arXiv subjects
Publications and source records attributed to Gregory Gutin.
Huang, Ma, Shapira, Sudakov and Yuster (Comb. Prob. Comput. 2013) proved that every Eulerian digraph of average out-degree $d$ has a directed cycle of length at least $\sqrt{d}.$ We improve the lower bound from $\sqrt{d}$ to $\sqrt{2d}-3/2.$
We study the colorful sum of radii problem, where the input is a point set $P$ partitioned into classes $P_1, P_2, \dots, P_ω$, along with per-class outlier bounds $m_1, m_2, \dots, m_ω$, summing to $m$. The goal is to select a subset $\mathcal{C} \subseteq P$ of $k$ centers and assign points to centers in $\mathcal{C}$, allowing up to $m_i$ unassigned points (outliers) from each class $P_i$, while minimizing the sum of cluster radii. The radius of a cluster is defined as the maximum distance from any point in the cluster to its center. The classical (non-colorful) version of the sum of radii problem is known to be NP-hard, even on weighted planar graphs. The colorful sum of radii is introduced by Chekuri et al. (2022), who provide an $O(\log ω)$-approximation algorithm. In this paper, we present the first constant-factor approximation algorithms for the colorful sum of radii running in FPT (fixed-parameter tractable) time. Our contributions are twofold: We design an iterative covering algorithm that achieves a $(2+\varepsilon)$-approximation with running time exponential in both $k$ and $m$; We further develop a $(7+\varepsilon)$-approximation algorithm by leveraging a colorful $k$-center subroutine, improving the running time by removing the exponential dependency on $m$.
An oriented multigraph is a directed multigraph without directed 2-cycles. Let ${\rm fas}(D)$ denote the minimum size of a feedback arc set in an oriented multigraph $D$. The degree of a vertex is the sum of its out- and in-degrees. In several papers, upper bounds for ${\rm fas}(D)$ were obtained for oriented multigraphs $D$ with maximum degree upper-bounded by a constant. Hanauer (2017) conjectured that ${\rm fas}(D)\le 2.5n/3$ for every oriented multigraph $D$ with $n$ vertices and maximum degree at most 5. We prove a strengthening of the conjecture: ${\rm fas}(D)\le m/3$ holds for every oriented multigraph $D$ with $m$ arcs and maximum degree at most 5. This bound is tight and improves a bound of Berger and Shor (1990,1997). It would be interesting to determine $c$ such that ${\rm fas}(D)\le cn$ for every oriented multigraph $D$ with $n$ vertices and maximum degree at most 5 such that the bound is tight. We show that $\frac{5}{7}\le c \le \frac{24}{29} < \frac{2.5}{3}$.
While there have been many results on lower bounds for Max Cut in unweighted graphs, there are only few results for lower bounds for Max Cut in weighted graphs. In this paper, we launch an extensive study of lower bounds for Max Cut in weighted graphs. We introduce a new approach for obtaining lower bounds for Weighted Max Cut. Using it, Probabilistic Method, Vizing's chromatic index theorem, and other tools, we obtain several lower bounds for arbitrary weighted graphs, weighted graphs of bounded girth and triangle-free weighted graphs. We pose conjectures and open questions.
An oriented graph $D$ is converse invariant if, for any tournament $T$, the number of copies of $D$ in $T$ is equal to that of its converse $-D$. El Sahili and Ghazo Hanna [J. Graph Theory 102 (2023), 684-701] showed that any oriented graph $D$ with maximum degree at most 2 is converse invariant. They proposed a question: Can we characterize all converse invariant oriented graphs? In this paper, we introduce a digraph polynomial and employ it to give a necessary condition for an oriented graph to be converse invariant. This polynomial serves as a cornerstone in proving all the results presented in this paper. In particular, we characterize all orientations of trees with diameter at most 3 that are converse invariant. We also show that all orientations of regular graphs are not converse invariant if $D$ and $-D$ have different degree sequences. In addition, in contrast to the findings of El Sahili and Ghazo Hanna, we prove that every connected graph $G$ with maximum degree at least $3$, admits an orientation $D$ of $G$ such that $D$ is not converse invariant. We pose one conjecture.
In this paper we consider stable matchings subject to assignment constraints. These are matchings that require certain assigned pairs to be included, insist that some other assigned pairs are not, and, importantly, are stable. Our main contribution is an algorithm, based on the iterated deletion of unattractive alternatives, that determines if assignment constraints are compatible with stability. Whenever there is a stable matching that satisfies the assignment constraints, our algorithm outputs all of them (each in polynomial time per solution). This provides market designers with (i) a tool to test the feasibility of stable matchings subject to assignment constraints, and (ii) a tool to implement them when feasible.
An oriented graph is called $k$-anti-traceable if the subdigraph induced by every subset with $k$ vertices has a hamiltonian anti-directed path. In this paper, we consider an anti-traceability conjecture. In particular, we confirm this conjecture holds when $k\leq 4$. We also show that every sufficiently large $k$-anti-traceable oriented graph admits an anti-path that contains $n-o(n)$ vertices.
Role mining is a technique used to derive a role-based authorization policy from an existing policy. Given a set of users $U$, a set of permissions $P$ and a user-permission authorization relation $\mahtit{UPA}\subseteq U\times P$, a role mining algorithm seeks to compute a set of roles $R$, a user-role authorization relation $\mathit{UA}\subseteq U\times R$ and a permission-role authorization relation $\mathit{PA}\subseteq R\times P$, such that the composition of $\mathit{UA}$ and $\mathit{PA}$ is close (in some appropriate sense) to $\mathit{UPA}$. In this paper, we first introduce the Generalized Noise Role Mining problem (GNRM) -- a generalization of the MinNoise Role Mining problem -- which we believe has considerable practical relevance. Extending work of Fomin et al., we show that GNRM is fixed parameter tractable, with parameter $r + k$, where $r$ is the number of roles in the solution and $k$ is the number of discrepancies between $\mathit{UPA}$ and the relation defined by the composition of $\mathit{UA}$ and $\mathit{PA}$. We further introduce a bi-objective optimization variant of GNRM, where we wish to minimize both $r$ and $k$ subject to upper bounds $r\le \bar{r}$ and $k\le \bar{k}$, where $\bar{r}$ and $\bar{k}$ are constants. We show that the Pareto front of this bi-objective optimization problem (BO-GNRM) can be computed in fixed-parameter tractable time with parameter $\bar{r}+\bar{k}$. We then report the results of our experimental work using the integer programming solver Gurobi to solve instances of BO-GNRM. Our key findings are that (a) we obtained strong support that Gurobi's performance is fixed-parameter tractable, (b) our results suggest that our techniques may be useful for role mining in practice, based on our experiments in the context of three well-known real-world authorization policies.
We survey the field of algorithms and complexity for graph problems parameterized above or below guaranteed values, a research area which was pioneered by Venkatesh Raman. Those problems seek, for a given graph $G$, a solution whose value is at least $g(G)+k$ or at most $g(G)-k$, where $g(G)$ is a guarantee on the value that any solution on $G$ takes. The goal is to design algorithms which find such solution in time whose complexity in $k$ is decoupled from that in the guarantee, or to rule out the existence of such algorithms by means of intractability results. We discuss a large number of algorithms and intractability results, and complement them by several open problems.
A bisection in a graph is a cut in which the number of vertices in the two parts differ by at most 1. In this paper, we give lower bounds for the maximum weight of bisections of edge-weighted graphs with bounded maximum degree. Our results improve a bound of Lee, Loh, and Sudakov (J. Comb. Th. Ser. B 103 (2013)) for (unweighted) maximum bisections in graphs whose maximum degree is either even or equals 3, and for almost all graphs. We show that a tight lower bound for maximum size of bisections in 3-regular graphs obtained by Bollobás and Scott (J. Graph Th. 46 (2004)) can be extended to weighted subcubic graphs. We also consider edge-weighted triangle-free subcubic graphs and show that a much better lower bound (than for edge-weighted subcubic graphs) holds for such graphs especially if we exclude $K_{1,3}$. We pose three conjectures.
Given a natural number $k\ge 2$, we consider the $k$-submodular cover problem ($k$-SC). The objective is to find a minimum cost subset of a ground set $\mathcal{X}$ subject to the value of a $k$-submodular utility function being at least a certain predetermined value $τ$. For this problem, we design a bicriteria algorithm with a cost at most $O(1/ε)$ times the optimal value, while the utility is at least $(1-ε)τ/r$, where $r$ depends on the monotonicity of $g$.
In 1981, Bermond and Thomassen conjectured that for any positive integer $k$, every digraph with minimum out-degree at least $2k-1$ admits $k$ vertex-disjoint directed cycles. In this short paper, we verify the Bermond-Thomassen conjecture for triangle-free multipartite tournaments and 3-partite tournaments. Furthermore, we characterize 3-partite tournaments with minimum out-degree at least $2k-1$ ($k\geq 2$) such that in each set of $k$ vertex-disjoint directed cycles, every cycle has the same length.
In order to coordinate players in a game must first identify a target pattern of behaviour. In this paper we investigate the difficulty of identifying prominent outcomes in two kinds of binary action coordination problems in social networks: pure coordination games and anti-coordination games. For both environments, we determine the computational complexity of finding a strategy profile that (i) maximises welfare, (ii) maximises welfare subject to being an equilibrium, and (iii) maximises potential. We show that the complexity of these objectives can vary with the type of coordination problem. Objectives (i) and (iii) are tractable problems in pure coordination games, but for anti-coordination games are NP-hard. Objective (ii), finding the best Nash equilibrium, is NP-hard for both. Our results support the idea that environments in which actions are strategic complements (e.g., technology adoption) facilitate successful coordination more readily than those in which actions are strategic substitutes (e.g., public good provision).
A graph $H$ is a clique graph if $H$ is a vertex-disjoin union of cliques. Abu-Khzam (2017) introduced the $(a,d)$-{Cluster Editing} problem, where for fixed natural numbers $a,d$, given a graph $G$ and vertex-weights $a^*:\ V(G)\rightarrow \{0,1,\dots, a\}$ and $d^*{}:\ V(G)\rightarrow \{0,1,\dots, d\}$, we are to decide whether $G$ can be turned into a cluster graph by deleting at most $d^*(v)$ edges incident to every $v\in V(G)$ and adding at most $a^*(v)$ edges incident to every $v\in V(G)$. Results by Komusiewicz and Uhlmann (2012) and Abu-Khzam (2017) provided a dichotomy of complexity (in P or NP-complete) of $(a,d)$-{Cluster Editing} for all pairs $a,d$ apart from $a=d=1.$ Abu-Khzam (2017) conjectured that $(1,1)$-{Cluster Editing} is in P. We resolve Abu-Khzam's conjecture in affirmative by (i) providing a serious of five polynomial-time reductions to $C_3$-free and $C_4$-free graphs of maximum degree at most 3, and (ii) designing a polynomial-time algorithm for solving $(1,1)$-{Cluster Editing} on $C_3$-free and $C_4$-free graphs of maximum degree at most 3.
We introduce and study a new optimization problem on digraphs, termed Maximum Weighted Digraph Partition (MWDP) problem. We prove three complexity dichotomies for MWDP: on arbitrary digraphs, on oriented digraphs, and on symmetric digraphs. We demonstrate applications of the dichotomies for binary-action polymatrix games and several graph theory problems.
This paper considers incentives to provide goods that are partially shareable along social links. We introduce a model in which each individual in a social network not only decides how much of a shareable good to provide, but also decides which subset of neighbours to nominate as co-beneficiaries. An outcome of the model specifies an endogenously generated subnetwork of the original network and a public goods game occurring over the realised subnetwork. We prove the existence of specialised pure strategy Nash equilibria: those in which some individuals contribute while the remaining individuals free ride. We then consider how the set of efficient specialised equilibria vary as the constraints on sharing are relaxed and we show that, paradoxically, an increase in shareability may decrease efficiency.
We investigate the difficulty of finding economically efficient solutions to coordination problems on graphs. Our work focuses on two forms of coordination problem: pure-coordination games and anti-coordination games. We consider three objectives in the context of simple binary-action polymatrix games: (i) maximizing welfare, (ii) maximizing potential, and (iii) finding a welfare-maximizing Nash equilibrium. We introduce an intermediate, new graph-partition problem, termed Maximum Weighted Digraph Partition, which is of independent interest, and we provide a complexity dichotomy for it. This dichotomy, among other results, provides as a corollary a dichotomy for Objective (i) for general binary-action polymatrix games. In addition, it reveals that the complexity of achieving these objectives varies depending on the form of the coordination problem. Specifically, Objectives (i) and (ii) can be efficiently solved in pure-coordination games, but are NP-hard in anti-coordination games. Finally, we show that objective (iii) is NP-hard even for simple non-trivial pure-coordination games.
Witnessing the advancing scale and complexity of chip design and benefiting from high-performance computation technologies, the simulation of Very Large Scale Integration (VLSI) Circuits imposes an increasing requirement for acceleration through parallel computing with GPU devices. However, the conventional parallel strategies do not fully align with modern GPU abilities, leading to new challenges in the parallelism of VLSI simulation when using GPU, despite some previous successful demonstrations of significant acceleration. In this paper, we propose a novel approach to accelerate 4-value logic timing-aware gate-level logic simulation using waveform-based GPU parallelism. Our approach utilizes a new strategy that can effectively handle the dependency between tasks during the parallelism, reducing the synchronization requirement between CPU and GPU when parallelizing the simulation on combinational circuits. This approach requires only one round of data transfer and hence achieves one-pass parallelism. Moreover, to overcome the difficulty within the adoption of our strategy in GPU devices, we design a series of data structures and tune them to dynamically allocate and store new-generated output with uncertain scale. Finally, experiments are carried out on industrial-scale open-source benchmarks to demonstrate the performance gain of our approach compared to several state-of-the-art baselines.