Cuts in Optimization and Approximation: Generalized MIR Cutting Plane and Integrality Gap Bounds for the Planar Multicut Problem

Abstract

This thesis has two main themes, both centered around the role of cuts in integer programming and approximation algorithms.

In the first part, we investigate the complexity of split closures in mixed-integer sets in~$\mathbb{R}^3$ with two integral variables and one continuous variable. While it is known that in~$\mathbb{R}^2$ the split closure admits a polynomial-size description, extending this result to the mixed-integer setting in~$\mathbb{R}^3$ poses new challenges. For a rational polyhedron~$P \subseteq \mathbb{R}^3$ and integer index set~$I = \{1,2\}$, we denote by~$P_I := conv(P \cap (\mathbb{Z}^2 \times \mathbb{R}))$ the mixed-integer hull of $P$ with respect to $I$, and by~$P^{split}_I$ the intersection of all split cuts with respect to $I$. We make progress toward understanding the structure of~$P^{split}_I$ by proving the existence of a polyhedron~$Q$ such that~$P_I \subseteq Q \subseteq P^{split}_I$, where~$Q$ can be described by a polynomial number of inequalities relative to the input size of~$P$. In addition, we introduce a generalized mixed-integer rounding (MIR) procedure that begins with systems of two equations, rather than a single equation as in the classical setting. This leads to a new family of valid inequalities for mixed-integer sets, whose derivation relies on structural results for split closures in~$\mathbb{R}^3$.

The second part focuses on approximation algorithms for the minimum multicut problem in planar graphs and certain subclasses thereof. We analyze the integrality gap of the natural LP relaxation within the framework of small-diameter decompositions. By introducing novel tools, we improve the best known lower bound on this integrality gap from $2$ to $\tfrac{20}{9}$ for the family of cactus graphs, which in turn yields an improved lower bound of the integrality gap for planar graphs. Complementing this, we establish an upper bound of $3.44$ for cactus graphs, improving the previous bound of $4$. We further obtain tight bounds in special subclasses: the integrality gap is exactly $2$ for unicyclic graphs and for path cactus graphs. Extending the study to outerplanar graphs, we prove an upper bound of $8$ on the integrality gap by designing a randomized rounding algorithm that transforms the optimal fractional solution of the LP relaxation into a feasible multicut whose expected cost is at most $8$ times the cost of the optimal fractional solution. Together, these results sharpen our understanding of the minimum multicut problem and its LP relaxation in planar graphs.