We study implementation details for dynamic programming over tree decompositions. Firstly, a fact that is overlooked in many papers and books on this subject is that it is not clear how to test adjacency between two vertices in time bounded by a function of k, where k is the width of the given tree decomposition. This is necessary to obtain linear time dynamic programming algorithms. We address...

In the matroid secretary problem we are given a stream of elements in random order and asked to choose a set of elements that maximizes the total value of the set, subject to being an independent set of a matroid given in advance. The difficulty comes from the assumption that decisions are irrevocable: if we choose to accept an element when it is presented by the stream then we can never get ri...

Remarkable breakthroughs have been made recently in obtaining approximate solutions to some fundamental NP-hard problems, namely Max-Cut, Max k-Cut, Max-Sat, Max-Dicut, Max-bisection, k-vertex coloring, maximum independent set, etc. All these breakthroughs involve polynomial time randomized algorithms based upon semidefinite programming, a technique pioneered by Goemans and Williamson. In this ...

The topological ordering algorithm sorts nodes of a directed graph such that the order of the tail of each arc is lower than the order of its head. In this paper, we introduce the notion of covering between nodes of a directed graph. Then, we apply the topological orderingalgorithm on graphs containing the covering nodes. We show that there exists a cut set withforward arcs in these...

If C is a clutter with n vertices and q edges whose clutter matrix has column vectorsA = {v1, . . . , vq}, we call C an Ehrhart clutter if {(v1, 1), . . . , (vq, 1)} ⊂ {0, 1} n+1 is a Hilbert basis. Letting A(P ) be the Ehrhart ring of P = conv(A), we are able to show that if C is a uniform unmixed MFMC clutter, then C is an Ehrhart clutter and in this case we provide sharp upper bounds on the ...

In optimization problems such as integer programs or their relaxations, one encounters feasible regions of the form {x ∈ R+ : Rx ∈ S} where R is a general real matrix and S ⊂ R is a specific closed set with 0 / ∈ S. For example, in a relaxation of integer programs introduced in [ALWW2007], S is of the form Z − b where b 6∈ Z . One would like to generate valid inequalities that cut off the infea...

The class Max (r, 2)-CSP consists of constraint satisfaction problems with at most two r-valued variables per clause. For instances with n variables and m binary clauses, we present an O(nr)-time algorithm. It is the fastest algorithm for most problems in the class (including Max Cut, Max 2-Sat, and weighted Maximum Independent Set), and in combination with “Generalized CSPs” introduced in a co...

We show a method resulting in the improvement of several polynomial-space, exponentialtime algorithms. An instance of the problem Max (r, 2)-CSP, or simply Max 2-CSP, is parametrized by the domain size r (often 2), the number of variables n (vertices in the constraint graph G), and the number of constraints m (edges in G). When G is cubic, and omitting sub-exponential terms here for clarity, we...

This paper studies the problem of inferring a partition (or a graph cut) of an undirected deterministic graph where the labels of some nodes are observed thereby bridging a gap between graph theory and probabilistic inference techniques. Given a weighted graph, we focus on the rules of weighted neighbors to predict the label of a particular node. A maximum margin and maximal average margin base...

Consider a situation in which one has a suboptimal solution S to a maximization problem which only constitutes a weak approximation to the problem. Suppose that even though the value of S is small compared to an optimal solution OPT to the problem, S happens to be structurally similar to OPT . A natural question to ask in this scenario is whether there is a way of improving the value of S based...