Petr Kolman

Petr Kolman, Hans Raj Tiwary

A cut in a graph $G$ is called a {\em bond} if both parts of the cut induce connected subgraphs in $G$, and the {\em bond polytope} is the convex hull of all bonds. Computing the maximum weight bond is an NP-hard problem even for planar graphs. However, the problem is solvable in linear time on $(K_5 \setminus e)$-minor-free graphs, and in more general, on graphs of bounded treewidth, essentially due to clique-sum decomposition into simpler graphs. We show how to obtain the bond polytope of graphs that are $1$- or $2$-sum of graphs $G_1$ and $ G_2$ from the bond polytopes of $G_1,G_2$. Using this we show that the extension complexity of the bond polytope of $(K_5 \setminus e)$-minor-free graphs is linear. Prior to this work, a linear size description of the bond polytope was known only for $3$-connected planar $(K_5 \setminus e)$-minor-free graphs, essentially only for wheel graphs. We also describe an elementary linear time algorithm for the \MaxBond problem on $(K_5\setminus e)$-minor-free graphs. Prior to this work, a linear time algorithm in this setting was known. However, the hidden constant in the big-Oh notation was large because the algorithm relies on the heavy machinery of linear time algorithms for graphs of bounded treewidth, used as a black box.

Hans Raj Tiwary, Petr Kolman

We introduce a variant of the multiway cut that we call the min-max connected multiway cut. Given a graph $G=(V,E)$ and a set $Γ\subseteq V$ of $t$ terminals, partition $V$ into $t$ parts such that each part is connected and contains exactly one terminal; the objective is to minimize the maximum weight of the edges leaving any part of the partition. This problem is a natural modification of the standard multiway cut problem and it differs from it in two ways: first, the cost of a partition is defined to be the maximum size of the boundary of any part, as opposed to the sum of all boundaries, and second, the subgraph induced by each part is required to be connected. Although the modified objective function has been considered before in the literature under the name min-max multiway cut, the requirement on each component to be connected has not been studied as far as we know. Our main motivation for studying this problems is its close connection with the spanning tree congestion problem that has been extensively studied but on which little progress has been made. We show various hardness results for this problem, including a proof of weak NP-hardness of the weighted version of the problem on graphs with tree-width two, and $W[1]$-hardness for the problem when parameterized by the tree-width of the graph. Complementing our lower bounds tightly, we also provide a pseudopolynomial time algorithm as well as an FPTAS for the weighted problem on graphs of bounded tree-width. As a consequence of our investigation we also show that the (unconstrained) min-max multiway cut problem is NP-hard even for three terminals, strengthening the known results.

Petr Kolman

The Spanning Tree Congestion (STC) problem is the following NP-hard problem: given a graph $G$, construct a spanning tree $T$ of $G$ minimizing its maximum edge congestion where the congestion of an edge $e\in T$ is the number of edges $uv$ in $G$ such that the unique path between $u$ and $v$ in $T$ passes through $e$; the optimal value for a given graph $G$ is denoted $STC(G)$. It is known that every spanning tree is an $n/2$-approximation for the STP problem. A long-standing problem is to design a better approximation algorithm. Our contribution towards this goal is an $O(Δ\cdot\log^{3/2}n)$-approximation algorithm where $Δ$ is the maximum degree in $G$ and $n$ the number of vertices. For graphs with a maximum degree bounded by a polylog of the number of vertices, this is an exponential improvement over the previous best approximation. Our main tool for the algorithm is a new lower bound on the spanning tree congestion which is of independent interest. Denoting by $hb(G)$ the hereditary bisection of $G$ which is the maximum bisection width over all subgraphs of $G$, we prove that for every graph $G$, $STC(G)\geq Ω(hb(G)/Δ)$.