In a public goods game, every player chooses whether or not to buy a good that all neighboring players will have access to. We consider a setting in which the good is indivisible, neighboring players are out-neighbors in a directed graph, and there is a capacity constraint on their number, k, that can benefit from the good. This means that each player makes a two-pronged decision: decide whether or not to buy and, conditional on buying, choose which k out-neighbors to share access. We examine both pure and mixed Nash equilibria in the model from the perspective of existence, computation, and efficiency. We perform a comprehensive study for these three dimensions with respect to both sharing capacity (k) and the network structure (the underlying directed graph), and establish sharp complexity dichotomies for each.

• Constant FPT Approximation Algorithms for Clustering • Colorful Sum of Radii with outliers • From an algorithm for Colorful k-center to one for Colorful Sum of Radii We study the colorful sum of radii problem, where the input is a point set P partitioned into classes P 1 , P 2 , ⋯ , P ω , along with per-class outlier bounds m 1 , m 2 , ⋯ , m ω , summing to m . The goal is to select a subset C ⊆ P of k centers and assign points to centers in 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. In this paper, we present the first constant-factor approximation algorithms for the colorful sum of radii running in fixed-parameter tractable time. Our contributions are twofold: We design an iterative covering algorithm that achieves a ( 2 + ε ) -approximation with running time exponential in both k and m , where ϵ > 0 is an arbitrary constant; we further develop a ( 7 + ε ) -approximation algorithm running in time exponential only in k by leveraging a colorful k -center subroutine.

Witnessing the advancements in the scale and complexity of chip design, along with the benefits from high-performance computing technologies, the simulation of Very Large Scale Integration (VLSI) circuits increasingly demands acceleration through parallel computing with GPU devices. However, conventional parallel strategies fail to fully leverage modern GPU capabilities, introducing new challenges in GPU-based parallelism for VLSI simulations despite previous demonstrations of significant acceleration. In this paper, we propose a novel approach for accelerating the simulation of 4-value logic timing-aware gate-level circuits through waveform-based GPU parallelism. Our approach introduces an innovative strategy that effectively manages task dependencies during the parallelism of combinational circuits, significantly reducing the synchronization requirement between CPU and GPU. The proposed approach achieves one-pass parallelism by requiring only a single round of data transfer. Moreover, to address the implementation challenges associated with our strategy on GPU devices, we have developed and optimized a series of data structures that dynamically allocate and store newly generated outputs of uncertain scale. Finally, we conduct experiments on industrial-scale open-source benchmarks to demonstrate our approach’s performance gains over several state-of-the-art baselines.

Role mining is a technique that is 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 \(\mathit {UPA} \subseteq U \times P\) , 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 UA and PA is close (in some appropriate sense) to UPA . Role mining is therefore a core problem in the specification of role-based authorization policies. Role mining is known to be hard in general and exact solutions are often impossible to obtain, so there exists an extensive literature on variants of the role mining problem that seek to find approximate solutions and algorithms that use heuristics to find reasonable solutions efficiently. In this article, 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. In particular, GNRM can produce “security-aware” or “availability-aware” solutions. Extending the 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 \(\bar{r} +\bar{k}\) . From a practical perspective, a solution to BO-GNRM gives security managers the opportunity to identify a mined policy offering the best tradeoff between the number of policy discrepancies and the number of roles. 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, and (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 observed that, in many cases, our solver is capable of obtaining optimal solutions when the values of either k or r are small.

Abstract. 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, Bollobás, Gyárfás, Lehel, and Scott [ J. Graph Theory, 55 (2007), pp. 1–13] 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.

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 MWDP, 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.

Gerke et al. (2019) introduced a game-theoretic model to study public good provision in social networks when there are constraints on sharing. This model generates a purely graph-theoretic problem termed exact capacitated domination. 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 closely related problem of deciding whether a capacitated graph has a unique D-set is co-NP-complete.

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)→{0,1,…,a} and d∗:V(G)→{0,1,…,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∈V(G) and adding at most a∗(v) edges incident to every v∈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 series of five polynomial-time reductions to C3-free and C4-free graphs of maximum degree at most 3, and (ii) designing a polynomial-time algorithm for solving (1,1)-Cluster Editing on C3-free and C4-free graphs of maximum degree at most 3.

We introduce and study two natural generalizations of the Connected Vertex Cover (VC) problem: the p-Edge-Connected and p-Vertex-Connected VC problem (where p≥2 is a fixed integer). We obtain an 2O(pk)nO(1)-time algorithm for p-Edge-Connected VC and an 2O(k2)nO(1)-time algorithm for p-Vertex-Connected VC. Thus, like Connected VC, both constrained VC problems are FPT. Furthermore, like Connected VC, neither problem admits a polynomial kernel unless NP ⊆ coNP/poly, which is highly unlikely. We prove however that both problems admit time efficient polynomial sized approximate kernelization schemes. 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 subgraphs of a p-vertex/edge-connected graph by Nishizeki and Poljak (1994) and Nagamochi and Ibaraki (1992).

Abstract. While there have been many results on lower bounds for Max Cut in unweighted graphs, the only lower bound for noninteger weights is that by Poljak and Turzík [ Discrete Math., 58 (1986), pp. 99–104]. 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, the 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 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 for finding 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.

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.

Role mining seeks to compute a set of roles R, a user-role authorization relation UA and a permission-role authorization relation PA, given a user-permission authorization relation UPA, and is therefore a core problem in the specification of role-based authorization policies. Role mining is known to be hard in general and exact solutions are often impossible to obtain, so there exists an extensive literature on variants of the role mining problem that seek to find approximate solutions and algorithms that use heuristics to find reasonable solutions efficiently.

Abstract A directed graph D is semicomplete if for every pair x, y of vertices of D, there is at least one arc between x and y. Thus, a tournament is a semicomplete digraph. In the Directed Component Order Connectivity (DCOC) problem, given a digraph $$D=(V,A)$$ D = ( V , A ) and a pair of natural numbers k and $$\ell $$ ℓ , we are to decide whether there is a subset X of V of size k such that the largest strongly connected component in $$D-X$$ D - X has at most $$\ell $$ ℓ vertices. Note that DCOC reduces to the Directed Feedback Vertex Set problem for $$\ell =1.$$ ℓ = 1 . We study the parameterized complexity of DCOC for general and semicomplete digraphs with the following parameters: $$k, \ell ,\ell +k$$ k , ℓ , ℓ + k and $$n-\ell $$ n - ℓ . In particular, we prove that DCOC with parameter k on semicomplete digraphs can be solved in time $$O^*(2^{16k})$$ O ∗ ( 2 16 k ) but not in time $$O^*(2^{o(k)})$$ O ∗ ( 2 o ( k ) ) unless the Exponential Time Hypothesis (ETH) fails. The upper bound $$O^*(2^{16k})$$ O ∗ ( 2 16 k ) implies the upper bound $$O^*(2^{16(n-\ell )})$$ O ∗ ( 2 16 ( n - ℓ ) ) for the parameter $$n-\ell .$$ n - ℓ . We complement the latter by showing that there is no algorithm of time complexity $$O^*(2^{o({n-\ell })})$$ O ∗ ( 2 o ( n - ℓ ) ) unless ETH fails. Finally, we improve (in dependency on $$\ell $$ ℓ ) the upper bound of Göke, Marx and Mnich (2019) for the time complexity of DCOC with parameter $$\ell +k$$ ℓ + k on general digraphs from

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.) Our analysis is based on the novel concept of a user profile. We also introduce a new type of resiliency problem in the context of workflow satisfiability, show how it can be addressed using Valued APEP, and use this to build a set of benchmark instances for Valued APEP. We describe two different formulations of this problem using mixed integer programming and report the results of computational experiments which solve the problem using these formulations as input to a general-purpose solver. Our results show that the formulation which employs the user profile concept, has FPT-like running time and usually significantly outperforms our naive formulation of the problem.

As an algorithmic framework, message passing is extremely powerful and has wide applications in the context of different disciplines including communications, coding theory, statistics, signal processing, artificial intelligence and combinatorial optimization. In this paper, we investigate the performance of a message-passing algorithm called min-sum belief propagation (BP) for the vertex-disjoint shortest$k$-path problem ($k$-VDSP) on weighted directed graphs, and derive the iterative message-passing update rules. As the main result of this paper, we prove that for a weighted directed graph$G$of order$n$, BP algorithm converges to the unique optimal solution of$k$-VDSP on$G$within$O(n^{2}w_{max})$iterations, provided that the weight$w_{e}$is nonnegative integral for each arc$e\in E(G)$, where$w_{max}=\max \{w_{e}: e\in E(G)\}$. To the best of our knowledge, this is the first instance where BP algorithm is proved correct for NP-hard problems. Additionally, we establish the extensions of$k$-VDSP to the case of multiple sources or sinks.

Let G be a graph on n vertices. For i ∈ {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). An 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. We also prove that for a prescribed edge e of G, it is NP-hard to obtain a 0-perfect forest containing e, but we can find a 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.

Problems of satisfiability and resiliency in workflows have been widely studied in the last decade. Recent work has shown that many such problems may be viewed as special cases of the authorization policy existence problem (APEP), which returns an authorization policy if one exists and "No'' otherwise. A solution may not exist because of the restrictions imposed by the base authorization relation and constraints that form part of the input to APEP.

Golovach, Paulusma, and Song [ Inform. and Comput., 237 (2014), pp. 204--214] 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 $\lambda_P: X \rightarrow Q$ for $X \subseteq V(G),$ decide whether $\lambda_P$ can be extended to a proper coloring of $G$ using only colors from $Q$. For problem 1 we design an ${\mathcal O}^*(2^k)$-time randomized algorithm and for problem 2 we obtain a kernel with at most $3k$ vertices. Banik et al. [in Proceedings of IWOCA 2019, Springer, Berlin, 2019, pp. 61--69] proved 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 ${\mathcal O}(k^2)$ vertices and colors and a compression to a variation of the problem with ${\mathcal O}(k)$ vertices and ${\mathcal O}(k^2)$ colors.

Abasi et al. (2014) introduced the following two problems. In the r -S imple k -P ath problem, given a digraph G on n vertices and positive 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 )-M onomial D etection problem, given an arithmetic circuit that succinctly encodes some polynomial P on n variables and positive integers k , r , decide whether P has a monomial of total degree k where the degree of each variable is at most r . Abasi et al. obtained randomized algorithms of running time 4 ( k / r )log r ⋅ n O (1) for both problems. Gabizon et al. (2015) designed deterministic 2 O (( k / r )log r ) ⋅ n O (1) -time algorithms for both problems (however, for the ( r , k )-M onomial D etection problem the input circuit is restricted to be non-canceling). Gabizon et al. also studied the following problem. In the P -S et ( r , q )-P acking P roblem , given a universe V , positive integers ( p , q , r ), and a collection H of sets of size P whose elements belong to V , decide whether there exists a subcollection H ′ of H of size q where each element occurs in at most r sets of H ′ . Gabizon et al. obtained a deterministic 2 O (( pq / r )log r ) ⋅ n O (1) -time algorithm for P -S et ( r , q )-P acking . The above results prove that the three problems are single-exponentially fixed-parameter tractable (FPT) parameterized by the product of two parameters, that is, k / r and log r , where k = pq for P -S et ( r , q )-P acking . Abasi et al. and Gabizon et al. asked whether the log r factor in the exponent can be avoided. Bonamy et al. (2017) answered the question for ( r , k )-M onomial D etection by proving that unless the Exponential Time Hypothesis (ETH) fails there is no 2 o (( k / r ) log r ) ⋅ ( n + log k ) O (1) -time algorithm for ( r , k )-M onomial D etection , i.e., ( r , k )-M onomial D etection is unlikely to be single-exponentially FPT when parameterized by k / r alone. The question remains open for r -S imple k -P ath and P -S et ( r , q )-P acking . We consider the question from a wider perspective: are the above problems FPT when parameterized by k / r only, i.e., whether there exists a computable function f such that the problems admit a f ( k / r )( n +log k ) O (1) -time algorithm? Since r can be substantially larger than the input size, the algorithms of Abasi et al. and Gabizon et al. do not even show that any of these three problems is in XP parameterized by k / r alone. 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

User authorization queries in the context of role-based access control have attracted considerable interest in the past 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 article, 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 fixed-parameter tractable (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.

A directed graph D is semicomplete if for every pair x,y of vertices of D, there is at least one arc between x and y. Thus, a tournament is a semicomplete digraph. In the Directed Component Order Connectivity (DCOC) problem, given a digraph D = (V,A) and a pair of natural numbers k and 𝓁, we are to decide whether there is a subset X of V of size k such that the largest strong connectivity component in D-X has at most 𝓁 vertices. Note that DCOC reduces to the Directed Feedback Vertex Set problem for 𝓁 = 1. We study parameterized complexity of DCOC for general and semicomplete digraphs with the following parameters: k, 𝓁, 𝓁+k and n-𝓁. In particular, we prove that DCOC with parameter k on semicomplete digraphs can be solved in time O^*(2^(16k)) but not in time O^*(2^o(k)) unless the Exponential Time Hypothesis (ETH) fails. The upper bound O^*(2^(16k)) implies the upper bound O^*(2^(16(n-𝓁))) for the parameter n-𝓁. We complement the latter by showing that there is no algorithm of time complexity O^*(2^o(n-𝓁)) unless ETH fails. Finally, we improve (in dependency on 𝓁) the upper bound of Göke, Marx and Mnich (2019) for the time complexity of DCOC with parameter 𝓁+k on general digraphs from O^*(2^O(k𝓁 log (k𝓁))) to O^*(2^O(klog (k𝓁))). Note that Drange, Dregi and van 't Hof (2016) proved that even for the undirected version of DCOC on split graphs there is no algorithm of running time O^*(2^o(klog 𝓁)) unless ETH fails and it is a long-standing problem to decide whether Directed Feedback Vertex Set admits an algorithm of time complexity O^*(2^o(klog k)).

There has been a considerable interest in recent years in the problem of workflow satisfiability which seeks an allocation of authorised users to every step of the workflow, subject to workflow specification constraints. Unfortunately, the workflow satisfiability problem (WSP) where arbitrary constraints are allowed, is computationally intractable. Wang and Li (2010) were the first to study WSP in the framework of parameterized complexity (with the parameter being the number of steps). Wang and Li proved that the WSP for arbitrary constraints is intractable even in the framework of parameterized complexity, i.e., it is highly unlikely to be fixed-parameter tractable (FPT). Extending the work of Wang and Li (2013) and Crampton et al. (2013), Cohen et al. (2014) introduced the family of user-independent (UI) constraints, which are constraints whose satisfiability does not depend on the identities of the users. Cohen et al. proved that WSP with UI constraints is FPT. Karapetyan et al. (2019) employed these ideas in practically efficient solution methods for WSP with UI constraints, including methods based on SAT and CSP general purpose solvers. While the family of UI constraints includes the most common constraints used in practice, some real-world cases are outside of the family. In this paper, we generalise the concept of authorizations by making them context-dependent and show how to absorb some non-UI constraints into context-dependent authorizations. This allows us to extend algorithms and their implementations developed for WSP with UI constraints to arbitrary constraints. We carry out computational experiments with a general-purpose SAT solver, SAT4J, to test practicality of solving WSP with UI and non-UI constraints using our approach.

Constraints such as separation-of-duty are widely used to specify requirements that supplement basic authorization policies. However, the existence of constraints (and authorization policies) may mean that a user is unable to fulfill her/his organizational duties because access to resources has been denied. In short, there is a tension between the need to protect resources (using policies and constraints) and the availability of resources. Recent work on workflow satisfiability and resiliency in access control asks whether this tension compromises the ability of an organization to achieve its objectives. In this paper, we develop a new method of specifying constraints which subsumes much related work and allows a wider range of constraints to be specified. The use of such constraints leads naturally to a range of questions related to “policy existence”, where a positive answer means that an organization's objectives can be realized. We analyze the complexity of these policy existence questions and, for particular sub-classes of constraints defined by our language, develop fixed-parameter tractable algorithms to solve them.11.An extended abstract of this paper appeared in the Proceedings of the Seventh ACM Conference on Data and Application Security and Privacy [1]. Research was partially supported by Leverhulme Trust grant RPG-2018-161 and Royal Society Wolfson Research Merit Award.

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.

Abasi et al. (2014) introduced the following two problems. In the r-Simple k-Path problem, given a digraph G on n vertices and positive 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 succinctly encodes some polynomial P on n variables and positive integers k, r, decide whether P has a monomial of total degree k where the degree of each variable is at most r. Abasi et al. obtained randomized algorithms of running time 4(k/r)log r ·nO(1) for both problems. Gabizon et al. (2015) designed deterministic 2O((k/r)log r) · nO(1)-time algorithms for both problems (however, for the (r, k)-Monomial Detection problem the input circuit is restricted to be noncanceling). Gabizon et al. also studied the following problem. In the p-Set (r, q)-Packing problem, given a universe V, positive integers p, q, r, and a collection ℋ of sets of size p whose elements belong to V, decide whether there exists a subcollection ℋ' of ℋ of size q where each element occurs in at most r sets of ℋ'. Gabizon et al. obtained a deterministic 2O((pq/r)log r) ·nO(1)-time algorithm for p-Set (r, q)-Packing. The above results prove that the three problems are single-exponentially fixed-parameter tractable (FPT) when parameterized by the product of two parameters, that is, k/r and log r, where k = pq for p-Set (r, q)-Packing. Abasi et al. and Gabizon et al. asked whether the log r factor in the exponent can be avoided. Bonamy et al. (2017) answered the question for (r, k)-Monomial Detection by proving that unless the Exponential Time Hypothesis (ETH) fails there is no 2o((k/r) log r) · (n + log k)O(1)-time algorithm for (r, k)-Monomial Detection, i.e. (r, k)-Monomial Detection is highly unlikely to be single-exponentially FPT when parameterized by k/r alone. The question remains open for r-Simple k-Path and p-Set (r, q)-Packing. We consider the question from a wider perspective: are the above problems FPT when parameterized by k/r only, i.e. whether there exists a computable function f such that the problems admit a f(k/r)(n + log k)O(1)-time algorithm? Since r can be substantially larger than the input size, the algorithms of Abasi et al. and Gabizon zon et al. do not even show that any of these three problems is in XP parameterized by k/r alone. We resolve the wider question by (a) obtaining a 2O((k/r)2 log(k/r)) · (n + log k)O(1)-time algorithm for r-Simple k-Peth on digraphs and a 2O(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 (in contrast, we prove that p-Multiset (r, q)-Packing is W[1]-hard), and (c) proving that (r, k)-Monomial Detrction is para-NP-hard even if only two distinct variables are in polynomial P and the circuit is noncanceling. For the special case of (r, k)-Monomial Detection here k is polynomially bounded by the input size (which is in XP), we show W[1]-hardness. Along the way to solve 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. All our algorithms are deterministic.

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.

We study several problems related to graph modification under connectivity constraints from the perspective of parameterized complexity. In particular, we study (a) (Weighted) Biconnectivity Deletion, where we are tasked with deleting k edges while preserving biconnectivity in an undirected graph, and (b) Path-contraction Preserving Strong Connectivity, where we want to maintain strong connectivity of a digraph while path-contracting k arcs. The parameterized tractability of this last problem was posed in Bang-Jensen and Yeo (2008) [1] as an open question and we answer it here in the negative. On the other hand, we show that preserving (weighted) biconnectivity is fixed-parameter tractable (FPT) and the unweighted case even admits a randomized polynomial kernel. Finally, we show that the most general case of the (unweighted) problem where one would like to preserve ρ-vertex connectivity for any ρ is (non-uniformly) FPT parameterized by k and ρ.

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 and using it prove that the problem of Bang-Jensen et al. (2016) is FPT for all digraphs.

Constraints such as separation-of-duty are widely used to specify requirements that supplement basic authorization policies. However, the existence of constraints (and authorization policies) may mean that a user is unable to fulfill her/his organizational duties because access to resources is denied. In short, there is a tension between the need to protect resources (using policies and constraints) and the availability of resources. Recent work on workflow satisfiability and resiliency in access control asks whether this tension compromises the ability of an organization to achieve its objectives. In this paper, we develop a new method of specifying constraints which subsumes much related work and allows a wider range of constraints to be specified. The use of such constraints leads naturally to a range of questions related to "policy existence", where a positive answer means that an organization's objectives can be realized. We provide an overview of our results establishing that some policy existence questions, notably for those instances that are restricted to user-independent constraints, are fixed-parameter tractable.

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 means that it is 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. In this paper, we present the first fixed parameter algorithms to solve WSP for workflow specifications of this type. Moreover, we significantly extend the range of constraints that can be used in workflow specifications of this type.

Let c, k be two positive integers. Given a graph $$G=(V,E)$$ , the c-Load Coloring problem asks whether there is a c-coloring $$\varphi : V \rightarrow [c]$$ such that for every $$i \in [c]$$ , there are at least k edges with both endvertices colored i. Gutin and Jones (Inf Process Lett 114:446–449, 2014) studied this problem with $$c=2$$ . They showed 2-Load Coloring to be fixed-parameter tractable (FPT) with parameter k by obtaining a kernel with at most 7k vertices. In this paper, we extend the study to any fixed c by giving both a linear-vertex and a linear-edge kernel. In the particular case of $$c=2$$ , we obtain a kernel with less than 4k vertices and less than $$6k+(3+\sqrt{2})\sqrt{k}+4$$ edges. These results imply that for any fixed $$c\ge 2$$ , c-Load Coloring is FPT and the optimization version of c-Load Coloring (where k is to be maximized) has an approximation algorithm with a constant ratio.

We may enforce an information flow policy by encrypting a protected resource and ensuring that only users authorized by the policy are able to decrypt the resource. In most schemes in the literature that use symmetric cryptographic primitives, each user is assigned a single secret and derives decry ption keys using this secret and publicly available information. Recent work has challenged this approach by developing schemes, based on a chain partition of the information flow policy, that do not require public information for key derivation, the trade-off being that a user may need to be assigned more than one secret. In general, many different chain partitions exist for the same policy and, until now, it was not known how to compute an appropriate one. In this paper, we introduce the notion of a tree partition, of which chain partitions are a special case. We show how a tree partition may be used to define a cryptographic enforcement scheme and prove that such schemes can be instantiated in such a way as to preserve the strongest security properties known for cryptographic enforcement schemes. We establish a number of results linking the amount of secret material that needs to be distributed to users with a weighted acyclic graph derived from the tree partition. These results enable us to develop efficient algorithms for deriving tree and chain partitions that minimize the total amount of secret material that needs to be distributed.

A computerized workflow management system may enforce a security policy, specified in terms of authorized actions and constraints, thereby restricting which users can perform particular steps in a workflow. The existence of a security policy may mean that a workflow is unsatisfiable, in the sense that it is impossible to find a valid plan (an assignment of steps to authorized users such that all constraints are satisfied). Work in the literature focuses on the workflow satisfiability problem, a decision problem that outputs a valid plan if the instance is satisfiable (and a negative result otherwise). In this paper, we introduce the Bi-Objective Workflow Satisfiability Problem (BO-WSP), which enables us to solve optimization problems related to workflows and security policies. In particular, we are able to compute a “least bad” plan when some components of the security policy may be violated. In general, BO-WSP is intractable from both the classical and parameterized complexity point of view (where the parameter is the number of steps). We prove that computing a Pareto front for BO-WSP is fixed-parameter tractable (FPT) if we restrict our attention to user-independent constraints. This result has important practical consequences, since most constraints of practical interest in the literature are user-independent. Our proof is constructive and defines an algorithm, the implementation of which we describe and evaluate. We also present a second algorithm to compute a Pareto front which solves multiples instances of a related problem using mixed integer programming (MIP). We compare the performance of both our algorithms on synthetic instances, and show that the FPT algorithm outperforms the MIP-based one by several orders of magnitude on most instances. Finally, we study the important question of workflow resiliency and prove new results establishing that known decision problems are fixed-parameter tractable when restricted to user-independent constraints. We then propose a new way of modeling the availability of users and demonstrate that many questions related to resiliency in the context of this new model may be reduced to instances of BO-WSP.

Resiliency is a relatively new topic in the context of access control. Informally, it refers to the extent to which a multi-user computer system, subject to an authorization policy, is able to continue functioning if a number of authorized users are unavailable. Several interesting problems connected to resiliency were introduced by Li, Wang and Tripunitara [13], many of which were found to be intractable. In this paper, we show that these resiliency problems have unexpected connections with the workflow satisfiability problem (WSP). In particular, we show that an instance of the resiliency checking problem (RCP) may be reduced to an instance of WSP. We then demonstrate that recent advances in our understanding of WSP enable us to develop fixed-parameter tractable algorithms for RCP. Moreover, these algorithms are likely to be useful in practice, given recent experimental work demonstrating the advantages of bespoke algorithms to solve WSP. We also generalize RCP in several different ways, showing in each case how to adapt the reduction to WSP. Li et al also showed that the coexistence of resiliency policies and static separation-of-duty policies gives rise to further interesting questions. We show how our reduction of RCP to WSP may be extended to solve these problems as well and establish that they are also fixed-parameter tractable.

In the mixed Chinese postman problem (MCPP), given a weighted mixed graph $G$ (it may have both edges and arcs), our aim is to find a closed walk of minimum weight traversing each edge and arc at least once. The MCPP parameterized by the number of edges in $G$ or the number of arcs in $G$ is fixed-parameter tractable as proved by van Bevern et al. in 2014 and Gutin, Jones, and Sheng in 2014, respectively. Solving an open question of van Bevern et al., we show that somewhat unexpectedly the MCPP parameterized by the (undirected) treewidth of $G$ is W[1]-hard. In fact, we prove that even the unweighted MCPP parameterized by the pathwidth of $G$ is W[1]-hard. On the positive side, we show that MCPP parameterized by treedepth is fixed-parameter tractable (even with arbitrary integer weights). We are unaware of any widely studied graph parameters between pathwidth and treedepth and so our results provide a close characterization of the complexity of MCPP.

In the traveling salesman problem (TSP), we are given a complete graph $K_n$ together with an integer weighting $w$ on the edges of $K_n$, and we are asked to find a Hamilton cycle of $K_n$ of minimum weight. Let $h(w)$ denote the average weight of a Hamilton cycle of $K_n$ for the weighting $w$. Vizing in 1973 asked whether there is a polynomial-time algorithm which always finds a Hamilton cycle of weight at most $h(w)$. He answered this question in the affirmative and subsequently Rublineckii, also in 1973, and others described several other TSP heuristics satisfying this property. In this paper, we prove a considerable generalization of Vizing's result: for each fixed $k$, we give an algorithm that decides whether, for any input edge weighting $w$ of $K_n$, there is a Hamilton cycle of $K_n$ of weight at most $h(w)-k$ (and constructs such a cycle if it exists). For $k$ fixed, the running time of the algorithm is polynomial in $n$, where the degree of the polynomial does not depend on $k$ (i.e., the generalized Vizing problem is fixed-parameter tractable with respect to the parameter $k$).

A workflow specification defines a set of steps, a set of users, and an access control policy. The policy determines which steps a user is authorized to perform and imposes constraints on which sets of users can perform which sets of steps. The workflow satisfiability problem (WSP) is the problem of determining whether there exists an assignment of users to workflow steps that satisfies the policy. Given the computational hardness of WSP and its importance in the context of workflow management systems, it is important to develop algorithms that are as efficient as possible to solve WSP. In this article, we study the fixed-parameter tractability of WSP in the presence of class-independent constraints, which enable us to (1) model security requirements based on the groups to which users belong and (2) generalize the notion of a user-independent constraint. Class-independent constraints are defined in terms of equivalence relations over the set of users. We consider sets of nested equivalence relations because this enables us to model security requirements in hierarchical organizations. We prove that WSP is fixed-parameter tractable (FPT) for class-independent constraints defined over nested equivalence relations and develop an FPT algorithm to solve WSP instances incorporating such constraints. We perform experiments to evaluate the performance of our algorithm and compare it with that of SAT4J, an off-the-shelf pseudo-Boolean SAT solver. The results of these experiments demonstrate that our algorithm significantly outperforms SAT4J for many instances of WSP.

A workflow specification defines sets of steps and users. An authorization policy determines for each user a subset of steps the user is allowed to perform. Other security requirements, such as separation-of-duty, impose constraints on which subsets of users may perform certain subsets of steps. The workflow satisfiability problem (WSP) is the problem of determining whether there exists an assignment of users to workflow steps that satisfies all such authorizations and constraints. An algorithm for solving WSP is important, both as a static analysis tool for workflow specifications, and for the construction of run-time reference monitors for workflow management systems. Given the computational difficulty of WSP, it is important, particularly for the second application, that such algorithms are as efficient as possible. We introduce class-independent constraints, enabling us to model scenarios where the set of users is partitioned into groups, and the identities of the user groups are irrelevant to the satisfaction of the constraint. We prove that solving WSP is fixed-parameter tractable (FPT) for this class of constraints and develop an FPT algorithm that is useful in practice. We compare the performance of the FPT algorithm with that of SAT4J (a pseudo-Boolean SAT solver) in computational experiments, which show that our algorithm significantly outperforms SAT4J for many instances of WSP. User-independent constraints, a large class of constraints including many practical ones, are a special case of class-independent constraints for which WSP was proved to be FPT (Cohen et al., J. Artif. Intel. Res. 2014). Thus our results considerably extend our knowledge of the fixed-parameter tractability of WSP.

A workflow is a collection of steps that must be executed in some specific order to achieve an objective. A computerised workflow management system may enforce authorisation policies and constraints, thereby restricting which users can perform particular steps in a workflow. The existence of policies and constraints may mean that a workflow is unsatisfiable, in the sense that it is impossible to find an authorised user for each step in the workflow and satisfy all constraints. In this paper, we consider the problem of finding the "least bad" assignment of users to workflow steps by assigning a weight to each policy and constraint violation. To this end, we introduce a framework for associating costs with the violation of workflow policies and constraints and define the valued workflow satisfiability problem (Valued WSP), whose solution is an assignment of steps to users of minimum cost. We establish the computational complexity of Valued WSP with user-independent constraints and show that it is fixed-parameter tractable. We then describe an algorithm for solving Valued WSP with user-independent constraints and evaluate its performance, comparing it to that of an off-the-shelf mixed integer programming package.

The Workflow Satisfiability Problem (WSP) is a problem of practical interest that arises whenever tasks need to be performed by authorized users, subject to constraints defined by business rules. We are required to decide whether there exists a plan - an assignment of tasks to authorized users - such that all constraints are satisfied. It is natural to see the WSP as a subclass of the Constraint Satisfaction Problem (CSP) in which the variables are tasks and the domain is the set of users. What makes the WSP distinctive is that the number of tasks is usually very small compared to the number of users, so it is appropriate to ask for which constraint languages the WSP is fixed-parameter tractable (FPT), parameterized by the number of tasks. This novel approach to the WSP, using techniques from CSP, has enabled us to design a generic algorithm which is FPT for several families of workflow constraints considered in the literature. Furthermore, we prove that the union of FPT languages remains FPT if they satisfy a simple compatibility condition. Lastly, we identify a new FPT constraint language, user-independent constraints, that includes many of the constraints of interest in business processing systems. We demonstrate that our generic algorithm has provably optimal running time O*(2^(klog k)), for this language, where k is the number of tasks.

A workflow specification defines a set of steps and the order in which those steps must be executed. Security requirements and business rules may impose constraints on which users are permitted to perform those steps. A workflow specification is said to be satisfiable if there exists an assignment of authorized users to workflow steps that satisfies all the constraints. An algorithm for determining whether such an assignment exists is important, both as a static analysis tool for workflow specifications, and for the construction of run-time reference monitors for workflow management systems. We develop new methods for determining workflow satisfiability based on the concept of constraint expressions, which were introduced recently by Khan and Fong. These methods are surprising versatile, enabling us to develop algorithms for, and determine the complexity of, a number of different problems related to workflow satisfiability.

A workflow specification defines a set of steps and the order in which these steps must be executed. Security requirements may impose constraints on which groups of users are permitted to perform subsets of these steps. A workflow specification is said to be satisfiable if there exists an assignment of users to workflow steps that satisfies all the constraints. An algorithm for determining whether such an assignment exists is important, both as a static analysis tool for workflow specifications and for the construction of runtime reference monitors for workflow management systems. Finding such an assignment is a hard problem in general, but work by Wang and Li [2010] using the theory of parameterized complexity suggests that efficient algorithms exist under reasonable assumptions about workflow specifications. In this article, we improve the complexity bounds for the workflow satisfiability problem. We also generalize and extend the types of constraints that may be defined in a workflow specification and prove that the satisfiability problem remains fixed-parameter tractable for such constraints. Finally, we consider preprocessing for the problem and prove that in an important special case, in polynomial time, we can reduce the given input into an equivalent one where the number of users is at most the number of steps. We also show that no such reduction exists for two natural extensions of this case, which bounds the number of users by a polynomial in the number of steps, provided a widely accepted complexity-theoretical assumption holds.

A workflow specification defines a set of steps and the order in which those steps must be executed. Security requirements may impose constraints on which groups of users are permitted to perform subsets of those steps. A workflow specification is said to be satisfiable if there exists an assignment of users to workflow steps that satisfies all the constraints. An algorithm for determining whether such an assignment exists is important, both as a static analysis tool for workflow specifications, and for the construction of run-time reference monitors for workflow management systems. Finding such an assignment is a hard problem in general, but work by Wang and Li in 2010 using the theory of parameterized complexity suggests that efficient algorithms exist under reasonable assumptions about workflow specifications. In this paper, we improve the complexity bounds for the workflow satisfiability problem. We also generalize and extend the types of constraints that may be defined in a workflow specification and prove that the satisfiability problem remains fixed-parameter tractable for such constraints.

An oriented graph is a directed graph without directed 2-cycles. Poljak and Turzík (1986) proved that every connected oriented graph $G$ on $n$ vertices and $m$ arcs contains an acyclic subgraph with at least $\frac{m}{2}+\frac{n-1}{4}$ arcs. Raman and Saurabh (2006) gave another proof of this result and left it as an open question to establish the parameterized complexity of the following problem: does $G$ have an acyclic subgraph with least $\frac{m}{2}+\frac{n-1}{4}+k$ arcs, where $k$ is the parameter? We answer this question by showing that the problem can be solved by an algorithm of runtime $(12k)!n^{O(1)}$. Thus, the problem is fixed-parameter tractable. We also prove that there is a polynomial time algorithm that either establishes that the input instance of the problem is a Yes-instance or reduces the input instance to an equivalent one of size $O(k^2)$.

We consider vectors from $\{0,1\}^n$. The weight of such a vector $v$ is the sum of the coordinates of $v$. The distance ratio of a set $L$ of vectors is ${\rm dr}(L):=\max \{\rho(x,y):\ x,y \in L\}/ \min \{\rho(x,y):\ x,y \in L,\ x\neq y\},$ where $\rho(x,y)$ is the Hamming distance between $x$ and $y$. We prove that (a) for every constant $\lambda>1$ there are no positive constants $\alpha$ and $C$ such that every set $K$ of at least $\lambda^p$ vectors with weight $p$ contains a subset $K'$ with $|K'|\ge |K|^{\alpha}$ and ${\rm dr}(K')\le C$; and (b) for a set $K$ of vectors with weight $p$, and a constant $C>2$, there exists $K'\subseteq K$ such that ${\rm dr}(K')\le C$ and $|K'| \ge |K|^\alpha$, where $\alpha = 1/ \lceil \log(p/2)/\log(C/2) \rceil$.

In the parameterized problem MaxLin2-AA[$k$], we are given a system with variables x_1,...,x_n consisting of equations of the form Product_{i in I}x_i = b, where x_i,b in {-1, 1} and I is a nonempty subset of {1,...,n}, each equation has a positive integral weight, and we are to decide whether it is possible to simultaneously satisfy equations of total weight at least W/2+k, where W is the total weight of all equations and k is the parameter (if k=0, the possibility is assured). We show that MaxLin2-AA[k] has a kernel with at most O(k^2 log k) variables and can be solved in time 2^{O(k log k)}(nm)^{O(1)}. This solves an open problem of Mahajan et al. (2006). The problem Max-r-Lin2-AA[k,r] is the same as MaxLin2-AA[k] with two differences: each equation has at most r variables and r is the second parameter. We prove a theorem on Max-$r$-Lin2-AA[k,r] which implies that Max-r-Lin2-AA[k,r] has a kernel with at most (2k-1)r variables, improving a number of results including one by Kim and Williams (2010). The theorem also implies a lower bound on the maximum of a function f that maps {-1,1}^n to the set of reals and whose Fourier expansion (which is a multilinear polynomial) is of degree r. We show applicability of the lower bound by giving a new proof of the Edwards-Erdös bound (each connected graph on n vertices and m edges has a bipartite subgraph with at least m/2 +(n-1)/4 edges) and obtaining a generalization.

Memetic algorithms are known to be a powerful technique in solving hard optimization problems. To design a memetic algorithm, one needs to make a host of decisions. Selecting the population size is one of the most important among them. Most of the algorithms in the literature fix the population size to a certain constant value. This reduces the algorithm's quality since the optimal population size varies for different instances, local search procedures, and runtimes. In this paper we propose an adjustable population size. It is calculated as a function of the runtime of the whole algorithm and the average runtime of the local search for the given instance. Note that in many applications the runtime of a heuristic should be limited and, therefore, we use this bound as a parameter of the algorithm. The average runtime of the local search procedure is measured during the algorithm's run. Some coefficients which are independent of the instance and the local search are to be tuned at the design time; we provide a procedure to find these coefficients. The proposed approach was used to develop a memetic algorithm for the multidimensional assignment problem (MAP). We show that our adjustable population size makes the algorithm flexible to perform efficiently for a wide range of running times and local searches and this does not require any additional tuning of the algorithm.

We present an exact algorithm that decides, for every fixed r ≥ 2 in time O(m) + 2O(k2) whether a given set of m clauses of size r admits a truth assignment that satisfies at least ((2r – 1)m + k)/2r clauses. Thus Max-r-Sat is fixed-parameter tractable when parameterized by the number of satisfied clauses above the tight lower bound (1 − 2−r)m. This solves an open problem of Mahajan, Raman and Sikdar (J. Comput. System Sci., 75, 2009). Our algorithm is based on a polynomial-time data reduction procedure that reduces a problem instance to an equivalent algebraically represented problem with O(k2) variables. This is done by representing the instance as an appropriate polynomial, and by applying a probabilistic argument combined with some simple tools from Harmonic analysis to show that if the polynomial cannot be reduced to one of size O(k2), then there is a truth assignment satisfying the required number of clauses. Combining another probabilistic argument with tools from graph matching theory and signed graphs, we show that if an instance of Max-2-Sat with m clauses has at least 3k variables after application of certain polynomial time reduction rules to it, then there is a truth assignment that satisfies at least (3m + k)/4 clauses. We also outline how the fixed-parameter tractability result on Max-r-Sat can be extended to a family of Boolean Constraint Satisfaction Problems.

The Directed Maximum Leaf Out-Branching problem is to find an out-branching (i.e., a rooted oriented spanning tree) in a given digraph with the maximum number of leaves. In this paper, we obtain two combinatorial results on the number of leaves in out-branchings. We show that (1) every strongly connected n-vertex digraph D with minimum in-degree at least 3 has an out-branching with at least $(n/4)^{1/3}-1$ leaves; (2) if a strongly connected digraph D does not contain an out-branching with k leaves, then the pathwidth of its underlying graph $\mathrm{UG}(D)$ is $O(k\log k)$, and if the digraph is acyclic with a single vertex of in-degree zero, then the pathwidth is at most $4k$. The last result implies that it can be decided in time $2^{O(k\log^2k)}\cdot n^{O(1)}$ whether a strongly connected digraph on n vertices has an out-branching with at least k leaves. On acyclic digraphs the running time of our algorithm is $2^{O(k\log k)}\cdot n^{O(1)}$.

We survey results and open questions on complexity of parameterized problems on digraphs. The problems include the feedback vertex and arc set problems, induced subdigraph problems and directed k-leaf problems. We also prove some new results on the topic. Most of these new results are on parameterizations of the backward paired comparison problem.

For digraphs D and H, a mapping $f:V(D)\rightarrow V(H)$ is a homomorphism of D to H if $uv\in A(D)$ implies $f(u)f(v)\in A(H)$. If, moreover, each vertex $u\in V(D)$ is associated with costs $c_i(u)$, $i\in V(H)$, then the cost of the homomorphism f is $\sum_{u\in V(D)}c_{f(u)}(u)$. For each fixed digraph H, we have the minimum cost homomorphism problem for H. The problem is to decide, for an input graph D with costs $c_i(u)$, $u\in V(D)$, $i\in V(H)$, whether there exists a homomorphism of D to H and, if one exists, to find one of minimum cost. Minimum cost homomorphism problems encompass (or are related to) many well-studied optimization problems. We describe a dichotomy of the minimum cost homomorphism problem for semicomplete bipartite digraphs H. This solves an open problem from an earlier paper. To obtain the dichotomy of this paper, we introduce and study a new notion, a k-Min-Max ordering of digraphs.

A digraph obtained by replacing each edge of a complete m-partite graph with an arc or a pair of mutually opposite arcs with the same end vertices is called a complete m-partite digraph. An $O ( n^3 )$ algorithm for finding a longest path in a complete m-partite $( m \geq 2 )$ digraph with n vertices is described in this paper. The algorithm requires time $O( n^{2.5} )$ in case of testing only the existence of a Hamiltonian path and finding it if one exists. It is simpler than the algorithm of Manoussakis and Tuza [SIAM J. Discrete Math., 3 (1990), pp. 537–543], which works only for $m = 2$. The algorithm implies a simple characterization of complete m-partite digraphs having Hamiltonian paths that was obtained for the first time in Gutin [Kibernetica (Kiev), 4 (1985), pp. 124–125] for $m = 2$ and in Gutin [Kibernetica (Kiev), 1(1988), pp. 107–108] for $ m \geq 2 $.