arXiv subjects

Iiro Kumpulainen

Publications and source records attributed to Iiro Kumpulainen.

7 recordsLinked to original sources

arXiv2025

Finding coherent node groups in directed graphs

Grouping the nodes of a graph into clusters is a standard technique for studying networks. We study a problem where we are given a directed network and are asked to partition the graph into a sequence of coherent groups. We assume that nodes in the network have features, and we measure the group coherence by comparing these features. Furthermore, we incorporate the cross edges by penalizing the forward cross edges and backward cross edges with different weights. If the weights are set to 0, then the problem is equivalent to clustering. However, if we penalize the backward edges, the order of discovered groups matters, and we can view our problem as a generalization of a classic segmentation problem. We consider a common iterative approach where we solve the groups given the centroids, and then find the centroids given the groups. We show that, unlike in clustering, the first subproblem is NP-hard. However, we show that we can solve the subproblem exactly if the underlying graph is a tree or if the number of groups is 2. For a general case, we propose an approximation algorithm based on linear programming. We propose 3 additional heuristics: (1) optimizing each pair of groups separately while keeping the remaining groups intact, (2) computing a spanning tree and then optimizing using only the edges in that, and (3) a greedy search moving nodes between the groups while optimizing the overall loss. We demonstrate with our experiments that the algorithms are practical and yield interpretable results.

cs.SI↗

arXiv2025

The Densest SWAMP problem: subhypergraphs with arbitrary monotonic partial edge rewards

We consider a generalization of the densest subhypergraph problem where nonnegative rewards are given for including partial hyperedges in a dense subhypergraph. Prior work addressed this problem only in cases where reward functions are convex, in which case the problem is poly-time solvable. We consider a broader setting where rewards are monotonic but otherwise arbitrary. We first prove hardness results for a wide class of non-convex rewards, then design a 1/k-approximation by projecting to the nearest set of convex rewards, where k is the maximum hyperedge size. We also design another 1/k-approximation using a faster peeling algorithm, which (somewhat surprisingly) differs from the standard greedy peeling algorithm used to approximate other variants of the densest subgraph problem. Our results include an empirical analysis of our algorithm across several real-world hypergraphs.

cs.DS↗

arXiv2025

Improved Hardness and Approximations for Cardinality-Based Minimum $s$-$t$ Cuts Problems in Hypergraphs

In hypergraphs, an edge that crosses a cut (i.e., a bipartition of nodes) can be split in several ways, depending on how many nodes are placed on each side of the cut. A cardinality-based splitting function assigns a nonnegative cost of $w_i$ for each cut hyperedge $e$ with exactly $i$ nodes on the side of the cut that contains the minority of nodes from $e$. The cardinality-based minimum $s$-$t$ cut aims to find an $s$-$t$ cut with minimum total cost. We answer a recently posed open question by proving that the problem becomes NP-hard outside the submodular region shown by~\cite{veldt2022hypergraph}. Our result also holds for $r$-uniform hypergraphs with $r \geq 4$. Specifically for $4$-uniform hypergraphs we show that the problem is NP-hard for all $w_2 > 2$, and additionally prove that the No-Even-Split problem is NP-hard. We then turn our attention to approximation strategies and approximation hardness results in the non-submodular case. We design a strategy for projecting non-submodular penalties to the submodular region, which we prove gives the optimal approximation among all such projection strategies. We also show that alternative approaches are unlikely to provide improved guarantees, by showing matching approximation hardness bounds assuming the Unique Games Conjecture and asymptotically tight approximation hardness bounds assuming $\text{P} \neq \text{NP}$.

cs.CC↗

arXiv2025

Max-Min Diversification with Asymmetric Distances

One of the most well-known and simplest models for diversity maximization is the Max-Min Diversification (MMD) model, which has been extensively studied in the data mining and database literature. In this paper, we initiate the study of the Asymmetric Max-Min Diversification (AMMD) problem. The input is a positive integer $k$ and a complete digraph over $n$ vertices, together with a nonnegative distance function over the edges obeying the directed triangle inequality. The objective is to select a set of $k$ vertices, which maximizes the smallest pairwise distance between them. AMMD reduces to the well-studied MMD problem in case the distances are symmetric, and has natural applications to query result diversification, web search, and facility location problems. Although the MMD problem admits a simple $\frac{1}{2}$-approximation by greedily selecting the next-furthest point, this strategy fails for AMMD and it remained unclear how to design good approximation algorithms for AMMD. We propose a combinatorial $\frac{1}{6k}$-approximation algorithm for AMMD by leveraging connections with the Maximum Antichain problem. We discuss several ways of speeding up the algorithm and compare its performance against heuristic baselines on real-life and synthetic datasets.

cs.DS↗

arXiv2025

Dense Subgraph Discovery Meets Strong Triadic Closure

Finding dense subgraphs is a core problem with numerous graph mining applications such as community detection in social networks and anomaly detection. However, in many real-world networks connections are not equal. One way to label edges as either strong or weak is to use strong triadic closure~(STC). Here, if one node connects strongly with two other nodes, then those two nodes should be connected at least with a weak edge. STC-labelings are not unique and finding the maximum number of strong edges is NP-hard. In this paper, we apply STC to dense subgraph discovery. More formally, our score for a given subgraph is the ratio between the sum of the number of strong edges and weak edges, weighted by a user parameter $λ$, and the number of nodes of the subgraph. Our goal is to find a subgraph and an STC-labeling maximizing the score. We show that for $λ= 1$, our problem is equivalent to finding the densest subgraph, while for $λ= 0$, our problem is equivalent to finding the largest clique, making our problem NP-hard. We propose an exact algorithm based on integer linear programming and four practical polynomial-time heuristics. We present an extensive experimental study that shows that our algorithms can find the ground truth in synthetic datasets and run efficiently in real-world datasets.

cs.DS↗

arXiv2024

From your Block to our Block: How to Find Shared Structure between Stochastic Block Models over Multiple Graphs

Stochastic Block Models (SBMs) are a popular approach to modeling single real-world graphs. The key idea of SBMs is to partition the vertices of the graph into blocks with similar edge densities within, as well as between different blocks. However, what if we are given not one but multiple graphs that are unaligned and of different sizes? How can we find out if these graphs share blocks with similar connectivity structures? In this paper, we propose the shared stochastic block modeling (SSBM) problem, in which we model $n$ graphs using SBMs that share parameters of $s$ blocks. We show that fitting an SSBM is NP-hard, and consider two approaches to fit good models in practice. In the first, we directly maximize the likelihood of the shared model using a Markov chain Monte Carlo algorithm. In the second, we first fit an SBM for each graph and then select which blocks to share. We propose an integer linear program to find the optimal shared blocks and to scale to large numbers of blocks, we propose a fast greedy algorithm. Through extensive empirical evaluation on synthetic and real-world data, we show that our methods work well in practice.

cs.SI↗

arXiv2023

Dense subgraphs induced by edge labels

Finding densely connected groups of nodes in networks is a widely used tool for analysis in graph mining. A popular choice for finding such groups is to find subgraphs with a high average degree. While useful, interpreting such subgraphs may be difficult. On the other hand, many real-world networks have additional information, and we are specifically interested in networks with labels on edges. In this paper, we study finding sets of labels that induce dense subgraphs. We consider two notions of density: average degree and the number of edges minus the number of nodes weighted by a parameter $α$. There are many ways to induce a subgraph from a set of labels, and we study two cases: First, we study conjunctive-induced dense subgraphs, where the subgraph edges need to have all labels. Secondly, we study disjunctive-induced dense subgraphs, where the subgraph edges need to have at least one label. We show that both problems are NP-hard. Because of the hardness, we resort to greedy heuristics. We show that we can implement the greedy search efficiently: the respective running times for finding conjunctive-induced and disjunctive-induced dense subgraphs are in $O(p \log k)$ and $O(p \log^2 k)$, where $p$ is the number of edge-label pairs and $k$ is the number of labels. Our experimental evaluation demonstrates that we can find the ground truth in synthetic graphs and that we can find interpretable subgraphs from real-world networks.

cs.SI↗