arXiv subjects

Gregory Gutin

Publications and source records attributed to Gregory Gutin.

At least 73 records · Page 4Linked to original sources

arXiv2020

Approximation algorithms for general cluster routing problem

Graph routing problems have been investigated extensively in operations research, computer science and engineering due to their ubiquity and vast applications. In this paper, we study constant approximation algorithms for some variations of the general cluster routing problem. In this problem, we are given an edge-weighted complete undirected graph $G=(V,E,c),$ whose vertex set is partitioned into clusters $C_{1},\dots ,C_{k}.$ We are also given a subset $V'$ of $V$ and a subset $E'$ of $E.$ The weight function $c$ satisfies the triangle inequality. The goal is to find a minimum cost walk $T$ that visits each vertex in $V'$ only once, traverses every edge in $E'$ at least once and for every $i\in [k]$ all vertices of $C_i$ are traversed consecutively.

cs.DS↗

arXiv2020

Proximity and Remoteness in Directed and Undirected Graphs

Let $D$ be a strongly connected digraph. The average distance $\barσ(v)$ of a vertex $v$ of $D$ is the arithmetic mean of the distances from $v$ to all other vertices of $D$. The remoteness $ρ(D)$ and proximity $π(D)$ of $D$ are the maximum and the minimum of the average distances of the vertices of $D$, respectively. We obtain sharp upper and lower bounds on $π(D)$ and $ρ(D)$ as a function of the order $n$ of $D$ and describe the extreme digraphs for all the bounds. We also obtain such bounds for strong tournaments. We show that for a strong tournament $T$, we have $π(T)=ρ(T)$ if and only if $T$ is regular. Due to this result, one may conjecture that every strong digraph $D$ with $π(D)=ρ(D)$ is regular. We present an infinite family of non-regular strong digraphs $D$ such that $π(D)=ρ(D).$ We describe such a family for undirected graphs as well.

math.CO↗

arXiv2019

Parameterized Pre-coloring Extension and List Coloring Problems

Golovach, Paulusma and Song (Inf. Comput. 2014) asked to determine the parameterized complexity of the following problems parameterized by $k$: (1) Given a graph $G$, a clique modulator $D$ (a clique modulator is a set of vertices, whose removal results in a clique) of size $k$ for $G$, and a list $L(v)$ of colors for every $v\in V(G)$, decide whether $G$ has a proper list coloring; (2) Given a graph $G$, a clique modulator $D$ of size $k$ for $G$, and a pre-coloring $λ_P: X \rightarrow Q$ for $X \subseteq V(G),$ decide whether $λ_P$ can be extended to a proper coloring of $G$ using only colors from $Q.$ For Problem 1 we design an $O^*(2^k)$-time randomized algorithm and for Problem 2 we obtain a kernel with at most $3k$ vertices. Banik et al. (IWOCA 2019) proved the the following problem is fixed-parameter tractable and asked whether it admits a polynomial kernel: Given a graph $G$, an integer $k$, and a list $L(v)$ of exactly $n-k$ colors for every $v \in V(G),$ decide whether there is a proper list coloring for $G.$ We obtain a kernel with $O(k^2)$ vertices and colors and a compression to a variation of the problem with $O(k)$ vertices and $O(k^2)$ colors.

cs.DS↗

arXiv2019

Pattern-Based Approach to the Workflow Satisfiability Problem with User-Independent Constraints

The fixed parameter tractable (FPT) approach is a powerful tool in tackling computationally hard problems. In this paper, we link FPT results to classic artificial intelligence (AI) techniques to show how they complement each other. Specifically, we consider the workflow satisfiability problem (WSP) which asks whether there exists an assignment of authorised users to the steps in a workflow specification, subject to certain constraints on the assignment. It was shown by Cohen et al. (JAIR 2014) that WSP restricted to the class of user-independent constraints (UI), covering many practical cases, admits FPT algorithms, i.e. can be solved in time exponential only in the number of steps $k$ and polynomial in the number of users $n$. Since usually $k << n$ in WSP, such FPT algorithms are of great practical interest. We present a new interpretation of the FPT nature of the WSP with UI constraints giving a decomposition of the problem into two levels. Exploiting this two-level split, we develop a new FPT algorithm that is by many orders of magnitude faster than the previous state-of-the-art WSP algorithm and also has only polynomial-space complexity. We also introduce new pseudo-Boolean (PB) and Constraint Satisfaction (CSP) formulations of the WSP with UI constraints which efficiently exploit this new decomposition of the problem and raise the novel issue of how to use general-purpose solvers to tackle FPT problems in a fashion that meets FPT efficiency expectations. In our computational study, we investigate, for the first time, the phase transition (PT) properties of the WSP, under a model for generation of random instances. We show how PT studies can be extended, in a novel fashion, to support empirical evaluation of scaling of FPT algorithms.

cs.AI↗

arXiv2019

Arc-disjoint in- and out-branchings rooted at the same vertex in compositions of digraphs

A digraph $D=(V, A)$ has a good pair at a vertex $r$ if $D$ has a pair of arc-disjoint in- and out-branchings rooted at $r$. Let $T$ be a digraph with $t$ vertices $u_1,\dots , u_t$ and let $H_1,\dots H_t$ be digraphs such that $H_i$ has vertices $u_{i,j_i},\ 1\le j_i\le n_i.$ Then the composition $Q=T[H_1,\dots , H_t]$ is a digraph with vertex set $\{u_{i,j_i}\mid 1\le i\le t, 1\le j_i\le n_i\}$ and arc set $$A(Q)=\cup^t_{i=1}A(H_i)\cup \{u_{ij_i}u_{pq_p}\mid u_iu_p\in A(T), 1\le j_i\le n_i, 1\le q_p\le n_p\}.$$ When $T$ is arbitrary, we obtain the following result: every strong digraph composition $Q$ in which $n_i\ge 2$ for every $1\leq i\leq t$, has a good pair at every vertex of $Q.$ The condition of $n_i\ge 2$ in this result cannot be relaxed. When $T$ is semicomplete, we characterize semicomplete compositions with a good pair, which generalizes the corresponding characterization by Bang-Jensen and Huang (J. Graph Theory, 1995) for quasi-transitive digraphs. As a result, we can decide in polynomial time whether a given semicomplete composition has a good pair rooted at a given vertex.

cs.DM↗

arXiv2019

Bounded and Approximate Strong Satisfiability in Workflows

There has been a considerable amount of interest in recent years in the problem of workflow satisfiability, which asks whether the existence of constraints in a workflow specification makes it impossible to allocate authorized users to each step in the workflow. Recent developments have seen the workflow satisfiability problem (WSP) studied in the context of workflow specifications in which the set of steps may vary from one instance of the workflow to another. This, in turn, means that some constraints may only apply to certain workflow instances. Inevitably, WSP becomes more complex for such workflow specifications. Other approaches have considered the possibility of associating costs with the violation of `soft' constraints and authorizations. Workflow satisfiability in this context becomes a question of minimizing the cost of allocating users to steps in the workflow. In this paper, we introduce new problems, which we believe to be of practical relevance, that combine these approaches. In particular, we consider the question of whether, given a workflow specification with costs and a `budget', all possible workflow instances have an allocation of users to steps that does not exceed the budget. We design a fixed-parameter tractable algorithm to solve this problem parameterized by the total number of steps, release points and xor branchings.

cs.DS↗

arXiv2019

Arc-disjoint Strong Spanning Subdigraphs of Semicomplete Compositions

A strong arc decomposition of a digraph $D=(V,A)$ is a decomposition of its arc set $A$ into two disjoint subsets $A_1$ and $A_2$ such that both of the spanning subdigraphs $D_1=(V,A_1)$ and $D_2=(V,A_2)$ are strong. Let $T$ be a digraph with $t$ vertices $u_1,\dots , u_t$ and let $H_1,\dots H_t$ be digraphs such that $H_i$ has vertices $u_{i,j_i},\ 1\le j_i\le n_i.$ Then the composition $Q=T[H_1,\dots , H_t]$ is a digraph with vertex set $\cup_{i=1}^t V(H_i)=\{u_{i,j_i}\mid 1\le i\le t, 1\le j_i\le n_i\}$ and arc set \[ \left(\cup^t_{i=1}A(H_i) \right) \cup \left( \cup_{u_iu_p\in A(T)} \{u_{ij_i}u_{pq_p} \mid 1\le j_i\le n_i, 1\le q_p\le n_p\} \right). \] We obtain a characterization of digraph compositions $Q=T[H_1,\dots H_t]$ which have a strong arc decomposition when $T$ is a semicomplete digraph and each $H_i$ is an arbitrary digraph. Our characterization generalizes a characterization by Bang-Jensen and Yeo (2003) of semicomplete digraphs with a strong arc decomposition and solves an open problem by Sun, Gutin and Ai (2018) on strong arc decompositions of digraph compositions $Q=T[H_1,\dots , H_t]$ in which $T$ is semicomplete and each $H_i$ is arbitrary. Our proofs are constructive and imply the existence of a polynomial algorithm for constructing a \good{} decomposition of a digraph $Q=T[H_1,\dots , H_t]$, with $T$ semicomplete, whenever such a decomposition exists.

cs.DM↗

arXiv2018

Arc-disjoint strong spanning subdigraphs in compositions and products of digraphs

A digraph $D=(V,A)$ has a good decomposition if $A$ has two disjoint sets $A_1$ and $A_2$ such that both $(V,A_1)$ and $(V,A_2)$ are strong. Let $T$ be a digraph with $t$ vertices $u_1,\dots , u_t$ and let $H_1,\dots H_t$ be digraphs such that $H_i$ has vertices $u_{i,j_i},\ 1\le j_i\le n_i.$ Then the composition $Q=T[H_1,\dots , H_t]$ is a digraph with vertex set $\{u_{i,j_i}\mid 1\le i\le t, 1\le j_i\le n_i\}$ and arc set $$A(Q)=\cup^t_{i=1}A(H_i)\cup \{u_{ij_i}u_{pq_p}\mid u_iu_p\in A(T), 1\le j_i\le n_i, 1\le q_p\le n_p\}.$$ For digraph compositions $Q=T[H_1,\dots H_t]$, we obtain sufficient conditions for $Q$ to have a good decomposition and a characterization of $Q$ with a good decomposition when $T$ is a strong semicomplete digraph and each $H_i$ is an arbitrary digraph with at least two vertices. For digraph products, we prove the following: (a) if $k\geq 2$ is an integer and $G$ is a strong digraph which has a collection of arc-disjoint cycles covering all vertices, then the Cartesian product digraph $G^{\square k}$ (the $k$th powers with respect to Cartesian product) has a good decomposition; (b) for any strong digraphs $G, H$, the strong product $G\boxtimes H$ has a good decomposition.

cs.DM↗

arXiv2018

Strong Subgraph Connectivity of Digraphs: A Survey

In this survey we overview known results on the strong subgraph $k$-connectivity and strong subgraph $k$-arc-connectivity of digraphs. After an introductory section, the paper is divided into four sections: basic results, algorithms and complexity, sharp bounds for strong subgraph $k$-(arc-)connectivity, minimally strong subgraph $(k, \ell)$-(arc-) connected digraphs. This survey contains several conjectures and open problems for further study.

cs.DM↗

arXiv2018

Lower and Upper Bound for Computing the Size of All Second Neighbourhoods

We consider the problem of computing the size of each $r$-neighbourhood for every vertex of a graph. Specifically, we ask whether the size of the closed second neighbourhood can be computed in subquadratic time. Adapting the SETH reductions by Abboud et al. (2016) that exclude subquadratic algorithms to compute the radius of a graph, we find that a subquadratic algorithm would violate the SETH. On the other hand, a linear fpt-time algorithm by Demaine et al. (2014) parameterized by a certain `sparseness parameter' of the graph is known, where the dependence on the parameter is exponential. We show here that a better dependence is unlikely: for any~$δ< 2$, no algorithm running in time $O(2^{o({\rm vc}(G))} \, n^δ)$, where~${\rm vc}(G)$ is the vertex cover number, is possible unless the SETH fails. We supplement these lower bounds with algorithms that solve the problem in time~$O(2^{{\rm vc}(G)/2} {\rm vc}(G)^2 \cdot n)$ and $O(2^w w \cdot n)$.

cs.DS↗

arXiv2018

Strong subgraph $k$-arc-connectivity

Two previous papers, arXiv:1803.00284 and arXiv:1803.00281, introduced and studied strong subgraph $k$-connectivity of digraphs obtaining characterizations, lower and upper bounds and computational complexity results for the new digraph parameter. The parameter is an analog of well-studied generalized $k$-connectivity of undirected graphs. In this paper, we introduce the concept of strong subgraph $k$-arc-connectivity of digraphs, which is an analog of generalized $k$-edge-connectivity of undirected graphs. We also obtain characterizations, lower and upper bounds and computational complexity results for this digraph parameter. Several of our results differ from those obtained for strong subgraph $k$-connectivity.

cs.DM↗

arXiv2018

Parameterized Resiliency Problems via Integer Linear Programming

We introduce an extension of decision problems called resiliency problems. In resiliency problems, the goal is to decide whether an instance remains positive after any (appropriately defined) perturbation has been applied to it. To tackle these kinds of problems, some of which might be of practical interest, we introduce a notion of resiliency for Integer Linear Programs (ILP) and show how to use a result of Eisenbrand and Shmonin (Math. Oper. Res., 2008) on Parametric Linear Programming to prove that ILP Resiliency is fixed-parameter tractable (FPT) under a certain parameterization. To demonstrate the utility of our result, we consider natural resiliency versions of several concrete problems, and prove that they are FPT under natural parameterizations. Our first results concern a four-variate problem which generalizes the Disjoint Set Cover problem and which is of interest in access control. We obtain a complete parameterized complexity classification for every possible combination of the parameters. Then, we introduce and study a resiliency version of the Closest String problem, for which we extend an FPT result of Gramm et al. (Algorithmica, 2003). We also consider problems in the fields of scheduling and social choice. We believe that many other problems can be tackled by our framework.

cs.DS↗

arXiv2018

Alternative parameterizations of Metric Dimension

A set of vertices $W$ in a graph $G$ is called resolving if for any two distinct $x,y\in V(G)$, there is $v\in W$ such that ${\rm dist}_G(v,x)\neq{\rm dist}_G(v,y)$, where ${\rm dist}_G(u,v)$ denotes the length of a shortest path between $u$ and $v$ in the graph $G$. The metric dimension ${\rm md}(G)$ of $G$ is the minimum cardinality of a resolving set. The Metric Dimension problem, i.e. deciding whether ${\rm md}(G)\le k$, is NP-complete even for interval graphs (Foucaud et al., 2017). We study Metric Dimension (for arbitrary graphs) from the lens of parameterized complexity. The problem parameterized by $k$ was proved to be $W[2]$-hard by Hartung and Nichterlein (2013) and we study the dual parameterization, i.e., the problem of whether ${\rm md}(G)\le n- k,$ where $n$ is the order of $G$. We prove that the dual parameterization admits (a) a kernel with at most $3k^4$ vertices and (b) an algorithm of runtime $O^*(4^{k+o(k)}).$ Hartung and Nichterlein (2013) also observed that Metric Dimension is fixed-parameter tractable when parameterized by the vertex cover number $vc(G)$ of the input graph. We complement this observation by showing that it does not admit a polynomial kernel even when parameterized by $vc(G) + k$. Our reduction also gives evidence for non-existence of polynomial Turing kernels.

cs.DS↗

arXiv2018

Strong subgraph $k$-connectivity bounds

Let $D=(V,A)$ be a digraph of order $n$, $S$ a subset of $V$ of size $k$ and $2\le k\leq n$. Strong subgraphs $D_1, \dots , D_p$ containing $S$ are said to be internally disjoint if $V(D_i)\cap V(D_j)=S$ and $A(D_i)\cap A(D_j)=\emptyset$ for all $1\le i<j\le p$. Let $κ_S(D)$ be the maximum number of internally disjoint strong digraphs containing $S$ in $D$. The strong subgraph $k$-connectivity is defined as $$κ_k(D)=\min\{κ_S(D)\mid S\subseteq V, |S|=k\}.$$ A digraph $D=(V, A)$ is called minimally strong subgraph $(k,\ell)$-connected if $κ_k(D)\geq \ell$ but for any arc $e\in A$, $κ_k(D-e)\leq \ell-1$. In this paper, we first give a sharp upper bound for the parameter $κ_k(D)$ and then study the minimally strong subgraph $(k,\ell)$-connected digraphs.

cs.DM↗

arXiv2018

Strong Subgraph $k$-connectivity

Generalized connectivity introduced by Hager (1985) has been studied extensively in undirected graphs and become an established area in undirected graph theory. For connectivity problems, directed graphs can be considered as generalizations of undirected graphs. In this paper, we introduce a natural extension of generalized $k$-connectivity of undirected graphs to directed graphs (we call it strong subgraph $k$-connectivity) by replacing connectivity with strong connectivity. We prove NP-completeness results and the existence of polynomial algorithms. We show that strong subgraph $k$--connectivity is, in a sense, harder to compute than generalized $k$-connectivity. However, strong subgraph $k$-connectivity can be computed in polynomial time for semicomplete digraphs and symmetric digraphs. We also provide sharp bounds on strong subgraph $k$-connectivity and pose some open questions.

cs.DM↗

arXiv2018

$k$-Distinct In- and Out-Branchings in Digraphs

An out-branching and an in-branching of a digraph $D$ are called $k$-distinct if each of them has $k$ arcs absent in the other. Bang-Jensen, Saurabh and Simonsen (2016) proved that the problem of deciding whether a strongly connected digraph $D$ has $k$-distinct out-branching and in-branching is fixed-parameter tractable (FPT) when parameterized by $k$. They asked whether the problem remains FPT when extended to arbitrary digraphs. Bang-Jensen and Yeo (2008) asked whether the same problem is FPT when the out-branching and in-branching have the same root. By linking the two problems with the problem of whether a digraph has an out-branching with at least $k$ leaves (a leaf is a vertex of out-degree zero), we first solve the problem of Bang-Jensen and Yeo (2008). We then develop a new digraph decomposition called the rooted cut decomposition and using it we prove that the problem of Bang-Jensen et al. (2016) is FPT for all digraphs. We believe that the \emph{rooted cut decomposition} will be useful for obtaining other results on digraphs.

cs.DS↗

arXiv2018

Searching for Maximum Out-Degree Vertices in Tournaments

A vertex $x$ in a tournament $T$ is called a king if for every vertex $y$ of $T$ there is a directed path from $x$ to $y$ of length at most 2. It is not hard to show that every vertex of maximum out-degree in a tournament is a king. However, tournaments may have kings which are not vertices of maximum out-degree. A binary inquiry asks for the orientation of the edge between a pair of vertices and receives the answer. The cost of finding a king in an unknown tournament is the number of binary inquiries required to detect a king. For the cost of finding a king in a tournament, in the worst case, Shen, Sheng and Wu (SIAM J. Comput., 2003) proved a lower and upper bounds of $Ω(n^{4/3})$ and $O(n^{3/2})$, respectively. In contrast to their result, we prove that the cost of finding a vertex of maximum out-degree is ${n \choose 2} -O(n)$ in the worst case.

cs.DS↗

arXiv2017

Designing Deterministic Polynomial-Space Algorithms by Color-Coding Multivariate Polynomials

In recent years, several powerful techniques have been developed to design {\em randomized} polynomial-space parameterized algorithms. In this paper, we introduce an enhancement of color coding to design deterministic polynomial-space parameterized algorithms. Our approach aims at reducing the number of random choices by exploiting the special structure of a solution. Using our approach, we derive the following deterministic algorithms (see Introduction for problem definitions). 1. Polynomial-space $O^*(3.86^k)$-time (exponential-space $O^*(3.41^k)$-time) algorithm for {\sc $k$-Internal Out-Branching}, improving upon the previously fastest {\em exponential-space} $O^*(5.14^k)$-time algorithm for this problem. 2. Polynomial-space $O^*((2e)^{k+o(k)})$-time (exponential-space $O^*(4.32^k)$-time) algorithm for {\sc $k$-Colorful Out-Branching} on arc-colored digraphs and {\sc $k$-Colorful Perfect Matching} on planar edge-colored graphs. To obtain our polynomial space algorithms, we show that $(n,k,αk)$-splitters ($α\ge 1$) and in particular $(n,k)$-perfect hash families can be enumerated one by one with polynomial delay.

cs.DS↗