On tradeoffs between width- and fill-like graph parameters

TL;DR

This paper investigates the tradeoffs in optimizing graph parameters related to width and fill-in, providing approximation bounds and showing that simultaneous optimality for these parameters is often impossible.

Contribution

It introduces bounds for approximating interval supergraphs minimizing pathwidth and profile, and demonstrates the inherent tradeoffs in optimizing clique size and edges in chordal supergraphs.

Findings

Existence of interval supergraphs with bounded clique number and edges

Simultaneous minimization of width and fill-in parameters is often impossible

Approximation bounds for pathwidth, profile, treewidth, and fill-in

Abstract

In this work we consider two two-criteria optimization problems: given an input graph, the goal is to find its interval (or chordal) supergraph that minimizes the number of edges and its clique number simultaneously. For the interval supergraph, the problem can be restated as simultaneous minimization of the pathwidth $pw(G)$ and the profile $p(G)$ of the input graph $G$. We prove that for an arbitrary graph $G$ and an integer $t\in\{1,\ldots,pw(G)+1\}$, there exists an interval supergraph $G'$ of $G$ such that for its clique number it holds $\omega(G')\leq(1+\frac{2}{t})(pw(G)+1)$ and the number of its edges is bounded by $|E(G')|\leq(t+2)p(G)$. In other words, the pathwidth and the profile of a graph can be simultaneously minimized within the factors of $1+\frac{2}{t}$ (plus a small constant) and $t+2$, respectively. Note that for a fixed $t$, both upper bounds provide constant factor approximations. On the negative side, we show an example that proves that, for some graphs, there is no solution in which both parameters are optimal. In case of finding a chordal supergraph, the two corresponding graph parameters that reflect its clique size and number of edges are the treewidth and fill-in. We obtain that the treewidth and the fill-in problems are also `orthogonal' in the sense that for some graphs, a solution that minimizes one of those parameters cannot minimize the other. As a motivating example, we recall graph searching games which illustrates a need of simultaneous minimization of these pairs of graph parameters.