In this study, we investigate a scheduling problem on identical machines in which jobs require initial setup before execution. We assume that an algorithm can dynamically form a batch (i.e., a collection of jobs to be processed together) from the remaining jobs. The setup time is modeled as a known monotone function of the set of jobs within a batch, while the execution time of each job remains unknown until completion. This uncertainty poses significant challenges for minimizing the makespan. We address these challenges by considering two scenarios: each job batch must be assigned to a single machine, or a batch may be distributed across multiple machines. For both scenarios, we analyze settings with and without preemption. Across these four settings, we design online algorithms that achieve asymptotically optimal competitive ratios with respect to both the number of jobs and the number of machines.

We study the fair division of indivisible items with subsidies among n agents, where the absolute marginal valuation of each item is at most one. Under monotone nondecreasing valuations (where each item is a good), Brustle et al. [9] demonstrated that a maximum subsidy of 2 ( n − 1 ) and a total subsidy of 2 ( n − 1 ) 2 are sufficient to guarantee the existence of an envy-freeable allocation. In this paper, we improve upon these bounds, even in a wider model. Namely, we show that, given an EF1 allocation, we can compute in polynomial time an envy-free allocation with a subsidy of at most n − 1 per agent and a total subsidy of at most n ( n − 1 ) / 2 . Moreover, when the valuations are monotone nondecreasing, we provide a polynomial-time algorithm that computes an envy-free allocation with a subsidy of at most n − 1.5 per agent and a total subsidy of at most ( n 2 − n − 1 ) / 2 .

Consider a directed, rooted graph \(G=(V\cup\{r\},E)\) where each vertex in \(V\) has a partial order preference over its incoming edges. The preferences of a vertex naturally extend to preferences over arborescences rooted at \(r\) . We present a polynomial-time algorithm that decides whether a given input instance admits a popular arborescence, i.e., one for which there is no “more popular” arborescence. In fact, our algorithm solves the more general popular common base problem in the intersection of two matroids: we are given an arbitrary matroid \(M=(E,\mathcal{I})\) and a partition matroid \(M_{\text{part}}\) over \(E\) , where partition classes correspond to a set \(V\) of agents with \(|V|={\rm rank}(M)\) and each agent has a partial order preference over its associated partition class; the problem asks for a common base of \(M\) and \(M_{\text{part}}\) such that there is no “more popular” common base. Our algorithm is combinatorial, and can be regarded as a primal–dual algorithm. It searches for a solution along with its dual certificate, a chain of subsets of \(E\) , witnessing its popularity. Our generalized results, expressed in terms of matroids, demonstrate that the identification of agents with vertices of the graph in the popular arborescence problem is not essential. We also study the related popular common independent set problem. For the case with weak rankings, we formulate the popular common independent set polytope, and thus show that a minimum-cost popular common independent set can be computed efficiently. By contrast, we prove that it is \(\mathsf{NP}\) -hard to compute a minimum-cost popular arborescence, even when rankings are strict.

We study the fair division of indivisible items with subsidies among n agents, where the absolute marginal valuation of each item is at most one. Under monotone valuations (where each item is a good), it is known that a maximum subsidy of 2(n-1) and a total subsidy of 2(n-1)² are sufficient to guarantee the existence of an envy-freeable allocation. In this paper, we improve upon these bounds, even in a wider model. Namely, we show that, given an EF1 allocation, we can compute in polynomial time an envy-free allocation with a subsidy of at most n-1 per agent and a total subsidy of at most n(n-1)/2. Moreover, we present further improved bounds for monotone valuations.

We consider composition orderings for linear functions of one variable. Given $n$ linear functions $f_1,\dots,f_n$ and a constant $c$, the objective is to find a permutation $σ$ that minimizes/maximizes $f_{σ(n)}\circ\dots\circ f_{σ(1)}(c)$. It was first studied in the area of time-dependent scheduling, and known to be solvable in $O(n\log n)$ time if all functions are nondecreasing. In this paper, we present a complete characterization of optimal composition orderings for this case, by regarding linear functions as two-dimensional vectors. We also show several interesting properties on optimal composition orderings such as the equivalence between local and global optimality. Furthermore, by using the characterization above, we provide a fixed-parameter tractable (FPT) algorithm for the composition ordering problem for general linear functions, with respect to the number of decreasing linear functions. We next deal with matrix multiplication orderings as a generalization of composition of linear functions. Given $n$ matrices $M_1,\dots,M_n\in\mathbb{R}^{m\times m}$ and two vectors $w,y\in\mathbb{R}^m$, where $m$ denotes a positive integer, the objective is to find a permutation $σ$ that minimizes/maximizes $w^\top M_{σ(n)}\dots M_{σ(1)} y$. The problem is also viewed as a generalization of flow shop scheduling through a limit. By this extension, we show that the multiplication ordering problem for $2\times 2$ matrices is solvable in $O(n\log n)$ time if all the matrices are simultaneously triangularizable and have nonnegative determinants, and FPT with respect to the number of matrices with negative determinants, if all the matrices are simultaneously triangularizable. As the negative side, we finally prove that three possible natural generalizations are NP-hard: 1) when $m=2$, 2) when $m\geq 3$, and 3) the target version of the problem.

Our input is a directed, rooted graph G = (V ∪ {r}, E) where each vertex in V has a partial order preference over its incoming edges. The preferences of a vertex extend naturally to preferences over arborescences rooted at r. We seek a popular arborescence in G, i.e., one for which there is no “more popular” arborescence. Popular arborescences have applications in liquid democracy or collective decision making; however, they need not exist in every input instance. The popular arborescence problem is to decide if a given input instance admits a popular arborescence or not. We show a polynomial-time algorithm for this problem, whose computational complexity was not known previously.

Given a hypergraph H⊆2V on a finite base set V, a vertex v∈V is called the private vertex of a hyperedge H∈H if H is the only hyperedge of H containing it. A hypergraph is called irredundant if every edge of it has a private vertex, and it is called redundant otherwise. Motivated by some graph domination problems, Uno (2015) posed the problems of generating all minimal redundant and maximal irredundant subhypergraphs of a given hypergraph. Here we prove that these are NP-hard generation problems, and present positive results for certain special cases.

Abstract. Horn functions form a subclass of Boolean functions possessing interesting structural and computational properties. These functions play a fundamental role in algebra, artificial intelligence, combinatorics, computer science, database theory, and logic. In the present paper, we introduce the subclass of hypergraph Horn functions that generalizes matroids and equivalence relations. We provide multiple characterizations of hypergraph Horn functions in terms of implicate-duality and the closure operator, which are, respectively, regarded as generalizations of matroid duality and the Mac Lane–Steinitz exchange property of matroid closure. We also study algorithmic issues on hypergraph Horn functions and show that the recognition problem (i.e., deciding if a given definite Horn CNF represents a hypergraph Horn function) and key realization (i.e., deciding if a given hypergraph is realized as a key set by a hypergraph Horn function) can be done in polynomial time, while implicate sets can be generated with polynomial delay.

In fair division problems, we are given a set S of m items and a set N of n agents with individual preferences, and the goal is to find an allocation of items among agents so that each agent finds the allocation fair. There are several established fairness concepts and envy-freeness is one of the most extensively studied ones. However envy-free allocations do not always exist when items are indivisible and this has motivated relaxations of envy-freeness: envy-freeness up to one item (EF1) and envy-freeness up to any item (EFX) are two well-studied relaxations. We consider the problem of finding EF1 and EFX allocations for utility functions that are not necessarily monotone, and propose four possible extensions of different strength to this setting. In particular, we present a polynomial time algorithm for finding an EF1 allocation for two agents with arbitrary utility functions. An example is given showing that EFX allocations need not exist for two agents with non-monotone, non-additive, identical utility functions. However, when all agents have monotone (not necessarily additive) identical utility functions, we give a pseudo-polynomial time algorithm that always finds an EFX allocation of chores. As a step toward understanding the general case, we discuss two subclasses of utility functions: Boolean utilities that are {0,+1}-valued functions, and negative Boolean utilities that are {0,−1}-valued functions. For the latter, we give a polynomial time algorithm that finds an EFX allocation when the utility functions are identical.

We consider the many-to-many bipartite matching problem in the presence of two-sided preferences and two-sided lower quotas. The input to our problem is a bipartite graph G=(A U B, E), where each vertex in A U B specifies a strict preference ordering over its neighbors. Each vertex has an upper quota and a lower quota denoting the maximum and minimum number of vertices that can be assigned to it from its neighborhood. In the many-to-many setting with two-sided lower quotas, informally, a critical matching is a matching which fulfils vertex lower quotas to the maximum possible extent. This is a natural generalization of the definition of critical matching in the one-to-one setting [Kavitha T., FSTTCS 2021]. Our goal in the given problem is to find a popular matching in the set of critical matchings. A matching is popular in a given set of matchings if it remains undefeated in a head-to-head election with any matching in that set. Here, vertices cast votes between pairs of matchings. We show that there always exists a matching that is popular in the set of critical matchings. We present an efficient algorithm to compute such a matching of the largest size. We prove the popularity of our matching using a dual certificate.

Linear programming (LP) problems with gainfree Leontief substitution systems have been intensively studied in economics and operations research, and include the feasibility problem of a class of Horn systems, which arises in, e.g., polyhedral combinatorics and logic. This subclass of LP problems admits a strongly polynomial time algorithm, where devising such an algorithm for general LP problems is one of the major theoretical open questions in mathematical optimization and computer science. Recently, much attention has been paid to devising certifying algorithms in software engineering, since those algorithms enable one to confirm the correctness of outputs of programs with simple computations. In this paper, we provide the first combinatorial (and strongly polynomial time) certifying algorithm for LP problems with gainfree Leontief substitution systems. As a by-product, we answer affirmatively an open question whether the feasibility problem of the class of Horn systems admits a combinatorial certifying algorithm.

In this paper, we study the minimum k-partition problem of submodular functions, i.e., given a finite set V and a submodular function f: 2V → ℝ, computing a k-partition {V1,…, Vk} of V with minimum . The problem is a natural generalization of the minimum k-cut problem in graphs and hypergraphs. It is known that the problem is NP-hard for general k, and solvable in polynomial time for k ≤ 3. In this paper, we construct the first polynomial-time algorithm for the minimum 4-partition problem. * Authors are ordered alphabetically.

Tree-based phylogenetic networks provide a powerful model for representing complex data or non-tree-like evolution. Such networks consist of an underlying evolutionary tree called a "support tree" (also known as a "subdivision tree") together with extra arcs added between the edges of that tree. However, a tree-based network can have exponentially many support trees, and this leads to a variety of computational problems. Recently, Hayamizu established a theory called the structure theorem for rooted binary phylogenetic networks and provided linear-time and linear-delay algorithms for different problems, such as counting, optimization, and enumeration of support trees. However, in practice, it is often more useful to search for both optimal and near-optimal solutions than to calculate only an optimal solution. In the present paper, we thus consider the following problem: Given a tree-based phylogenetic network N where each arc is weighted by its probability, compute the ranking of top- k support trees of N according to their likelihood values. We provide a linear-delay (and hence optimal) algorithm for this problem.

Motivated by the serious problem that hospitals in rural areas suffer from a shortage of residents, we study the Hospitals/Residents model in which hospitals are associated with lower quotas and the objective is to satisfy them as much as possible. When preference lists are strict, the number of residents assigned to each hospital is the same in any stable matching because of the well-known rural hospitals theorem; thus there is no room for algorithmic interventions. However, when ties are introduced to preference lists, this will no longer apply because the number of residents may vary over stable matchings. In this paper, we formulate an optimization problem to find a stable matching with the maximum total satisfaction ratio for lower quotas. We first investigate how the total satisfaction ratio varies over choices of stable matchings in four natural scenarios and provide the exact values of these maximum gaps. Subsequently, we propose a strategy-proof approximation algorithm for our problem; in one scenario it solves the problem optimally, and in the other three scenarios, which are NP-hard, it yields a better approximation factor than that of a naive tie-breaking method. Finally, we show inapproximability results for the above-mentioned three NP-hard scenarios.

Horn functions form an important subclass of Boolean functions and appear in many different areas of computer science and mathematics as a general tool to describe implications and dependencies. Finding minimum sized representations for such functions with respect to most commonly used measures is a computationally hard problem admitting a $2^{\log^{1-o(1)}n}$ inapproximability bound. In this paper we consider the natural class of key Horn functions representing keys of relational databases. For this class, the minimization problems for most measures remain NP-hard. In this paper we provide logarithmic factor approximation algorithms for key Horn functions with respect to all such measures.

Given a relational database, a key is a set of attributes such that a value assignment to this set uniquely determines the values of all other attributes. The database uniquely defines a pure Horn function h, representing the functional dependencies. If the knowledge of the attribute values in set A determines the value for attribute v, then A→v is an implicate of h. If K is a key of the database, then K→v is an implicate of h for all attributes v. Keys of small sizes play a crucial role in various problems. We present structural and complexity results on the set of minimal keys of pure Horn functions. We characterize Sperner hypergraphs for which there is a unique pure Horn function with the given hypergraph as the set of minimal keys. Furthermore, we show that recognizing such hypergraphs is co-NP-complete already when every hyperedge has size two. On the positive side, we identify several classes of graphs for which the recognition problem can be decided in polynomial time. We also present an algorithm that generates the minimal keys of a pure Horn function with polynomial delay, improving on earlier results. By establishing a connection between keys and target sets, our approach can be used to generate all minimal target sets with polynomial delay when the thresholds are bounded by a constant. As a byproduct, our proof shows that the Minimum Key problem is at least as hard as the Minimum Target Set Selection problem with bounded thresholds.

Given a CNF formula Φ with clauses C1,…,Cm and variables V={x1,…,xn}, a truth assignment a:V→{0,1} of Φ leads to a clause sequence σΦ(a)=(C1(a),…,Cm(a))∈{0,1}m where Ci(a)=1 if clause Ci evaluates to 1 under assignment a, otherwise Ci(a)=0. The set of all possible clause sequences carries a lot of information on the formula, e.g. SAT, MAX-SAT and MIN-SAT can be encoded in terms of finding a clause sequence with extremal properties. We consider a problem posed at Dagstuhl Seminar 19211 “Enumeration in Data Management” (2019) about the generation of all possible clause sequences of a given CNF with bounded dimension. We prove that the problem can be solved in incremental polynomial time. We further give an algorithm with polynomial delay for the class of tractable CNF formulas. We also consider the generation of maximal and minimal clause sequences, and show that generating maximal clause sequences is NP-hard, while minimal clause sequences can be generated with polynomial delay.

We give an incremental polynomial time algorithm for enumerating the vertices of any polyhedron $P=P(A,\b1)=\{x\in \mathbb{R}^n \mid Ax\geq \b1,~x\geq \b0\}$, when $A$ is a totally unimodular matrix. Our algorithm is based on decomposing the hypergraph transversal problem for unimodular hypergraphs using Seymour's decomposition of totally unimodular matrices and may be of independent interest.

If $k

In this paper, we introduce online knapsack problems with a resource buffer. In the problems, we are given a knapsack with capacity $1$, a buffer with capacity $R\ge 1$, and items that arrive one by one. Each arriving item has to be taken into the buffer or discarded on its arrival irrevocably. When every item has arrived, we transfer a subset of items in the current buffer into the knapsack. Our goal is to maximize the total value of the items in the knapsack. We consider four variants depending on whether items in the buffer are removable (i.e., we can remove items in the buffer) or non-removable, and proportional (i.e., the value of each item is proportional to its size) or general. For the general&non-removable case, we observe that no constant competitive algorithm exists for any $R\ge 1$. For the proportional&non-removable case, we show that a simple greedy algorithm is optimal for every $R\ge 1$. For the general&removable and the proportional&removable cases, we present optimal algorithms for small $R$ and give asymptotically nearly optimal algorithms for general $R$.

In this paper, we study several classes of satisfiability preserving assignments to the constraint satisfaction problem. In particular, we consider fixable, autark and satisfying assignments. Since it is in general NP-hard to find a nontrivial (i.e., nonempty) satisfiability preserving assignment, we introduce linear satisfiability preserving assignments, which are defined by polyhedral cones in an associated vector space. The vector space is obtained by the identification, introduced by Kullmann, of assignments with real vectors. We consider arbitrary polyhedral cones, where only restricted classes of cones for autark assignments are considered in the literature. We reveal that cones in certain classes are maximal as a convex subset of the set of the associated vectors, which can be regarded as extensions of Kullmann's results for autark assignments of CNFs. As algorithmic results, we present a pseudo-polynomial time algorithm that computes a linear fixable assignment for a given integer linear system, which implies the well known pseudo-polynomial solvability for integer linear systems such as two-variable-per-inequality, Horn and q-Horn systems.

We consider two-player zero-sum stochastic mean payoff games with perfect information. We show that any such game, with a constant number of random positions and polynomially bounded positive transition probabilities, admits a polynomial time approximation scheme, both in the relative and absolute sense.

In this paper, we introduce maximum composition ordering problems. The input is n real functions $$f_1,\dots ,f_n:\mathbb {R}\rightarrow \mathbb {R}$$ and a constant $$c\in \mathbb {R}$$ . We consider two settings: total and partial compositions. The maximum total composition ordering problem is to compute a permutation $$\sigma :[n]\rightarrow [n]$$ which maximizes $$f_{\sigma (n)}\circ f_{\sigma (n-1)}\circ \dots \circ f_{\sigma (1)}(c)$$ , where $$[n]=\{1,\dots ,n\}$$ . The maximum partial composition ordering problem is to compute a permutation $$\sigma :[n]\rightarrow [n]$$ and a nonnegative integer $$k~(0\le k\le n)$$ which maximize $$f_{\sigma (k)}\circ f_{\sigma (k-1)}\circ \dots \circ f_{\sigma (1)}(c)$$ . We propose $$\mathrm {O}(n\log n)$$ time algorithms for the maximum total and partial composition ordering problems for monotone linear functions $$f_i$$ , which generalize linear deterioration and shortening models for the time-dependent scheduling problem. We also show that the maximum total composition ordering problem can be solved in polynomial time if $$f_i$$ is of the form $$\max \{a_ix+b_i,d_i,x\}$$ for some constants $$a_i\,(\ge 0)$$ , $$b_i$$ and $$d_i$$ . As a corollary, we show that the two-valued free-order secretary problem can be solved in polynomial time. We finally prove that there exists no constant-factor approximation algorithm for the problems, even if $$f_i$$ ’s are monotone, piecewise linear functions with at most two pieces, unless P $$=$$ NP.

In this paper, we study several classes of satisfiability preserving assignments to the constraint satisfaction problem (CSP). In particular, we consider fixable, autark and satisfying assignments. Since it is in general NP-hard to find a nontrivial (i.e., nonempty) satisfiability preserving assignment, we introduce linear satisfiability preserving assignments, which are defined by polyhedral cones in an associated vector space. The vector space is obtained by the identification, introduced by Kullmann, of assignments with real vectors. We consider arbitrary polyhedral cones, where only restricted classes of cones for autark assignments are considered in the literature. We reveal that cones in certain classes are maximal as a convex subset of the set of the associated vectors, which can be regarded as extensions of Kullmann's results for autark assignments of CNFs. As algorithmic results, we present a pseudo-polynomial time algorithm that computes a linear fixable assignment for a given integer linear system, which implies the well known pseudo-polynomial solvability for integer linear systems such as two-variable-per-inequality (TVPI), Horn and q-Horn systems.

This paper studies optimal matroid partitioning problems for various objective functions. In the problem, we are given a finite set $E$ and $k$ weighted matroids $(E, \mathcal{I}_i, w_i)$, $i = 1, \dots, k$, and our task is to find a minimum partition $(I_1,\dots,I_k)$ of $E$ such that $I_i \in \mathcal{I}_i$ for all $i$. For each objective function, we give a polynomial-time algorithm or prove NP-hardness. In particular, for the case when the given weighted matroids are identical and the objective function is the sum of the maximum weight in each set (i.e., $\sum_{i=1}^k\max_{e\in I_i}w_i(e)$), we show that the problem is strongly NP-hard but admits a PTAS.

In this paper, we study the effect of surrogate objective functions in optimization problems. We introduce surrogate ratio as a measure of such effect, where the surrogate ratio is the ratio between the optimal values of the original and surrogate objective functions. We prove that the surrogate ratio is at most mu^{|1/p - 1/q|} when the objective functions are p- and q-norms, and the feasible region is a mu-dimensional space (i.e., a subspace of R^mu), a mu-intersection of matroids, or a mu-extendible system. We also show that this is the best possible bound. In addition, for mu-systems, we demonstrate that the ratio becomes mu^{1/p} when p < q and unbounded if p > q. Here, a mu-system is an independence system such that for any subset of ground set the ratio of the cardinality of the largest to the smallest maximal independent subset of it is at most mu. We further extend our results to the surrogate ratios for approximate solutions.

In this paper, we study the parameterized complexity of the linear complementarity problem (LCP), which is one of the most fundamental mathematical optimization problems. The parameters we focus on are the sparsities of the input and the output of the LCP: the maximum numbers of nonzero entries per row/column in the coefficient matrix and the number of nonzero entries in a solution. Our main result is to present a fixed-parameter algorithm for the LCP with all the parameters. We also show that if we drop any of the three parameters, then the LCP is fixed-parameter intractable. In addition, we discuss the nonexistence of a polynomial kernel for the LCP.

We consider finite Markov decision processes (MDPs) with undiscounted total effective payoff. We show that there exist uniformly optimal pure stationary strategies that can be computed by solving a polynomial number of linear programs. We apply this result to two-player zero-sum stochastic games with perfect information and undiscounted total effective payoff, and derive the existence of a saddle point in uniformly optimal pure stationary strategies.

In a weighted independence system, an independent set is said to be $\alpha$-robust if, for all $p$, the total weight of its heaviest $p$ elements is at least $\alpha$ times the maximum weight of a $p$-independent set. Here a $p$-independent set is an independent set with at most $p$ elements. The set of matchings in a weighted graph is a typical example of a weighted independence system, and Hassin and Rubinstein [SIAM J. Discrete Math., 15 (2002), pp. 530--537] showed that every graph has a $\frac{1}{\sqrt2}$-robust matching and it can be found by a $k$th power algorithm in polynomial time. In this paper, we show that it can be extended to the matroid intersection problem; i.e., there always exists a $\frac{1}{\sqrt2}$-robust matroid intersection, which is polynomially computable. We also study the time complexity of the robust matching problem. We show that a 1-robust matching can be computed in polynomial time (if one exists), and, for any fixed number $\alpha$ with $\frac{1}{\sqrt2}<\alpha<1$, the problem to determine whether a given weighted graph has an $\alpha$-robust matching is NP-complete. These together with the positive result for $\alpha=\frac{1}{\sqrt2}$ in [R. Hassin and S. Rubinstein, SIAM J. Discrete Math., 15 (2002), pp. 530--537] give us a sharp border for the complexity for the robust matching problem. Moreover, we show that the problem is strongly NP-complete when $\alpha$ is a part of the input. Finally, we show the limitations of the $k$th power algorithm for robust matchings; i.e., for any $\epsilon>0$, there exists a weighted graph such that no $k$th power algorithm outputs a $(\frac{1}{\sqrt2}+\epsilon)$-approximation for computing the most robust matching.

An independence system $\mathcal{F}$ is one of the most fundamental combinatorial concepts, which includes a variety of objects in graphs and hypergraphs such as matchings, stable sets, and matroids. We discuss the robustness for independence systems, which is a natural generalization of the greedy property of matroids. For a real number $\alpha> 0$, a set $X\in\mathcal{F}$ is said to be $\alpha$-robust if for any $k$, it includes an $\alpha$-approximation of the maximum $k$-independent set, where a set $Y$ in $\mathcal{F}$ is called $k$-independent if the size $|Y|$ is at most $k$. In this paper, we show that every independence system has a $1/\sqrt{\mu(\mathcal{F})}$-robust independent set, where $\mu(\mathcal{F})$ denotes the exchangeability of $\mathcal{F}$. Our result contains a classical result for matroids and the ones of Hassin and Rubinstein [SIAM J. Discrete Math., 15 (2002), pp. 530--537] for matchings and Fujita, Kobayashi, and Makino [SIAM J. Discrete Math., 27 (2013), pp. 1234--1256] for matroid $2$-intersections, and provides better bounds for the robustness for many independence systems such as $b$-matchings, hypergraph matchings, matroid $p$-intersections, and unions of vertex disjoint paths. Furthermore, we provide bounds of the robustness for nonlinear weight functions such as submodular and convex quadratic functions. We also extend our results to independence systems in the integral lattice with separable concave weight functions.

We study the notion of regret ratio proposed in [19] Nanongkai et al. [VLDB10] to deal with multi-criteria decision making in database systems. The regret minimization query proposed in [19] Nanongkai et al. was shown to have features of both skyline and top-k: it does not need information from the user but still controls the output size. While this approach is suitable for obtaining a reasonably small regret ratio, it is still open whether one can make the regret ratio arbitrarily small. Moreover, it remains open whether reasonable questions can be asked to the users in order to improve efficiency of the process.

In this paper, we consider solving the integer linear systems, i.e., given a matrix A in R^{m*n}, a vector b in R^m, and a positive integer d, to compute an integer vector x in D^n such that Ax <= b, where m and n denote positive integers, R denotes the set of reals, and D={0,1,..., d-1}. The problem is one of the most fundamental NP-hard problems in computer science. For the problem, we propose a complexity index h which is based only on the sign pattern of A. For a real r, let ILS_=(r) denote the family of the problem instances I with h(I)=r. We then show the following trichotomy: - ILS_=(r) is linearly solvable, if r < 1, - ILS_=(r) is weakly NP-hard and pseudo-polynomially solvable, if r = 1, and - ILS_=(r) is strongly NP-hard, if r > 1. This, for example, includes the existing results that quadratic systems and Horn systems can be solved in pseudo-polynomial time.

We prove strong ${\mathbb {NP}}$ -completeness for the four variants of caching with multi-size pages. These four variants are obtained by choosing either the fault cost or the bit cost model, and by combining it with either a forced or an optional caching policy. This resolves two questions in the area of paging and caching that were open since the 1990s.

In this article, we investigate deductive inference for interiors and exteriors of Horn knowledge bases, where interiors and exteriors were introduced by Makino and Ibaraki [1996] to study stability properties of knowledge bases. We present a linear time algorithm for deduction for interiors and show that deduction is coNP-complete for exteriors. Under model-based representation, we show that the deduction problem for interiors is NP-complete while the one for exteriors is coNP-complete. As for Horn envelopes of exteriors, we show that it is linearly solvable under model-based representation, while it is coNP-complete under formula-based representation. We also discuss polynomially solvable cases for all the intractable problems.

We study nonadaptive broadcasting in trees, a process of sending a message from one vertex in a tree to all other vertices. In the nonadaptive model, each vertex has a specified, ordered list of its neighbors. After receiving a broadcast message, a vertex sends the message to its neighbors, one after another, in the order specified by the list. The broadcast is completed when all vertices have received the message. We obtain lower and upper bounds on the minimum time required to complete a nonadaptive broadcast in a tree and improved upper bounds for general graphs. We give a polynomial time algorithm for determining the minimum nonadaptive broadcast time of any given tree. We also show how to construct the largest possible trees having a given nonadaptive broadcast time. © 2010 Wiley Periodicals, Inc. NETWORKS, Vol. 57(2), 157–168 2011

Given the prime conjunctive normal form (CNF) representation $\phi$ of a monotone Boolean function $f:\{0,1\}^n\to\{0,1\}$, the dualization problem calls for finding the corresponding prime disjunctive normal form representation $\psi$ of f. A very simple method works by multiplying out the clauses of $\phi$ from left to right in some order, simplifying whenever possible by using the absorption law. We show that for any monotone CNF $\phi$, left-to-right multiplication can be done in subexponential time, and for many interesting subclasses of monotone CNFs such as those with bounded size, bounded degree, bounded intersection, bounded conformality, and read-once formula, it can be done in polynomial or quasi-polynomial time.

Given a system $(V,f,d)$ on a finite set V consisting of two set functions $f:2^V\to\mathbb{R}$ and $d:2^V\to\mathbb{R}$, we consider the problem of finding a set $R\subseteq V$ of minimum cardinality such that $f(X)\ge d(X)$ for all $X\subseteq V-R$, where the problem can be regarded as a natural generalization of the source location problems and the external network problems in (undirected) graphs and hypergraphs. We give a structural characterization of minimal deficient sets of $(V,f,d)$ under certain conditions. We show that all such sets form a tree hypergraph if f is posimodular and d is modulotone (i.e., each nonempty subset X of V has an element $v\in X$ such that $d(Y)\ge d(X)$ for all subsets Y of X that contain v) and that, conversely, any tree hypergraph can be represented by minimal deficient sets of $(V,f,d)$ for a posimodular function f and a modulotone function d. By using this characterization, we present a polynomial-time algorithm if, in addition, f is submodular and d is given by either $d(X)=\max\{p(v)\mid v\in X\}$ for a function $p:V\to\RR_+$ or $d(X)=\max\{r(v,w)\mid v\in X,w\in V-X\}$ for a function $r:V^2\to\mathbb{R}_+$. Our result provides first polynomial-time algorithms for the source location problem in hypergraphs and the external network problems in graphs and hypergraphs. We also show that the problem is intractable, even if f is submodular and $d\equiv\mathbf{0}$.

We provide a characterization of Horn cores for formulas in conjunctive normal form (CNF) and, based on it, a novel algorithm for computing Horn cores of disjunctions of Horn CNFs that has appealing properties (e.g., it is polynomial for a bounded disjunction). Furthermore, we show that recognizing the Horn envelope of a disjunction of two Horn CNFs is intractable, and that computing a compact Horn CNF for it (that is irredundant and prime) is not feasible in polynomial total time unless P=NP; this answers an open problem.

Let G=(V,E) be an undirected graph, and let B⊆V×V be a collection of vertex pairs. We give an incremental polynomial time algorithm to generate all minimal edge sets X⊆E such that every pair (s,t)∈B of vertices is disconnected in (V,E ∖ X), generalizing well-known efficient algorithms for generating all minimal s-t cuts, for a given pair s,t of vertices. We also present an incremental polynomial time algorithm for generating all minimal subsets X⊆E such that no (s,t)∈B is a bridge in (V,X∪B). Both above problems are special cases of a more general problem that we call generating cut conjunctions for matroids: given a matroid M on ground set S=E∪B, generate all minimal subsets X⊆E such that no element b∈B is spanned by E ∖ X. Unlike the above special cases, corresponding to the cycle and cocycle matroids of the graph (V,E∪B), the more general problem of generating cut conjunctions for vectorial matroids turns out to be NP-hard.

Abduction is a fundamental mode of reasoning with applications in many areas of AI and Computer Science. The computation of abductive explanations is an important computational problem, which is at the core of early systems such as the ATMS and Clause Management Systems and is intimately related to prime implicate generation in propositional logic. Many algorithms have been devised for computing some abductive explanation, and the complexity of the problem has been well studied. However, little attention has been paid to the problem of computing multiple explanations, and in particular all explanations for an abductive query. We fill this gap and consider the computation of all explanations of an abductive query from a propositional Horn theory, or of a polynomial subset of them. Our study pays particular attention to the form of the query, ranging from a literal to a compound formula, to whether explanations are based on a set of abducible literals and to the representation of the Horn theory, either by a Horn conjunctive normal form (CNF) or model-based in terms of its characteristic models. For these combinations, we present either tractability results in terms of polynomial total-time algorithms, intractability results in terms of nonexistence of such algorithms (unless P = NP), or semi-tractability results in terms of solvability in quasi-polynomial time, established by polynomial-time equivalence to the problem of dualizing a monotone CNF expression. Our results complement previous results in the literature, and refute a longstanding conjecture by Selman and Levesque. They elucidate the complexity of generating all abductive explanations and shed light on related problems such as generating sets of restricted prime implicates of a Horn theory. The algorithms for tractable cases can be readily applied for generating a polynomial subset of explanations in polynomial time.

Let M be a matroid defined by an independence oracle on ground set S, and let $A\subseteq S$. We present an incremental polynomial-time algorithm for enumerating all minimal (maximal) subsets of S which span (do not span) A. Special cases of these problems include the generation of bases, circuits, hyperplanes, flats of given rank, circuits through a given element, generalized Steiner trees, and multiway cuts in graphs, as well as some other applications. We also consider some tractable and NP-hard generation problems related to systems of polymatroid inequalities and (generalized) packing and spanning in matroids.

We consider the problem of dualizing a monotone CNF (equivalently, computing all minimal transversals of a hypergraph) whose associated decision problem is a prominent open problem in NP-completeness. We present a number of new polynomial time, respectively, output-polynomial time results for significant cases, which largely advance the tractability frontier and improve on previous results. Furthermore, we show that duality of two monotone CNFs can be disproved with limited nondeterminism. More precisely, this is feasible in polynomial time with O(log 2 n /\log log n) suitably guessed bits. This result sheds new light on the complexity of this important problem.

This paper considers the problem of dualizing a monotone CNF (equivalently, computing all minimal transversals of a hypergraph), whose associated decision problem is a prominent open problem in NP-completeness. We present a number of new polynomial time resp. output-polynomial time results for significant cases, which largely advance the tractability frontier and improve on previous results. Furthermore, we show that duality of two monotone CNFs can be disproved with limited nondeterminism (more precisely, in polynomial time with $O(\log^2 n)$ suitably guessed bits). This result sheds new light on the complexity of this important problem.

We consider the problem of enumerating all minimal integer solutions of a monotone system of linear inequalities. We first show that, for any monotone system of r linear inequalities in n variables, the number of maximal infeasible integer vectors is at most rn times the number of minimal integer solutions to the system. This bound is accurate up to a polylog(r) factor and leads to a polynomial-time reduction of the enumeration problem to a natural generalization of the well-known dualization problem for hypergraphs, in which dual pairs of hypergraphs are replaced by dual collections of integer vectors in a box. We provide a quasi-polynomial algorithm for the latter dualization problem. These results imply, in particular, that the problem of incrementally generating all minimal integer solutions to a monotone system of linear inequalities can be done in quasi-polynomial time.

In this paper, we study issues on disjunctions of propositional Horn theories. In particular, we consider the problems of deciding whether a disjunction of Horn theories is Horn, and, if not, computing a Horn core (i.e., a maximal Horn theory included in this disjunction) and the Horn envelope (i.e., the minimum Horn theory including the disjunction), where a Horn core and the Horn envelope are important approximations of the original theory in artificial intelligence. The problems are investigated for two different representations of Horn theories, namely, for Horn conjunctive normal forms (CNFs) and characteristic models. While the problems are shown to be intractable in general, in the case of bounded disjunctions, we present polynomial time algorithms for testing the Horn property in both representations and for computing a Horn core in the CNF representation. Even in the case of bounded disjunction, no polynomial algorithm exists (unless P=NP) for computing a Horn core in the characteristic model representation. Computing the Horn envelope is polynomial in the characteristic model representation, while it is exponential in the CNF representation, even for bounded disjunction.

A coterie under an underlying set U is a family of subsets of U such that every pair of subsets has at least one element in common, but neither is a subset of the other. A coterie C under U is said to be nondominated (ND) if there is no other coterie D under U such that, for every $Q \in C$, there exists $Q' \in D$ satisfying $Q' \subseteq Q$. We introduce the operation $\sigma$ which transforms a ND coterie to another ND coterie. A regular coterie is a natural generalization of a vote-assignable coterie. We show that any regular ND coterie C can be transformed to any other regular ND coterie D by judiciously applying the $\sigma$ operation to C at most |C|+|D|-2 times. As another application of the $\sigma$ operation, we present an incrementally polynomial-time algorithm for generating all regular ND coteries. We then introduce the concept of a g-regular functional as a generalization of availability. We show how to construct an optimum coterie C with respect to a g-regular functional in O(n 3 |C|) time, where n =|U|. Finally, we discuss the structures of optimum coteries with respect to a g-regular functional.

A coterie is a family of subsets such that every pair of subsets in it has at least one element in common but neither is a subset of the other. We introduce an operator σ, which transforms a ND (non-dominated; see the Introduction for definition) coterie to another ND coterie. A “regular” coterie is a natural generalization of a “vote-assignable” coterie, which is used in some practical applications. We show that any regular ND coterie C can be transformed to any other regular ND coterie D by judiciously applying σ operations to C at most |C| + |D| - 2 times.

We consider two generalizations of the notion of transversal to a finite hypergraph, the so-called multiple and partial transversals. Multiple transversals naturally arise in 0-1 programming, while partial transversals are related to data mining and machine learning. We show that for an arbitrary hypergraph the families of multiple and partial transversals are both dual-bounded in the sense that the size of the corresponding dual hypergraph is bounded by a polynomial in the cardinality and the length of description of the input hypergraph. Our bounds are based on new inequalities of extremal set theory and threshold Boolean logic, which may be of independent interest. We also show that the problems of generating all multiple and all partial transversals for a given hypergraph are polynomial-time reducible to the generation of all ordinary transversals for another hypergraph, i.e., to the well-known dualization problem for hypergraphs. As a corollary, we obtain incremental quasi-polynomial-time algorithms for both of the above problems, as well as for the generation of all the minimal binary solutions for an arbitrary monotone system of linear inequalities.

Given a partially defined Boolean function (pdBf) (T,F), we investigate in thispaper how to find a Horn extension $f: \{0,1\}^n \mapsto \{0,1\}$, which is consistent with (T,F), where $T \subseteq \{0,1\}^n$ denotes a set of true Boolean vectors (or positive examples) and $F \subseteq \{0,1\}^n$ denotes a set of false Boolean vectors (or negative examples). Given a pdBf (T,F), it is known that the existence of a Horn extension can be checked in polynomial time. As there are many Horn extensions, however, we consider those extensions f which have maximal and minimal sets T(f) of the true vectors of f, respectively. For a pdBf (T,F), there always exists the unique maximal (i.e., maximum) Horn extension, but there are in general many minimal Horn extensions. We first show that a polynomial time membership oracle can be constructed for the maximum extension, even if its disjunctive normal form (DNF) can be very long. Our main contribution is to show that checking if a given Horn DNF represents a minimal extension and generating a Horn DNF of a minimal Horn extension can both be done in polynomial time. We also can check in polynomial time if a pdBf (T,F) has the unique minimal Horn extension. However, the problems of finding a Horn extension f with the smallest |T(f)| and of obtaining a Horn DNF, whose number of literals is smallest, are both NP-hard.

Consider the problem of identifying $\min T(f)$ and $\max F(f)$ of a positive (i.e., monotone) Boolean function f by using membership queries only, where $\min T(f)\,(\max F(f))$ denotes the set of minimal true vectors (maximal false vectors) of f. It is known that an incrementally polynomial algorithm exists if and only if there is a polynomial time algorithm to check the existence of an unknown vector u for given sets $MT \subseteq \min T(f)$ and $MF \subseteq \max F(f)$; that is, $u \in \{0,1\}^n \setminus (\{v | v \geq w {\rm for some } w \in MT \} \cup \{v | v \leq w {\rm for some } w \in MF \})$. This paper introduces a measure for the difficulty to find an unknown vector, which is called the maximum latency. If the maximum latency is constant, then an unknown vector can be found in polynomial time and there is an incrementally polynomial algorithm for identification. Several subclasses of positive functions are shown to have constant maximum latency, e.g., 2-monotonic positive functions, $\Delta$-partial positive threshold functions, and matroid functions, while the class of general positive functions has $\lfloor n/4 \rfloor +1$ maximum latency and the class of positive k-DNF functions has $\Omega (\sqrt{n})$ maximum latency.