arXiv subjects

Gregory Gutin

Publications and source records attributed to Gregory Gutin.

At least 55 records · Page 3Linked to original sources

arXiv2023

Bounds on Maximum Weight Directed Cut

We obtain lower and upper bounds for the maximum weight of a directed cut in the classes of weighted digraphs and weighted acyclic digraphs as well as in some of their subclasses. We compare our results with those obtained for the maximum size of a directed cut in unweighted digraphs. In particular, we show that a lower bound obtained by Alon, Bollobas, Gyafas, Lehel and Scott (J Graph Th 55(1) (2007)) for unweighted acyclic digraphs can be extended to weighted digraphs with the maximum length of a cycle being bounded by a constant and the weight of every arc being at least one. We state a number of open problems.

math.CO↗

arXiv2022

Preference Swaps for the Stable Matching Problem

An instance $I$ of the Stable Matching Problem (SMP) is given by a bipartite graph with a preference list of neighbors for every vertex. A swap in $I$ is the exchange of two consecutive vertices in a preference list. A swap can be viewed as a smallest perturbation of $I$. Boehmer et al. (2021) designed a polynomial-time algorithm to find the minimum number of swaps required to turn a given maximal matching into a stable matching. We generalize this result to the many-to-many version of SMP. We do so first by introducing a new representation of SMP as an extended bipartite graph and subsequently by reducing the problem to submodular minimization. It is a natural problem to establish the computational complexity of deciding whether at most $k$ swaps are enough to turn $I$ into an instance where one of the maximum matchings is stable. Using a hardness result of Gupta et al. (2020), we prove that this problem is NP-hard and, moreover, this problem parameterised by $k$ is W[1]-hard. We also obtain a lower bound on the running time for solving the problem using the Exponential Time Hypothesis.

cs.DS↗

arXiv2022

Solving the Workflow Satisfiability Problem using General Purpose Solvers

The workflow satisfiability problem (WSP) is a well-studied problem in access control seeking allocation of authorised users to every step of the workflow, subject to workflow specification constraints. It was noticed that the number $k$ of steps is typically small compared to the number of users in the real-world instances of WSP; therefore $k$ is considered as the parameter in WSP parametrised complexity research. While WSP in general was shown to be W[1]-hard, WSP restricted to a special case of user-independent (UI) constraints is fixed-parameter tractable (FPT). However, restriction to the UI constraints might be impractical. To efficiently handle non-UI constraints, we introduce the notion of branching factor of a constraint. As long as the branching factors of the constraints are relatively small and the number of non-UI constraints is reasonable, WSP can be solved in FPT time. Extending the results from Karapetyan et al. (2019), we demonstrate that general-purpose solvers are capable of achieving FPT-like performance on WSP with arbitrary constraints when used with appropriate formulations. This enables one to tackle most of practical WSP instances. While important on its own, we hope that this result will also motivate researchers to look for FPT-aware formulations of other FPT problems.

cs.AI↗

arXiv2022

p-Edge/Vertex-Connected Vertex Cover: Parameterized and Approximation Algorithms

We introduce and study two natural generalizations of the Connected VertexCover (VC) problem: the $p$-Edge-Connected and $p$-Vertex-Connected VC problem (where $p \geq 2$ is a fixed integer). Like Connected VC, both new VC problems are FPT, but do not admit a polynomial kernel unless $NP \subseteq coNP/poly$, which is highly unlikely. We prove however that both problems admit time efficient polynomial sized approximate kernelization schemes. We obtain an $O(2^{O(pk)}n^{O(1)})$-time algorithm for the $p$-Edge-Connected VC and an $O(2^{O(k^2)}n^{O(1)})$-time algorithm for the $p$-Vertex-Connected VC. Finally, we describe a $2(p+1)$-approximation algorithm for the $p$-Edge-Connected VC. The proofs for the new VC problems require more sophisticated arguments than for Connected VC. In particular, for the approximation algorithm we use Gomory-Hu trees and for the approximate kernels a result on small-size spanning $p$-vertex/edge-connected subgraph of a $p$-vertex/edge-connected graph obtained independently by Nishizeki and Poljak (1994) and Nagamochi and Ibaraki (1992).

cs.DS↗

arXiv2022

Results on the Small Quasi-Kernel Conjecture

A {\em quasi-kernel} of a digraph $D$ is an independent set $Q\subseteq V(D)$ such that for every vertex $v\in V(D)\backslash Q$, there exists a directed path with one or two arcs from $v$ to a vertex $u\in Q$. In 1974, Chvátal and Lovász proved that every digraph has a quasi-kernel. In 1976, Erdős and Sźekely conjectured that every sink-free digraph $D=(V(D),A(D))$ has a quasi-kernel of size at most $|V(D)|/2$. In this paper, we give a new method to show that the conjecture holds for a generalization of anti-claw-free digraphs. For any sink-free one-way split digraph $D$ of order $n$, when $n\geq 3$, we show a stronger result that $D$ has a quasi-kernel of size at most $\frac{n+3}{2} - \sqrt{n}$, and the bound is sharp.

math.CO↗

arXiv2022

Exact capacitated domination: on the computational complexity of uniqueness

In this paper we consider a local service-requirement assignment problem named exact capacitated domination from an algorithmic point of view. This problem aims to find a solution (a Nash equilibrium) to a game-theoretic model of public good provision. In the problem we are given a capacitated graph, a graph with a parameter defined on each vertex that is interpreted as the capacity of that vertex. The objective is to find a DP-Nash subgraph: a spanning bipartite subgraph with partite sets D and P, called the D-set and P-set respectively, such that no vertex in P is isolated and that each vertex in D is adjacent to a number of vertices equal to its capacity. We show that whether a capacitated graph has a unique DP-Nash subgraph can be decided in polynomial time. However, we also show that the nearby problem of deciding whether a capacitated graph has a unique D-set is co-NP-complete.

math.CO↗

arXiv2022

Solving Routing Problems via Important Cuts

We introduce a novel approach of using important cuts which allowed us to design significantly faster fixed-parameter tractable (FPT) algorithms for the following routing problems: the Mixed Chinese Postman Problem parameterized by the number of directed edges (Gutin et al., JCSS 2017), the Minimum Shared Edges problem (MSE) parameterized by the number p of paths between two specified vertices (Fluschnik et al., JCSS 2019), and the Weighted Min Cut Prevention problem (Gruttemeier et al., WG 2021). The Minimum Vulnerability problem (MV) is a generalization of MSE (Assadi et al., Algorithmica 2014). The only known FPT algorithm for MV parameterized by p (the same parameter as for MSE) was for chordal graphs (Aoki et al., JCO 2018). We design an FPT algorithm for MV on all undirected graphs.

cs.DS↗

arXiv2022

Strong subgraph 2-arc-connectivity and arc-strong connectivity of Cartesian product of digraphs

Let $D=(V,A)$ be a digraph of order $n$, $S$ a subset of $V$ of size $k$ and $2\le k\leq n$. A strong subgraph $H$ of $D$ is called an $S$-strong subgraph if $S\subseteq V(H)$. A pair of $S$-strong subgraphs $D_1$ and $D_2$ are said to be arc-disjoint if $A(D_1)\cap A(D_2)=\emptyset$. Let $λ_S(D)$ be the maximum number of arc-disjoint $S$-strong subgraphs in $D$. The strong subgraph $k$-arc-connectivity is defined as $$λ_k(D)=\min\{λ_S(D)\mid S\subseteq V(D), |S|=k\}.$$ The parameter $λ_k(D)$ can be seen as a generalization of classical edge-connectivity of undirected graphs. In this paper, we first obtain a formula for the arc-connectivity of Cartesian product $λ(G\Box H)$ of two digraphs $G$ and $H$ generalizing a formula for edge-connectivity of Cartesian product of two undirected graphs obtained by Xu and Yang (2006). Then we study the strong subgraph 2-arc-connectivity of Cartesian product $λ_2(G\Box H)$ and prove that $ \min\left \{ λ\left ( G \right ) \left | H \right | , λ\left ( H \right ) \left |G \right |,δ^{+ } \left ( G \right )+ δ^{+ } \left ( H \right ),δ^{- } \left ( G \right )+ δ^{- } \left ( H \right ) \right \}\geλ_2(G\Box H)\ge λ_2(G)+λ_2(H)-1.$ The upper bound for $λ_2(G\Box H)$ is sharp and is a simple corollary of the formula for $λ(G\Box H)$. The lower bound for $λ_2(G\Box H)$ is either sharp or almost sharp i.e. differs by 1 from the sharp bound. We also obtain exact values for $λ_2(G\Box H)$, where $G$ and $H$ are digraphs from some digraph families.

math.CO↗

arXiv2021

Extended Path Partition Conjecture for Semicomplete and Acyclic Compositions

Let $D$ be a digraph and let $λ(D)$ denote the number of vertices in a longest path of $D$. For a pair of vertex-disjoint induced subdigraphs $A$ and $B$ of $D$, we say that $(A,B)$ is a partition of $D$ if $V(A)\cup V(B)=V(D).$ The Path Partition Conjecture (PPC) states that for every digraph, $D$, and every integer $q$ with $1\leq q\leqλ(D)-1$, there exists a partition $(A,B)$ of $D$ such that $λ(A)\leq q$ and $λ(B)\leqλ(D)-q.$ Let $T$ be a digraph with vertex set $\{u_1,\dots, u_t\}$ and for every $i\in [t]$, let $H_i$ be a digraph with vertex set $\{u_{i,j_i}\colon\, j_i\in [n_i]\}$. The {\em composition} $Q=T[H_1,\dots , H_t]$ of $T$ and $H_1,\ldots, H_t$ is a digraph with vertex set $\{u_{i,j_i}\colon\, i\in [t], j_i\in [n_i]\}$ and arc set $$A(Q)=\cup^t_{i=1}A(H_i)\cup \{u_{i,j_i}u_{p,q_p}\colon\, u_iu_p\in A(T), j_i\in [n_i], q_p\in [n_p]\}.$$ We say that $Q$ is acyclic {(semicomplete, respectively)} if $T$ is acyclic {(semicomplete, respectively)}. In this paper, we introduce a conjecture stronger than PPC using a property first studied by Bang-Jensen, Nielsen and Yeo (2006) and show that the stronger conjecture holds for wide families of acyclic and semicomplete compositions.

math.CO↗

arXiv2021

Packing Strong Subgraph in Digraphs

In this paper, we study two types of strong subgraph packing problems in digraphs, including internally disjoint strong subgraph packing problem and arc-disjoint strong subgraph packing problem. These problems can be viewed as generalizations of the famous Steiner tree packing problem and are closely related to the strong arc decomposition problem. We first prove the NP-completeness for the internally disjoint strong subgraph packing problem restricted to symmetric digraphs and Eulerian digraphs. Then we get inapproximability results for the arc-disjoint strong subgraph packing problem and the internally disjoint strong subgraph packing problem. Finally we study the arc-disjoint strong subgraph packing problem restricted to digraph compositions and obtain some algorithmic results by utilizing the structural properties.

math.CO↗

arXiv2021

Kings in Multipartite Hypertournaments

In his paper "Kings in Bipartite Hypertournaments" (Graphs $\&$ Combinatorics 35, 2019), Petrovic stated two conjectures on 4-kings in multipartite hypertournaments. We prove one of these conjectures and give counterexamples for the other.

math.CO↗

arXiv2021

Valued Authorization Policy Existence Problem: Theory and Experiments

Recent work has shown that many problems of satisfiability and resiliency in workflows may be viewed as special cases of the authorization policy existence problem (APEP), which returns an authorization policy if one exists and 'No' otherwise. However, in many practical settings it would be more useful to obtain a 'least bad' policy than just a 'No', where 'least bad' is characterized by some numerical value indicating the extent to which the policy violates the base authorization relation and constraints. Accordingly, we introduce the Valued APEP, which returns an authorization policy of minimum weight, where the (non-negative) weight is determined by the constraints violated by the returned solution. We then establish a number of results concerning the parameterized complexity of Valued APEP. We prove that the problem is fixed-parameter tractable (FPT) if the set of constraints satisfies two restrictions, but is intractable if only one of these restrictions holds. (Most constraints known to be of practical use satisfy both restrictions.) We also introduce a new type of resiliency for workflow satisfiability problem, show how it can be addressed using Valued APEP and use this to build a set of benchmark instances for Valued APEP. Following a set of computational experiments with two mixed integer programming (MIP) formulations, we demonstrate that the Valued APEP formulation based on the user profile concept has FPT-like running time and usually significantly outperforms a naive formulation.

cs.DS↗

arXiv2021

Perfect Forests in Graphs and Their Extensions

Let $G$ be a graph on $n$ vertices. For $i\in \{0,1\}$ and a connected graph $G$, a spanning forest $F$ of $G$ is called an $i$-perfect forest if every tree in $F$ is an induced subgraph of $G$ and exactly $i$ vertices of $F$ have even degree (including zero). A $i$-perfect forest of $G$ is proper if it has no vertices of degree zero. Scott (2001) showed that every connected graph with even number of vertices contains a (proper) 0-perfect forest. We prove that one can find a 0-perfect forest with minimum number of edges in polynomial time, but it is NP-hard to obtain a 0-perfect forest with maximum number of edges. Moreover, we show that to decide whether $G$ has a 0-perfect forest with at least $|V(G)|/2+k$ edges, where $k$ is the parameter, is W[1]-hard. We also prove that for a prescribed edge $e$ of $G,$ it is NP-hard to obtain a 0-perfect forest containing $e,$ but one can decide if there existsa 0-perfect forest not containing $e$ in polynomial time. It is easy to see that every graph with odd number of vertices has a 1-perfect forest. It is not the case for proper 1-perfect forests. We give a characterization of when a connected graph has a proper 1-perfect forest.

math.CO↗

arXiv2021

Towards Better Understanding of User Authorization Query Problem via Multi-variable Complexity Analysis

User authorization queries in the context of role-based access control have attracted considerable interest in the last 15 years. Such queries are used to determine whether it is possible to allocate a set of roles to a user that enables the user to complete a task, in the sense that all the permissions required to complete the task are assigned to the roles in that set. Answering such a query, in general, must take into account a number of factors, including, but not limited to, the roles to which the user is assigned and constraints on the sets of roles that can be activated. Answering such a query is known to be NP-hard. The presence of multiple parameters and the need to find efficient and exact solutions to the problem suggest that a multi-variate approach will enable us to better understand the complexity of the user authorization query problem (UAQ). In this paper, we establish a number of complexity results for UAQ. Specifically, we show the problem remains hard even when quite restrictive conditions are imposed on the structure of the problem. Our FPT results show that we have to use either a parameter with potentially quite large values or quite a restricted version of UAQ. Moreover, our second FPT algorithm is complex and requires sophisticated, state-of-the-art techniques. In short, our results show that it is unlikely that all variants of UAQ that arise in practice can be solved reasonably quickly in general.

cs.CR↗

arXiv2021

The Effect of Social Distancing on the Reach of an Epidemic in Social Networks

How does social distancing affect the reach of an epidemic in social networks? We present Monte Carlo simulation results of a capacity constrained Susceptible-Infected-Removed (SIR) model. The key modelling feature is that individuals are limited in the number of acquaintances that they can interact with, thereby constraining disease transmission to an infectious subnetwork of the original social network. While increased social distancing always reduces the spread of an infectious disease, the magnitude varies greatly depending on the topology of the network. Our results also reveal the importance of coordinating social distancing policies at the global level. In particular, the public health benefits from social distancing to a group (e.g., a country) may be completely undone if that group maintains connections with outside groups that are not following suit.

q-bio.PE↗

arXiv2020

The smallest number of vertices in a 2-arc-strong digraph which has no good pair

Bang-Jensen, Bessy, Havet and Yeo showed that every digraph of independence number at most 2 and arc-connectivity at least 2 has an out-branching $B^+$ and an in-branching $B^-$ which are arc-disjoint (such two branchings are called a {\it good pair}), which settled a conjecture of Thomassen for digraphs of independence number 2. They also proved that every digraph on at most 6 vertices and arc-connectivity at least 2 has a good pair and gave an example of a 2-arc-strong digraph $D$ on 10 vertices with independence number 4 that has no good pair. They asked for the smallest number $n$ of vertices in a 2-arc-strong digraph which has no good pair. In this paper, we prove that every digraph on at most 9 vertices and arc-connectivity at least 2 has a good pair, which solves this problem.

math.CO↗

arXiv2020

On $r$-Simple $k$-Path and Related Problems Parameterized by $k/r$

Abasi et al. (2014) and Gabizon et al. (2015) studied the following problems. In the $r$-Simple $k$-Path problem, given a digraph $G$ on $n$ vertices and integers $r,k$, decide whether $G$ has an $r$-simple $k$-path, which is a walk where every vertex occurs at most $r$ times and the total number of vertex occurrences is $k$. In the $(r,k)$-Monomial Detection problem, given an arithmetic circuit that encodes some polynomial $P$ on $n$ variables and integers $k,r$, decide whether $P$ has a monomial of degree $k$ where the degree of each variable is at most~$r$. In the $p$-Set $(r,q)$-Packing problem, given a universe $V$, positive integers $p,q,r$, and a collection $\cal H$ of sets of size $p$ whose elements belong to $V$, decide whether there exists a subcollection ${\cal H}'$ of $\cal H$ of size $q$ where each element occurs in at most $r$ sets of ${\cal H}'$. Abasi et al. and Gabizon et al. proved that the three problems are single-exponentially fixed-parameter tractable (FPT) when parameterized by $(k/r)\log r$, where $k=pq$ for $p$-Set $(r,q)$-Packing and asked whether the $\log r$ factor in the exponent can be avoided. We consider their question from a wider perspective: are the above problems FPT when parameterized by $k/r$ only? We resolve the wider question by (a) obtaining a $2^{O((k/r)^2\log(k/r))} (n+\log k)^{O(1)}$-time algorithm for $r$-Simple $k$-Path on digraphs and a $2^{O(k/r)} (n+\log k)^{O(1)}$-time algorithm for $r$-Simple $k$-Path on undirected graphs (i.e., for undirected graphs we answer the original question in affirmative), (b) showing that $p$-Set $(r,q)$-Packing is FPT, and (c) proving that $(r,k)$-Monomial Detection is para-NP-hard. For $p$-Set $(r,q)$-Packing, we obtain a polynomial kernel for any fixed $p$, which resolves a question posed by Gabizon et al. regarding the existence of polynomial kernels for problems with relaxed disjointness constraints.

cs.DS↗

arXiv2020

Note on semi-proper orientations of outerplanar graphs

A semi-proper orientation of a given graph $G$, denoted by $(D,w)$, is an orientation $D$ with a weight function $w: A(D)\rightarrow \mathbb{Z}_+$, such that the in-weight of any adjacent vertices are distinct, where the in-weight of $v$ in $D$, denoted by $w^-_D(v)$, is the sum of the weights of arcs towards $v$. The semi-proper orientation number of a graph $G$, denoted by $\overrightarrowχ_s(G)$, is the minimum of maximum in-weight of $v$ in $D$ over all semi-proper orientation $(D,w)$ of $G$. This parameter was first introduced by Dehghan (2019). When the weights of all edges eqaul to one, this parameter is equal to the proper orientation number of $G$. The optimal semi-proper orientation is a semi-proper orientation $(D,w)$ such that $\max_{v\in V(G)}w_D^-(v)=\overrightarrowχ_s(G)$. Araújo et al. (2016) showed that $\overrightarrowχ(G)\le 7$ for every cactus $G$ and the bound is tight. We prove that for every cactus $G$, $\overrightarrowχ_s(G) \le 3$ and the bound is tight. Araújo et al. (2015) asked whether there is a constant $c$ such that $\overrightarrowχ(G)\le c$ for all outerplanar graphs $G.$ While this problem remains open, we consider it in the weighted case. We prove that for every outerplanar graph $G,$ $\overrightarrowχ_s(G)\le 4$ and the bound is tight.

math.CO↗