A multifamily set representation of a finite simple graph G is a multifamily F of sets (not necessarily distinct) for which each set represents a vertex in G and two sets in F intersects if and only if the two corresponding vertices are adjacent. For a graph G, an edge clique covering (edge clique partition, respectively) Q is a set of cliques for which every edge is contained in at least (exac...

We introduce a parameterized version of set cover that generalizes several previously studied problems. Given a ground set V and a collection of subsets Si of V , a feasible solution is a partition of V such that each subset of the partition is included in one of the Si. The problem involves maximizing the mean subset size of the partition, where the mean is the generalized mean of parameter p,...

Abstract Motivated by applications in Topological Data Analysis, we consider decompositions of a simplicial complex induced cover its vertices. We study how the homotopy type such approximates itself. The difference between and an approximation is quantitatively measured means so called obstruction complexes. Our general machinery then specialized to clique complexes, Vietoris-Rips complexes me...

We study various optimization problems in t-subtree graphs, the intersection graphs of tsubtrees, where a t-subtree is the union of t disjoint subtrees of some tree. This graph class generalizes both the class of chordal graphs and the class of t-interval graphs, a generalization of interval graphs that has recently been studied from a combinatorial optimization point of view. We present approx...

Using ideas and results from polynomial approximation and exact computation we design approximation algorithms for several NP-hard combinatorial problems achieving ratios that cannot be achieved in polynomial time (unless a very unlikely complexity conjecture is confirmed) with worst-case complexity much lower (though super-polynomial) than that of an exact computation. We study in particular m...

We consider the problem of separating maximally violated inequalities for the precedence constrained knapsack problem. Though we consider maximally violated constraints in general, special emphasis is placed on induced cover inequalities and induced clique inequalities. Our contributions include a new partial characterization of maximally violated inequalities, a new safe shrinking technique, a...

For each m ≥ 1, we construct a graph G = (V, E) with ω(G) = m such that max i=1,2 ω(ind(Vi)) = m for arbitrary partition V = V1 ∪ V2, where ω(G) is the clique number of G and ind(Vi) is the induced graph of Vi. Using this result, we may show that for each m ≥ 2 there exists an exact m-cover of Z which is not the union of two 1-covers.

A total coloring of a graph G=(V,E) is an assignment colors to vertices and edges such that neither two adjacent nor incident get the same color, and, for each edge, end-points edge itself receive different colors. Any valid induces partition elements G into matchings, which are defined as subsets can take color. In this paper, we propose Integer Linear Programming models both Total Coloring Ma...