arXiv subjects

Petr Kolman

Publications and source records attributed to Petr Kolman.

8 recordsLinked to original sources

arXiv2026

Min-Max Connected Multiway Cut

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 $\Gamma\subseteq V$ 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 treewidth two, and $W[1]$-hardness for the problem when parameterized by the treewidth 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 treewidth. 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.

cs.DS↗

arXiv2026

Bond Polytope under Vertex- and Edge-sums

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.

math.CO↗

arXiv2026

Two Complexity Results on Spanning-Tree Congestion Problems

In the spanning-tree congestion problem ($\mathsf{STC}$), we are given a graph $G$, and the objective is to compute a spanning tree of $G$ that minimizes the maximum edge congestion. While $\mathsf{STC}$ is known to be $\mathbb{NP}$-hard, even for some restricted graph classes, several key questions regarding its computational complexity remain open, and we address some of these in our paper. (i) For graphs of degree at most $\Delta$, it is known that $\mathsf{STC}$ is $\mathbb{NP}$-hard when $\Delta\ge 8$. We provide a complete resolution of this variant, by showing that $\mathsf{STC}$ remains $\mathbb{NP}$-hard for each degree bound $\Delta\ge 3$. (ii) In the decision version of $\mathsf{STC}$, given an integer $K$, the goal is to determine whether the congestion of $G$ is at most $K$. We prove that this variant is polynomial-time solvable for $K$-edge-connected graphs.

cs.DS↗

arXiv2024

Approximation of Spanning Tree Congestion using Hereditary Bisection

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 $O(\Delta\cdot\log^{3/2}n)$-approximation algorithm where $\Delta$ 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 $G$, $STC(G)\geq \Omega(hb(G)/\Delta)$.

cs.DS↗

arXiv2019

On Polynomial-Time Combinatorial Algorithms for Maximum $L$-Bounded Flow

Given a graph $G=(V,E)$ with two distinguished vertices $s,t\in V$ and an integer $L$, an {\em $L$-bounded flow} is a flow between $s$ and $t$ that can be decomposed into paths of length at most $L$. In the {\em maximum $L$-bounded flow problem} the task is to find a maximum $L$-bounded flow between a given pair of vertices in the input graph. The problem can be solved in polynomial time using linear programming. However, as far as we know, no polynomial-time combinatorial algorithm for the $L$-bounded flow is known. The only attempt, that we are aware of, to describe a combinatorial algorithm for the maximum $L$-bounded flow problem was done by Koubek and Ří ha in 1981. Unfortunately, their paper contains substantional flaws and the algorithm does not work; in the first part of this paper, we describe these problems. In the second part of this paper we describe a combinatorial algorithm based on the exponential length method that finds a $(1+ε)$-approximation of the maximum $L$-bounded flow in time $O(ε^{-2}m^2 L\log L)$ where $m$ is the number of edges in the graph. Moreover, we show that this approach works even for the NP-hard generalization of the maximum $L$-bounded flow problem in which each edge has a length.

cs.DS↗

arXiv2017

On Algorithms for $L$-bounded Cut Problem

Given a graph $G=(V,E)$ with two distinguished vertices $s,t\in V$ and an integer parameter $L>0$, an {\em $L$-bounded cut} is a subset $F$ of edges (vertices) such that the every path between $s$ and $t$ in $G\setminus F$ has length more than $L$. The task is to find an $L$-bounded cut of minimum cardinality. Though the problem is very simple to state and has been studied since the beginning of the 70's, it is not much understood yet. The problem is known to be $\cal{NP}$-hard to approximate within a small constant factor even for $L\geq 4$ (for $L\geq 5$ for the vertex cuts). On the other hand, the best known approximation algorithm for general graphs has approximation ratio only $\mathcal{O}({n^{2/3}})$ in the edge case, and $\mathcal{O}({\sqrt{n}})$ in the vertex case, where $n$ denotes the number of vertices. We show that for planar graphs, it is possible to solve both the edge- and the vertex-version of the problem optimally in time $\mathcal{O}(L^{3L}n)$. That is, the problem is fixed parameter tractable (FPT) with respect to $L$ on planar graphs. Furthermore, we show that the problem remains FPT even for bounded genus graphs, a super class of planar graphs. Our second contribution deals with approximations of the vertex version of the problem. We describe an algorithm that for a given a graph $G$, its tree decomposition of treewidth $τ$ and vertices $s$ and $t$ computes a $τ$-approximation of the minimum $L$-bounded $s-t$ vertex cut; if the decomposition is not given, then the approximation ratio is $\mathcal{O}(τ\sqrt{\log τ})$. For graphs with treewidth bounded by $\mathcal{O}(n^{1/2-ε})$ for any $ε>0$, but not by a constant, this is the best approximation in terms of~$n$ that we are aware of.

cs.DS↗

arXiv2015

Extension Complexity, MSO Logic, and Treewidth

We consider the convex hull $P_{\varphi}(G)$ of all satisfying assignments of a given MSO formula $\varphi$ on a given graph $G$. We show that there exists an extended formulation of the polytope $P_{\varphi}(G)$ that can be described by $f(|\varphi|,\tau)\cdot n$ inequalities, where $n$ is the number of vertices in $G$, $\tau$ is the treewidth of $G$ and $f$ is a computable function depending only on $\varphi$ and $\tau.$ In other words, we prove that the extension complexity of $P_{\varphi}(G)$ is linear in the size of the graph $G$, with a constant depending on the treewidth of $G$ and the formula $\varphi$. This provides a very general yet very simple meta-theorem about the extension complexity of polytopes related to a wide class of problems and graphs. As a corollary of our main result, we obtain an analogous result % for the weaker MSO$_1$ logic on the wider class of graphs of bounded cliquewidth. Furthermore, we study our main geometric tool which we term the glued product of polytopes. While the glued product of polytopes has been known since the '90s, we are the first to show that it preserves decomposability and boundedness of treewidth of the constraint matrix. This implies that our extension of $P_\varphi(G)$ is decomposable and has a constraint matrix of bounded treewidth; so far only few classes of polytopes are known to be decomposable. These properties make our extension useful in the construction of algorithms.

cs.DS↗

arXiv2015

Extended Formulation for CSP that is Compact for Instances of Bounded Treewidth

In this paper we provide an extended formulation for the class of constraint satisfaction problems and prove that its size is polynomial for instances whose constraint graph has bounded treewidth. This implies new upper bounds on extension complexity of several important NP-hard problems on graphs of bounded treewidth.

cs.CC↗