arXiv subjects

Michał Szyfelbein

Publications and source records attributed to Michał Szyfelbein.

9 recordsLinked to original sources

arXiv2026

Min-Sum Set Cover on Parallel Machines

We consider a generalization of the Min-Sum Set Cover to the setup with $m$ set-sequences, or in scheduling terminology, $m$ parallel machines. We call this problem Parallel Min-Sum Set Cover. To obtain approximation algorithms for its numerous variants we use a crucial sub-problem called Parallel Densest Subfamily. We prove that an $α$-approximation algorithm for this task gives a $4\cdotα$-approximation for the Parallel Min-Sum Set Cover, which yields $\frac{4\cdot e}{e-1}+ε$ and $4\cdot \frac{e}{e-1}^2+ε$-approximation ratios for identical and unrelated machines, respectively. To obtain the latter result we give a new $\frac{e}{e-1}^2+ε$-approximation algorithm for the Maximum Coverage Multiple Knapsacks problem which is of independent interest. If the sets are precedence-constrained, for unit cost sets we give an $\mathcal{O}(k^{2/3})$ approximation ($k$ is the number of sets). For the case of out-forest precedence constraints we improve this bound to $\mathcal{O}(\log k)$ via a reduction to the Group Steiner Orienteering problem, and show this is tight, unless $NP\subseteq ZTIME(n^{\mathcal{O}(\text{poly}(\log n))})$.

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arXiv2026

Simpler Logarithmic Approximation Algorithms for the Optimal Decision Tree and Adaptive Set Cover

We study a well-known task of constructing a decision tree identifying an unknown hypothesis from a given ground set of hypotheses under both the average- and worst-case cost. The Optimal Decision Tree problem has been extensively studied in the literature and $\mathcal{O}(\log n)$-approximation guarantees are known for both cost criteria (here $n$ is the number of hypotheses). Although the algorithms achieving this approximation ratio are usually relatively simple, their analysis often turns out to be quite technical. Hereby, we show a new algorithm with a simplified analysis which simultaneously achieves an $\mathcal{O}(\log n)$-approximation for both cost criteria. Moreover, the leading constant under the $\mathcal{O}$-notation in the approximation ratio is relatively small, below $3.65$ for the worst-case cost and twice as much for the average-case cost, assuming base $2$ logarithm. Our algorithm for the Optimal Decision Tree behaves greedily with respect to the hereby introduced Separating Subfamily problem which asks for the cheapest subfamily of tests which partitions the hypotheses into pieces of small enough size. We show that the Separating Subfamily can itself be reduced to an instance of the well-known Maximum Coverage problem. At the core of our approach lies exploiting properties of cutting a clique into small pieces, where edges represent pairs of hypotheses to be separated. As an application of our result we also provide an $\mathcal{O}(\log (k\cdot n))$-approximation for the Adaptive Set Cover problem, where $k$ is the number of possible realizations.

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arXiv2026

Hierarchical $\mathcal{F}$-Clustering: Approximation and Hardness of Clustering into Trees and Bounded Diameter Graphs

Consider the following variation on the Hierarchical Clustering problem: Usually, while building a hierarchical clustering, one recursively partitions the data until each cluster becomes a singleton. We relax the halting condition of the recursive process to stop whenever the remaining cluster is a graph belonging to a class $\mathcal{F}$. We call this problem Hierarchical $\mathcal{F}$-Clustering and we measure the quality of any solution using adapted Dasgupta's clustering objective. We study two natural choices of $\mathcal{F}$: trees and graphs of bounded diameter. We present the first polynomial time $\mathcal{O}(\log n\cdot\log\log n)$ and $\mathcal{O}(\log n)$-approximation algorithms for clustering into trees and bounded diameter graphs respectively. Our main technical contribution is a framework for approximating such problems based on linear programming. In fact, we characterize graphs classes $\mathcal{F}$ for which our approach can be applied and show that it includes both trees and bounded diameter graphs. However, our ideas are not limited to them and might be useful for other structures as well. Broadly speaking, our framework applies whenever the corresponding flat clustering problem, which we call $p_{\mathcal{F}}$-Partitioning, admits a natural ILP formulation together with a rounding procedure with provable approximation guarantees. Intuitively, given a set of vertices called terminals, the problem is to find an edge set whose removal results in satisfying certain vertex-dependent structural predicate for each terminal. We then use these ingredients to build clustering trees with the aforementioned approximation guarantees. To complement these results, we show that both Hierarchical Clustering into trees and into bounded diameter graphs cannot be approximated within any constant factor under the Small Set Expansion Hypothesis.

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arXiv2026

Graph Partitioning with Demands: Generalized Conductance and its Applications

In this work, we study various graph partitioning problems under a general demand model. In each such task, we are given a graph $G=(V,E,c,w)$ with a capacity function $c\colon E\to \mathbb{N}$ and a demand function $w\colon V\times V\to \mathbb{N}$. Our main focus is the problem of finding a cut $(S, \bar{S})$ minimizing the quantity \[ ψ_w( S ) = \frac{c( S, \bar{S} )}{w( S, V )\cdot w( \bar{S}, V )}. \] Here, $c( S, \bar{S} )$ is the cost of edges between $S$ and the complement of $S$, $\bar{S}$, and $w( S, V )=w( S )+w( S, \bar{S} )$ is the sum of the internal demand within $S$, $w( S )$, and the demand between vertices of $S$ and $\bar{S}$, $w( S, \bar{S} )$. We call $ψ_w( S )$ the \emph{generalized conductance} of the cut $(S, \bar{S})$, and the task of minimizing $ψ_w( S )$ the Generalized Conductance Problem. Our main contribution is an algorithm with an $\mathcal{O}(\log n)$-approximation guarantee for this objective. Our result is achieved via a two-way reduction: first to the well-known Generalized $k$-Multicut Problem, and then to a constrained variant of the classic Sparsest-Cut Problem, with an additional upper-bound constraint on the amount of demand that may be cut. Moreover, we show that the above procedure can be used to obtain an $\mathcal{O}(\log n)$-bicriteria approximation for Graph Partitioning with Demands, where the goal is to find a minimum-cost subset of edges $C$ such that for every component $H$ of $G\setminus C$, $w( H )\leq ρ\cdot w( V )$. This, in turn, yields an $\mathcal{O}(\log n)$-approximation for Hierarchical Clustering with Demands, the problem of finding a hierarchy of cuts that partitions the graph into increasingly refined clusters. For multiplicative demand functions, we improve these guarantees to $\mathcal{O}(\sqrt{\log n})$ and for trees we get an $\mathcal{O}(1)$-approximation for all of our objectives.

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arXiv2026

Precedence-Constrained Decision Trees and Coverings

This work considers a number of optimization problems and reductive relations between them. The two main problems we are interested in are the Optimal Decision Tree and Set Cover. We study these two fundamental tasks under precedence constraints, that is, if a test (or set) $X$ is a predecessor of $Y$, then in any feasible decision tree $X$ needs to be an ancestor of $Y$ (or respectively, if $Y$ is added to set cover, then so must be $X$). For the Optimal Decision Tree we consider two optimization criteria: worst case identification time (height of the tree) or the average identification time. Similarly, for the Set Cover we study two cost measures: the size of the cover or the average cover time. Our approach is to develop a number of algorithmic reductions, where an approximation algorithm for one problem provides an approximation for another via a black-box usage of a procedure for the former. En route we introduce other optimization problems either to complete the `reduction landscape' or because they hold the essence of combinatorial structure of our problems. The latter is brought by a problem of finding a Maximum Density Precedence-Closed Subfamily, where the density is defined as the ratio of the number of items the family covers to its size. We provide $\mathcal{O}^*(\sqrt{m})$-approximation polynomial-time algorithms for all aforementioned problems. The picture is complemented by a number of hardness reductions that provide $\mathcal{O}(m^{1/12-ε})$-inapproximability results for the decision tree and covering problems. Besides giving a complete set of results for general precedence constraints, we also provide polylogarithmic approximation guarantees for two most typically studied and applicable graph types, outforests and inforests. By providing corresponding hardness results, we show most of these results to be tight.

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arXiv2026

Polylogarithmic Approximation for Covering and Connecting Multi-Interface Networks

We study problems related to connecting multi-interface networks of wireless devices. These problems can be modeled using graphs, where vertices represent the devices and edges represent potential communication links. Each vertex can activate multiple interfaces, and a connection between two vertices is established if they share at least one common active interface. However, activating an interface induces a cost that depends both on the type of the interface and on the vertex that activates it. We consider two problems arising in multi-interface networks: Coverage and Connectivity. In the Coverage problem, every connection defined in the network must be established, while in the Connectivity problem, it is only required that the established connections form a subgraph spanning the network. The solution should also minimize the maximum cost incurred by a node or the total cost incurred by all vertices. We model both problems using Integer Linear Programming (ILP) and we design approximation algorithms based on a randomized rounding of the solution of the linear programming relaxation. For the Coverage problem, this yields an $O(\log n)$-approximation algorithm, where $n$ is the number of vertices. This result is tight, since the problem generalizes Set Cover. This improves upon the $O(b\cdot\log n)$-approximation algorithm, where $b$ is a certain graph parameter which can be as large as $Ω(n)$ [Algorithmica '12]. The main result of our work is an $O(\log^2 n)$-approximation algorithm for the Connectivity, which is the first non-trivial approximation for this problem. The algorithm is based on a similar LP relaxation with additional cut constraints to ensure connectivity. The rounding procedure resembles the one for the Coverage but requires a more careful analysis to ensure that the connectivity constraints are satisfied.

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arXiv2026

Average Case Graph Searching in Non-Uniform Cost Models

We consider the following generalization of the classic Binary Search Problem: a searcher is required to find a hidden target vertex $x$ in a graph $G$, by iteratively performing queries about vertices. A query to $v$ incurs a cost $c(v, x)$ and responds whether $v=x$ and if not, returns the connected component in $G-v$ containing $x$. The goal is to design a search strategy that minimizes the average-case search cost. Firstly, we consider the case when the cost of querying a vertex is independent of the target. We develop a $\br{4+ε}$-approximation FPTAS for trees running in $O(n^4/ε^2)$ time and an $O({\sqrt{\log n}})$-approximation for general graphs. Additionally, we give an FPTAS parametrized by the number of non-leaf vertices of the graph. On the hardness side we prove that the problem is NP-hard even when the input is a tree with bounded degree or bounded diameter. Secondly, we consider trees and assume $c(v, x)$ to be a monotone non-decreasing function with respect to $x$, i.e.\ if $u \in P_{v, x}$ then $c(u, x) \leq c(v, x)$. We give a $2$-approximation algorithm which can also be easily altered to work for the worst-case variant. This is the first constant factor approximation algorithm for both criterions. Previously known results only regard the worst-case search cost and include a parametrized PTAS as well as a $4$-approximation for paths. At last, we show that when the cost function is an arbitrary function of the queried vertex and the target, then the problem does not admit any constant factor approximation under the UGC, even when the input tree is a star.

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arXiv2025

Approximating the Average-Case Graph Search Problem with Non-Uniform Costs

Consider the following generalization of the classic binary search problem: A searcher is required to find a hidden target vertex $x$ in a graph $G$. To do so, they iteratively perform queries to an oracle, each about a chosen vertex $v$. After each such call, the oracle responds whether the target was found and if not, the searcher receives as a reply the connected component in $G-v$ which contains $x$. Additionally, each vertex $v$ may have a different query cost $c(v)$ and a different weight $w(v)$. The goal is to find the optimal querying strategy which minimizes the weighted average-case cost required to find $x$. The problem is NP-hard even for uniform weights and query costs. Inspired by the progress on the edge query variant of the problem [SODA '17], we establish a connection between searching and vertex separation. By doing so, we provide an $O(\sqrt{\log n})$-approximation algorithm for general graphs and a $(4+ε)$-approximation algorithm for the case when the input is a tree.

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arXiv2025

Searching in trees with $k$-up-modular cost functions

Consider the following generalization of the classic binary search problem: a searcher is required to find a hidden vertex $x$ in a tree $T$. To do so, they iteratively perform queries to an oracle, each about a chosen vertex $v$. After each such call, the oracle responds whether the target was found and if not, the searcher receives as a reply the connected component of $T-v$ which contains $x$. Additionally, each vertex $v$ may have a different query cost $c(v)$. The goal is to find the optimal querying strategy which minimizes the worst case cost required to find $x$. The problem is known to be NP-hard even in restricted classes of trees such as bounded diameter spiders [Cicalese et al. 2016], and no constant factor approximation algorithm is known for general trees. Following the recent studies of [Dereniowski et al. 2022, Dereniowski et al. 2024], instead of restricted classes of trees, we explore restrictions on the cost function. We generalize the notion of up-monotonic functions and introduce the concept of \textit{$k$-up-modularity}. We show that an $O(\log\log n)$-approximate solution can be found within $k^{O(\log k)}\cdot\text{poly}(n)$ time.

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