Danil Sagunov

Russian Academy of Sciences | RAS · St.Petersburg Department of Steklov Mathematical Institute

We study two natural generalizations of q-Coloring. These problems can be seen as optimization problems and are mostly applied to graphs that are not properly colorable with q colors. One of them is known as Maximumq-Colorable Induced Subgraph, and asks to find the largest set of vertices inducing a q-colorable graph. While very natural, this gener...

We present a study of a few graph based problems motivated by fair allocation of resources in a social network. The central role in the paper is played by the following problem: What is the largest number of items we can allocate to the agents in the given social network so that each agent hides at most one item and overall at most k items are hidd...

A k-core of a graph G is the maximal induced subgraph in which every vertex has degree at least k. In the Edge k-Core optimization problem, we are given a graph G and integers k, b and p. The task is to ensure that the k-core of G has at least p vertices, by adding at most b edges. While Edge k-Core is known to be computationally hard in general, w...

In this paper we consider the Target Set Selection problem. The problem naturally arises in many fields like economy, sociology, medicine. In the Target Set Selection problem one is given a graph G with a function \({{\,\mathrm{thr}\,}}: V(G) \rightarrow {\mathbb {N}} \cup \{0\}\) and two integers \(k, \ell \). The goal of the problem is to activat...

There is a well-known approach to cope with NP-hard problems in practice: reduce the given problem to SAT or MAXSAT and run a SAT or a MaxSAT solver. This method is very efficient since SAT/MaxSAT solvers are extremely well-studied, as well as the complexity of these problems. At AAAI 2011, Li et al. proposed an alternative to this approach and sug...

In 1959, Erd\H{o}s and Gallai proved that every graph G with average vertex degree ad(G)\geq 2 contains a cycle of length at least ad(G). We provide an algorithm that for k\geq 0 in time 2^{O(k)} n^{O(1)} decides whether a 2-connected n-vertex graph G contains a cycle of length at least ad(G)+k. This resolves an open problem explicitly mentioned in...

We study two "above guarantee" versions of the classical Longest Path problem on undirected and directed graphs and obtain the following results. In the first variant of Longest Path that we study, called Longest Detour, the task is to decide whether a graph has an (s,t)-path of length at least dist_G(s,t)+k (where dist_G(s,t) denotes the length of...

This paper is motivated by seeking the exact complexity of resolution refutation of Tseitin formulas. We prove that the size of any regular resolution refutation of a Tseitin formula \( {\rm T}(G, c)\) based on a connected graph \({G} =(V, E)\) is at least \(2^{\Omega({\rm tw}(G)/ \log |V|)}\), where \({\rm tw}(G)\) denotes the treewidth of a graph...

Clique-width is one of the most important parameters that describes structural complexity of a graph. Probably, only treewidth is more studied graph width parameter. In this paper we study how clique-width influences the complexity of the Maximum Happy Vertices (MHV) and Maximum Happy Edges (MHE) problems. We answer a question of Choudhari and Redd...

In 1952, Dirac proved the following theorem about long cycles in graphs with large minimum vertex degrees: Every $n$-vertex $2$-connected graph $G$ with minimum vertex degree $\delta\geq 2$ contains a cycle with at least $\min\{2\delta,n\}$ vertices. In particular, if $\delta\geq n/2$, then $G$ is Hamiltonian. The proof of Dirac's theorem is constr...

We initiate the study of the Diverse Pair of (Maximum/ Perfect) Matchings problems which given a graph $G$ and an integer $k$, ask whether $G$ has two (maximum/perfect) matchings whose symmetric difference is at least $k$. Diverse Pair of Matchings (asking for two not necessarily maximum or perfect matchings) is NP-complete on general graphs if $k$...

In this paper, we study the Maximum Happy Vertices and the Maximum Happy Edges problems (MHV and MHE for short). Very recently, the problems attracted a lot of attention and were studied in Agrawal '18, Aravind et al. '16, Choudhari and Reddy '18, Misra and Reddy '18. Main focus of our work is lower bounds on the computational complexity of these p...

Clique-width is one of the most important parameters that describes structural complexity of a graph. Probably, only treewidth is more studied graph width parameter. In this paper we study how clique-width influences the complexity of the Maximum Happy Vertices (MHV) and Maximum Happy Edges (MHE) problems. We answer a question of Choudhari and Redd...

A popular model to measure network stability is the $k$-core, that is the maximal induced subgraph in which every vertex has degree at least $k$. For example, $k$-cores are commonly used to model the unraveling phenomena in social networks. In this model, users having less than $k$ connections within the network leave it, so the remaining users for...

We study the Maximum Happy Vertices and Maximum Happy Edges problems. The former problem is a variant of clusterization, where some vertices have already been assigned to clusters. The second problem gives a natural generalization of Multiway Uncut, which is the complement of the classical Multiway Cut problem. Due to their fundamental role in theo...

In this paper, we study the Maximum Happy Vertices and the Maximum Happy Edges problems (MHV and MHE for short). Very recently, the problems attracted a lot of attention and were studied in Agrawal ’17, Aravind et al. ’16, Choudhari and Reddy ’18, Misra and Reddy ’17. Main focus of our work is lower bounds on the computational complexity of these p...

We study the Maximum Happy Vertices and Maximum Happy Edges problems. The former problem is a variant of clusterization, where some vertices have already been assigned to clusters. The second problem gives a natural generalization of Multiway Uncut, which is the complement of the classical Multiway Cut problem. Due to their fundamental role in theo...

In this paper, we study the Maximum Happy Vertices and the Maximum Happy Edges problems (MHV and MHE for short). Very recently, the problems attracted a lot of attention and were studied in Agrawal '18, Aravind et al. '16, Choudhari and Reddy '18, Misra and Reddy '18. Main focus of our work is lower bounds on the computational complexity of these p...

In this paper we consider the Target Set Selection problem. The problem naturally arises in many fields like economy, sociology, medicine. In the Target Set Selection problem one is given a graph $G$ with a function $\operatorname{thr}: V(G) \to \mathbb{N} \cup \{0\}$ and integers $k, \ell$. The goal of the problem is to activate at most $k$ vertic...