a smalldistortion, i.e., make sure that distances between vertices inG closely match the distances between their geometric images. In this paper we develop efficient algorithms for embedding graphs low-dimensionally with a small distortion. Further algorithmic applications include: Given faithful low-dimensional representations of statistical data, it is possible to obtain meaningful and efficientclustering. This is one of the most basic tasks in pattern-recognition. For the (mostly heuristic) methods used in the practice of pattern-recognition, see [20], especially chapter 6. Our studies of multicommodity flows also imply that every embedding of (the metric of) ann-vertex, constant-degree expander into a Euclidean space (of any dimension) has distortion Ω(logn). This result is tight, and closes a gap left open by Bourgain [12].

Eigen values and expanders

Abstract: Linear expanders have numerous applications to theoretical computer science Here we show that a regular bipartite graph is an expanderif and only if the second largest eigenvalue of its adjacency matrix is well separated from the first This result, which has an analytic analogue for Riemannian manifolds enables one to generate expanders randomly and check efficiently their expanding properties It also supplies an efficient algorithm for approximating the expanding properties of a graph The exact determination of these properties is known to be coNP-complete

Randomized rounding: a technique for provably good algorithms and algorithmic proofs

Abstract: We study the relation between a class of 0–1 integer linear programs and their rational relaxations. We give a randomized algorithm for transforming an optimal solution of a relaxed problem into a provably good solution for the 0–1 problem. Our technique can be a of extended to provide bounds on the disparity between the rational and 0–1 optima for a given problem instance.