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Markus Chimani

Publications and source records attributed to Markus Chimani.

At least 19 recordsLinked to original sources
arXiv2026

Designing Capacitated Subnetworks for Shortest Path Routing

In pursuit of higher energy efficiency in computer networks, one subfield of green traffic engineering aims at reducing the size of a network during times of low traffic, while still guaranteeing the ability to route all occurring demands. In this setting, we have to simultaneously solve a network design problem (choosing connections to deactivate) and a routing problem (routing paths in the active subnetwork, adhering to some routing protocol). Interestingly, there seems to be no available method to tackle the problem as a whole for the simplest (and still most commonly used) routing paradigm: shortest path routing. State-of-the-art methods either do not consider capacities, or assume that the routing paths should not change when deactivating network connections, or separate the problem into its two constituents, first solving the network design problem (using some estimators in lieu of the precise routing protocol) and only then the actual routing problem. In this paper, we present an algorithm to tackle the full combined problem exactly via a novel integer linear program, modeling dynamically changing shortest paths. To solve it, we need to devise a special-purpose column generation method. To speed up the solution process, we further propose additional provably strengthening constraints. Now having the means to yield true optimal solutions for (small) practical instances, we can for the first time give an in-depth experimental evaluation that includes the absolute quality intrinsic to the above simplifying algorithms. It turns out that the arguably simplest method--first computing a routing, fixing it, and turning off all superfluous connections--yields solutions surprisingly close to the true optimum in practice. When considering multiple different traffic demands, a recent traffic-oblivious approach (TOCA) performs best, while being comparatively straightforward to implement.

cs.DS↗
arXiv2026

No Traffic to Cry: Traffic-Oblivious Link Deactivation for Green Traffic Engineering

As internet traffic grows, the underlying infrastructure consumes increasing amounts of energy. During off-peak hours, large parts of the networks remain underutilized, presenting significant potential for energy savings. Existing Green Traffic Engineering approaches attempt to leverage this potential by switching off those parts of the networks that are not required for the routing of specific traffic matrices. When traffic changes, the approaches need to adapt rapidly, which is hard to achieve given the complexity of the problem. We take a fundamentally different approach: instead of considering a specific traffic matrix, we rely on a traffic-oblivious routing scheme. We discuss the NP-hard problem of activating as few connections as possible while still guaranteeing that any down-scaled traffic matrix $\varrho\cdot T$ can be routed, where $\varrho \in (0,1)$ and $T$ is any traffic matrix routable in the original network. We present a $\max(\frac{1}{\varrho\cdot\lambda_{\text{min}}},2)$-approximation algorithm for this problem, with $\lambda_{\text{min}}$ denoting the minimum number of connections between any two connected routers. Additionally, we propose two post-processing heuristics to further improve solution quality. Our evaluation shows that we can quickly generate near-optimal solutions. By design, our method avoids the need for frequent reconfigurations and offers a promising direction to achieve practical energy savings in backbone networks.

cs.NI↗
arXiv2025

Simple Approximations for General Spanner Problems

Consider a graph with n nodes and m edges, independent edge weights and lengths, and arbitrary distance demands for node pairs. The spanner problem asks for a minimum-weight subgraph that satisfies these demands via sufficiently short paths w.r.t. the edge lengths. For multiplicative alpha-spanners (where demands equal alpha times the original distances) and assuming that each edge's weight equals its length, the simple Greedy heuristic by Alth\"ofer et al. (1993) is known to yield strong solutions, both in theory and practice. To obtain guarantees in more general settings, recent approximations typically abandon this simplicity and practicality. Still, so far, there is no known non-trivial approximation algorithm for the spanner problem in its most general form. We provide two surprisingly simple approximations algorithms. In general, our Augmented Greedy achieves the first unconditional approximation ratio of m, which is non-trivial due to the independence of weights and lengths. Crucially, it maintains all size and weight guarantees Greedy is known for, i.e., in the aforementioned multiplicative alpha-spanner scenario and even for additive +beta-spanners. Further, it generalizes some of these size guarantees to derive new weight guarantees. Our second approach, Randomized Rounding, establishes a graph transformation that allows a simple rounding scheme over a standard multicommodity flow LP. It yields an O(n log n)-approximation, assuming integer lengths and polynomially bounded distance demands. The only other known approximation guarantee in this general setting requires several complex subalgorithms and analyses, yet we match it up to a factor of O(n^{1/5-eps}) using standard tools. Further, on bounded-degree graphs, we yield the first O(log n) approximation ratio for constant-bounded distance demands (beyond multiplicative 2-spanners in unit-length graphs).

cs.DS↗
arXiv2025

Traffic-Oblivious Multi-Commodity Flow Network Design

We consider the Minimum Multi-Commodity Flow Subgraph (MMCFS) problem: given a directed graph $G$ with edge capacities $\mathit{cap}$ and a retention ratio $\alpha\in(0,1)$, find an edge-wise minimum subgraph $G' \subseteq G$ such that for all traffic matrices $T$ routable in $G$ using a multi-commodity flow, $\alpha\cdot T$ is routable in $G'$. This natural yet novel problem is motivated by recent research that investigates how the power consumption in backbone computer networks can be reduced by turning off connections during times of low demand without compromising the quality of service. Since the actual traffic demands are generally not known beforehand, our approach must be traffic-oblivious, i.e., work for all possible sets of simultaneously routable traffic demands in the original network. In this paper we present the problem, relate it to other known problems in literature, and show several structural results, including a reformulation, maximum possible deviations from the optimum, and NP-hardness (as well as a certain inapproximability) already on very restricted instances. The most significant contribution is a $\max(\frac{1}{\alpha}, 2)$-approximation based on a surprisingly simple LP-rounding scheme. We also give instances where this worst-case approximation ratio is met and thus prove that our analysis is tight.

cs.DS↗
arXiv2024

Two-edge-connected (not necessarily spanning) subgraphs and polyhedra

Given a graph $G$, we study the $2$-edge-connected subgraph polytope $\mathrm{TECSP}(G)$, which is given by the convex hull of the incidence vectors of all $2$-edge-connected subgraphs of $G$. We describe the lattice points of this polytope by linear inequalities which provides an ILP-algorithm for finding a $2$-edge-connected subgraph of maximum weight. Furthermore, we characterize when these inequalities define facets of $\mathrm{TECSP}(G)$. We also consider further types of supporting hyperplanes of $\mathrm{TECSP}(G)$ and study when they are facet-defining. Finally, we investigate the efficiency of our considered inequalities practically on some classes of graphs.

math.CO↗
arXiv2024

A Systematic Approach to Crossing Numbers of Cartesian Products with Paths

Determining the crossing numbers of Cartesian products of small graphs with arbitrarily large paths has been an ongoing topic of research since the 1970s. Doing so requires the establishment of coincident upper and lower bounds; the former is usually demonstrated by providing a suitable drawing procedure, while the latter often requires substantial theoretical arguments. Many such papers have been published, which typically focus on just one or two small graphs at a time, and use ad hoc arguments specific to those graphs. We propose a general approach which, when successful, establishes the required lower bound. This approach can be applied to the Cartesian product of any graph with arbitrarily large paths, and in each case involves solving a modified version of the crossing number problem on a finite number (typically only two or three) of small graphs. We demonstrate the potency of this approach by applying it to Cartesian products involving all 133 graphs $G$ of orders five or six, and show that it is successful in 128 cases. This includes 60 cases which a recent survey listed as either undetermined, or determined only in journals without adequate peer review.

math.CO↗
arXiv2024

The Price of Upwardness

Not every directed acyclic graph (DAG) whose underlying undirected graph is planar admits an upward planar drawing. We are interested in pushing the notion of upward drawings beyond planarity by considering upward $k$-planar drawings of DAGs in which the edges are monotonically increasing in a common direction and every edge is crossed at most $k$ times for some integer $k \ge 1$. We show that the number of crossings per edge in a monotone drawing is in general unbounded for the class of bipartite outerplanar, cubic, or bounded pathwidth DAGs. However, it is at most two for outerpaths and it is at most quadratic in the bandwidth in general. From the computational point of view, we prove that testing upward-$k$-planarity is NP-complete already for $k=1$ and even for restricted instances for which upward planarity testing is polynomial. On the positive side, we can decide in linear time whether a single-source DAG admits an upward 1-planar drawing in which all vertices are incident to the outer face.

cs.CG↗
arXiv2024

Crossing Numbers of Beyond Planar Graphs Re-revisited: A Framework Approach

Beyond planarity concepts (prominent examples include k-planarity or fan-planarity) apply certain restrictions on the allowed patterns of crossings in drawings. It is natural to ask, how much the number of crossings may increase over the traditional (unrestricted) crossing number. Previous approaches to bound such ratios, e.g. [arXiv:1908.03153, arXiv:2105.12452], require very specialized constructions and arguments for each considered beyond planarity concept, and mostly only yield asymptotically non-tight bounds. We propose a very general proof framework that allows us to obtain asymptotically tight bounds, and where the concept-specific parts of the proof typically boil down to a couple of lines. We show the strength of our approach by giving improved or first bounds for several beyond planarity concepts.

cs.CG↗
arXiv2024

Exact Minimum Weight Spanners via Column Generation

Given a weighted graph $G$, a minimum weight $\alpha$-spanner is a least-weight subgraph $H\subseteq G$ that preserves minimum distances between all node pairs up to a factor of $\alpha$. There are many results on heuristics and approximation algorithms, including a recent investigation of their practical performance [20]. Exact approaches, in contrast, have long been denounced as impractical: The first exact ILP (integer linear program) method [48] from 2004 is based on a model with exponentially many path variables, solved via column generation. A second approach [2], modeling via arc-based multicommodity flow, was presented in 2019. In both cases, only graphs with 40-100 nodes were reported to be solvable. In this paper, we briefly report on a theoretical comparison between these two models from a polyhedral point of view, and then concentrate on improvements and engineering aspects. We evaluate their performance in a large-scale empirical study. We report that our tuned column generation approach, based on multicriteria shortest path computations, is able to solve instances with over 16000 nodes within 13 minutes. Furthermore, now knowing optimal solutions for larger graphs, we are able to investigate the quality of the strongest known heuristic on reasonably sized instances for the first time.

cs.DS↗
arXiv2023

Green Traffic Engineering by Line Card Minimization

Green Traffic Engineering encompasses network design and traffic routing strategies that aim at reducing the power consumption of a backbone network. We argue that turning off linecards is the most effective approach to reach this goal. Thus, we investigate the problem of minimizing the number of active line cards in a network while simultaneously allowing a multi-commodity flow being routed and keeping the maximum link utilization below a certain threshold. In addition to proving this problem to be NP-hard, we present an optimal ILP-based algorithm as well as a heuristic based on 2-Segment Routing. Lastly, we evaluate both approaches on real-world networks obtained from the Repetita Framework and a globally operating Internet Service Provider. The results of this evaluation indicate that our heuristic is not only close to optimal but significantly faster than the optimal algorithm, making it viable in practice.

cs.NI↗
arXiv2023

On the connected blocks polytope

In this paper, we study the connected blocks polytope, which, apart from its own merits, can be seen as the generalization of certain connectivity based or Eulerian subgraph polytopes. We provide a complete facet description of this polytope, characterize its edges and show that it is Hirsch. We also show that connected blocks polytopes admit a regular unimodular triangulation by constructing a squarefree Gr\"obner basis. In addition, we prove that the polytope is Gorenstein of index $2$ and that its $h^\ast$-vector is unimodal.

math.CO↗
arXiv2023

Directed Capacity-Preserving Subgraphs: Hardness and Exact Polynomial Algorithms

We introduce and discuss the Minimum Capacity-Preserving Subgraph (MCPS) problem: given a directed graph and a retention ratio $\alpha \in (0,1)$, find the smallest subgraph that, for each pair of vertices $(u,v)$, preserves at least a fraction $\alpha$ of a maximum $u$-$v$-flow's value. This problem originates from the practical setting of reducing the power consumption in a computer network: it models turning off as many links as possible while retaining the ability to transmit at least $\alpha$ times the traffic compared to the original network. First we prove that MCPS is NP-hard already on a restricted set of directed acyclic graphs (DAGs) with unit edge capacities. Our reduction also shows that a closely related problem (which only considers the arguably most complicated core of the problem in the objective function) is NP-hard to approximate within a sublogarithmic factor already on DAGs. In terms of positive results, we present two algorithms that solve MCPS optimally on directed series-parallel graphs (DSPs): a simple linear-time algorithm for the special case of unit edge capacities and a cubic-time dynamic programming algorithm for the general case of non-uniform edge capacities. Further, we introduce the family of laminar series-parallel graphs (LSPs), a generalization of DSPs that also includes cyclic and very dense graphs. Their properties allow us to solve MCPS on LSPs by employing our DSP-algorithms as subroutines. In addition, we give a separate quadratic-time algorithm for MCPS on LSPs with unit edge capacities that also yields straightforward quadratic time algorithms for several related problems such as Minimum Equivalent Digraph and Directed Hamiltonian Cycle on LSPs.

cs.DS↗
arXiv2021

Properties of Large 2-Crossing-Critical Graphs

A $c$-crossing-critical graph is one that has crossing number at least $c$ but each of its proper subgraphs has crossing number less than $c$. Recently, a set of explicit construction rules was identified by Bokal, Oporowski, Richter, and Salazar to generate all large $2$-crossing-critical graphs (i.e., all apart from a finite set of small sporadic graphs). They share the property of containing a generalized Wagner graph $V_{10}$ as a subdivision. In this paper, we study these graphs and establish their order, simple crossing number, edge cover number, clique number, maximum degree, chromatic number, chromatic index, and treewidth. We also show that the graphs are linear-time recognizable and that all our proofs lead to efficient algorithms for the above measures.

cs.DM↗
arXiv2021

On the Dominant of the Multicut Polytope

Given a graph $G=(V,E)$ and a set $S \subseteq \binom{V}{2}$ of terminal pairs, the minimum multicut problem asks for a minimum edge set $δ\subseteq E$ such that there is no $s$-$t$-path in $G -δ$ for any $\{s,t\}\in S$. For $|S|=1$ this is the well known $s$-$t$-cut problem, but in general the minimum multicut problem is NP-complete, even if the input graph is a tree. The multicut polytope $\text{MultC}^\square (G,S)$ is the convex hull of all multicuts in $G$; the multicut dominant is given by $\text{MultC}(G,S)=\text{MultC}^\square (G,S)+\mathbb{R}^E$. The latter is the relevant object for the minimization problem. While polyhedra associated to several cut problems have been studied intensively there is only little knowledge for multicut. We investigate properties of the multicut dominant and in particular derive results on liftings of facet-defining inequalities. This yields a classification of all facet-defining path- and edge inequalities. Moreover, we investigate the effect of graph operations such as node splitting, edge subdivisions, and edge contractions on the multicut-dominant and its facet-defining inequalities. In addition, we introduce facet-defining inequalities supported on stars, trees, and cycles and show that the former two can be separated in polynomial time when the input graph is a tree.

math.CO↗
arXiv2021

Star-Struck by Fixed Embeddings: Modern Crossing Number Heuristics

We present a thorough experimental evaluation of several crossing minimization heuristics that are based on the construction and iterative improvement of a planarization, i.e., a planar representation of a graph with crossings replaced by dummy vertices. The evaluated heuristics include variations and combinations of the well-known planarization method, the recently implemented star reinsertion method, and a new approach proposed herein: the mixed insertion method. Our experiments reveal the importance of several implementation details such as the detection of non-simple crossings (i.e., crossings between adjacent edges or multiple crossings between the same two edges). The most notable finding, however, is that the insertion of stars in a fixed embedding setting is not only significantly faster than the insertion of edges in a variable embedding setting, but also leads to solutions of higher quality.

cs.DM↗
arXiv2021

A General Approach to Approximate Multistage Subgraph Problems

In a Subgraph Problem we are given some graph and want to find a feasible subgraph that optimizes some measure. We consider Multistage Subgraph Problems (MSPs), where we are given a sequence of graph instances (stages) and are asked to find a sequence of subgraphs, one for each stage, such that each is optimal for its respective stage and the subgraphs for subsequent stages are as similar as possible. We present a framework that provides a $(1/\sqrt{2\chi})$-approximation algorithm for the $2$-stage restriction of an MSP if the similarity of subsequent solutions is measured as the intersection cardinality and said MSP is preficient, i.e., we can efficiently find a single-stage solution that prefers some given subset. The approximation factor is dependent on the instance's intertwinement $\chi$, a similarity measure for multistage graphs. We also show that for any MSP, independent of similarity measure and preficiency, given an exact or approximation algorithm for a constant number of stages, we can approximate the MSP for an unrestricted number of stages. Finally, we combine and apply these results and show that the above restrictions describe a very rich class of MSPs and that proving membership for this class is mostly straightforward. As examples, we explicitly state these proofs for natural multistage versions of Perfect Matching, Shortest s-t-Path, Minimum s-t-Cut and further classical problems on bipartite or planar graphs, namely Maximum Cut, Vertex Cover, Independent Set, and Biclique.

cs.DS↗
arXiv2021

Spanner Approximations in Practice

A multiplicative $α$-spanner $H$ is a subgraph of $G=(V,E)$ with the same vertices and fewer edges that preserves distances up to the factor $α$, i.e., $d_H(u,v)\leqα\cdot d_G(u,v)$ for all vertices $u$, $v$. While many algorithms have been developed to find good spanners in terms of approximation guarantees, no experimental studies comparing different approaches exist. We implemented a rich selection of those algorithms and evaluate them on a variety of instances regarding, e.g., their running time, sparseness, lightness, and effective stretch.

cs.DS↗