Endre Boros

Rutgers, The State University of New Jersey | Rutgers · Department of Management Science and Information Systems

We study remoteness function R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}$$\end{document} of impartial games introduced by Smith in 1966. The player...

A hypergraph is conformal if it is the family of maximal cliques of a graph. In this paper we are interested in the problem of determining when is the family of minimal transversal of maximal cliques of a graph conformal. Such graphs are called clique dually conformal (CDC for short). As our main results, we completely characterize CDC graphs withi...

We prove that a deterministic n-person shortest path game has a Nash equlibrium in pure and stationary strategies if it is edge-symmetric (that is (u, v) is a move whenever (v, u) is, apart from moves entering terminal vertices) and the length of every move is positive for each player. Both conditions are essential, though it remains an open proble...

Given a hypergraph $\mathcal{H}$, the dual hypergraph of $\mathcal{H}$ is the hypergraph of all minimal transversals of $\mathcal{H}$. The dual hypergraph is always Sperner, that is, no hyperedge contains another. A special case of Sperner hypergraphs are the conformal Sperner hypergraphs, which correspond to the families of maximal cliques of grap...

We consider distributed approval voting schemes. Each voter i∈I\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i \in I$$\end{document} has αi\documentclass[12pt]{minima...

The Sprague–Grundy (SG) theory reduces the disjunctive compound of impartial games to the classical game of NIM. We generalize this concept by introducing hypergraph compounds of impartial games. An impartial game is called SG-decreasing if its SG value is decreased by every move. Extending the SG theory, we reduce hypergraph compounds of SG-decrea...

Hypergraph Horn functions were introduced as a subclass of Horn functions that can be represented by a collection of circular implication rules. These functions possess distinguished structural and computational properties. In particular, their characterizations in terms of implicate-duality and the closure operator provide extensions of matroid du...

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 ma...

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...

We present a model for layered security with applications to the protection of sites such as stadiums or large gathering places. We formulate the problem as one of maximizing the capture of illegal contraband. The objective function is indefinite and only limited information can be gained when the problem is solved by standard convex optimization m...

We prove that a deterministic n-person shortest path game has a Nash equlibrium in pure and stationary strategies, provided that the game is symmetric (that is (u,v) is a move whenever (v,u) is, apart from moves entering terminal vertices) and the length of every move is positive for each player. Both conditions are essential, though it remains an...

In many classes of discrete optimization problems, we do not know (yet) an efficient algorithm that would guarantee an optimal solution for all instances. However, for some instances we may be able to identify provably optimal values for some of its variables. Persistency and autarky are properties of partial assignments that allow us to argue abou...

We consider the game of proper Nim, in which two players alternately move by taking stones from n piles. In one move a player chooses a proper subset (at least one and at most n-1) of the piles and takes some positive number of stones from each pile of the subset. The player who cannot move is the loser. Jenkyns and Mayberry (Int J Game Theory 9(1)...

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 M...

Each voter $i \in I$ has $\alpha_i$ cards that (s)he distributes among the candidates $a \in A$ as a measure of approval. One (or several) candidate(s) who received the maximum number of cards is (are) elected. We provide polynomial algorithms to recognize voting forms and voting correspondences generated by such voting schemes in cases when either...

Given two graphs $G$ and $G^*$ with a one-to-one correspondence between their edges, when do $G$ and $G^*$ form a pair of dual graphs realizing the vertices and countries of a map embedded in a surface? A criterion was obtained by Jack Edmonds in 1965. Furthermore, let $\boldsymbol{d}=(d_1,\ldots,d_n)$ and $\boldsymbol{t}=(t_1,\ldots,t_m)$ be their...

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-...

The problem of minimizing a pseudo-Boolean function, that is, a real-valued function of 0–1 variables, arises in many applications. A quadratization is a reformulation of this nonlinear problem into a quadratic one, obtained by introducing a set of auxiliary binary variables. A desirable property for a quadratization is to introduce a small number...

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...

Given a CNF formula $\Phi$ with clauses $C_1,\ldots,C_m$ and variables $V=\{x_1,\ldots,x_n\}$, a truth assignment $a:V\rightarrow\{0,1\}$ of $\Phi$ leads to a clause sequence $\sigma_\Phi(a)=(C_1(a),\ldots,C_m(a))\in\{0,1\}^m$ where $C_i(a) = 1$ if clause $C_i$ evaluates to $1$ under assignment $a$, otherwise $C_i(a) = 0$. The set of all possible c...

A hypergraph is said to be 1‐Sperner if for every two hyperedges the smallest of their two set differences is of size one. We present several applications of 1‐Sperner hypergraphs to graphs. First, we consider several ways of associating hypergraphs to graphs, namely, vertex cover, clique, independent set, dominating set, and closed neighborhood hy...

A hypergraph is Sperner if no hyperedge contains another one. A Sperner hypergraph is equilizable (resp., threshold) if the characteristic vectors of its hyperedges are the (minimal) binary solutions to a linear equation (resp., inequality) with positive coefficients. These combinatorial notions have many applications and are motivated by the theor...

The classical game of {\sc Nim} can be naturally extended and played on an arbitrary hypergraph $\cH \subseteq 2^V \setminus \{\emptyset\}$ whose vertices $V = \{1, \ldots, n\}$ correspond to piles of stones. By one move a player chooses an edge $H$ of $\cH$ and reduces arbitrarily all piles $i \in H$. In 1901 Bouton solved the classical {\sc Nim}...

We consider two-person zero-sum stochastic mean payoff games with perfect information, or BWR-games, given by a digraph G=(V,E), with local rewards r:E→Z, and three types of positions: black V B , white V W , and random V R forming a partition of V. It is a long-standing open question whether a polynomial time algorithm for BWR-games exists, or not...

Horn functions form a 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 that remains hard even for the...

We say that a graph G has the CIS-property and call it a CIS-graph if every maximal clique and every maximal stable set of G intersects.By definition, G is a CIS-graph if and only if the complementary graph \(\overline {G}\) is a CIS-graph. Let us substitute a vertex v of a graph G′ by a graph G″ and denote the obtained graph by G. It is also easy...

We consider finite Markov decision processes with undiscounted total effective payoff. We show that there exist uniformly optimal pure and stationary strategies that can be computed by solving a polynomial number of linear programs. This implies that in a two-player zero-sum stochastic game with perfect information and with total effective payoff t...

A hypergraph H is said to be 1-Sperner if for every two hyperedges the smallest of their two set differences is of size one. The authors introduced this concept in 2015 and completely characterized this class of hypergraphs via a decomposition theorem. Here, we present several applications of 1-Sperner hypergraphs and their structure to graphs. In...

We consider a generalization of the classical game of $NIM$ called hypergraph $NIM$. Given a hypergraph $\cH$ on the ground set $V = \{1, \ldots, n\}$ of $n$ piles of stones, two players alternate in choosing a hyperedge $H \in \cH$ and strictly decreasing all piles $i\in H$. The player who makes the last move is the winner. Recently it was shown t...

We consider a generalization of the classical game of $NIM$ called hypergraph $NIM$. Given a hypergraph $\cH$ on the ground set $V = \{1, \ldots, n\}$ of $n$ piles of stones, two players alternate in choosing a hyperedge $H \in \cH$ and strictly decreasing all piles $i\in H$. The player who makes the last move is the winner. In this paper we give a...

We suggest a new algorithm for two-person zero-sum undiscounted stochastic games focusing on stationary strategies. Given a positive real $\epsilon$, let us call a stochastic game $\epsilon$-ergodic, if its values from any two initial positions differ by at most $\epsilon$. The proposed new algorithm outputs for every $\epsilon>0$ in finite time ei...

We give an example of a three-person deterministic graphical game that has no Nash equilibrium in pure stationary strategies. The game has seven positions, four outcomes (a unique cycle and three terminal positions), and its normal form is of size 2×2×4 only. Thus, the example strengthens significantly the one obtained in 2014 by Gurvich and Oudalo...

We consider two-person zero-sum stochastic mean payoff games with perfect information, or BWR-games, given by a digraph $G = (V, E)$, with local rewards $r: E \to \ZZ$, and three types of positions: black $V_B$, white $V_W$, and random $V_R$ forming a partition of $V$. It is a long-standing open question whether a polynomial time algorithm for BWR-...

A discrete function of $n$ variables is a mapping $g : X_1 \times \ldots \times X_n \rightarrow A$, where $X_1, \ldots, X_n$, and $A$ are arbitrary finite sets. Function $g$ is called {\em separable} if there exist $n$ functions $g_i : X_i \rightarrow A$ for $i = 1, \ldots, n$, such that for every input $x_1, \ldots ,x_n$ the function $g(x_1, \ldot...

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.

A pure Horn CNF is minimal if no shorter pure Horn CNF representing the same function exists, where the CNF length may mean several different things, e.g. the number of clauses, or the total number of literals (sum of clause lengths), or the number of distinct bodies (source sets). The corresponding minimization problems (a different problem for ea...

The Sprague-Grundy (SG) theory reduces the sum of impartial games to the classical game of $NIM$. We generalize the concept of sum and introduce $\cH$-combinations of impartial games for any hypergraph $\cH$. In particular, we introduce the game $NIM_\cH$ which is the $\cH$-combination of single pile $NIM$ games. An impartial game is called SG decr...

We give an example of a three person chess-like game that has no Nash equilibrium in pure stationary strategies. The game has seven positions, four outcomes (a unique cycle and three terminal positions), and its normal form is of size 2 x 2 x 4 only. Thus, our example strengthens significantly the one obtained in 2014 by Gurvich and Oudalov, which...

Very large nonlinear unconstrained binary optimization problems arise in a broad array of applications. Several exact or heuristic techniques have proved quite successful for solving many of these problems when the objective function is a quadratic polynomial. However, no similarly efficient methods are available for the higher degree case. Since h...

We consider Gillette’s two-person zero-sum stochastic games with perfect information. For each (Formula presented.) we introduce an effective reward function, called k-total. For (Formula presented.) and 1 this function is known as mean payoff and total reward, respectively. We restrict our attention to the deterministic case. For all k, we prove t...

An n-multigraph G=(V;Ei∣i∈I) is a complete graph G=(V,E) whose edges are covered by n=|I| sets, E=∪i∈IEi, some of which might be empty. If this cover is a partition, then G is called an n-graph. We say that an n-graph G′=(V;Ei′∣i∈I) is an edge subgraph of an n-multigraph G=(V;Ei∣i∈I) if Ei′⊆Ei for all i∈I. We denote by Δ the n-graph on three vertic...

In 1964 Shapley observed that a matrix has a saddle point in pure strategies whenever every its (Formula presented.) submatrix has one. In contrast, a bimatrix game may have no pure strategy Nash equilibrium (NE) even when every (Formula presented.) subgame has one. Nevertheless, Shapley’s claim can be extended to bimatrix games as follows. We part...

We introduce a new class of hypergraphs, the class of $1$-Sperner hypergraphs. A hypergraph ${\cal H}$ is said to be $1$-Sperner if every two distinct hyperedges $e,f$ of ${\cal H}$ satisfy $\min\{|e\setminus f|,|f\setminus e|\} = 1$. We prove a decomposition theorem for $1$-Sperner hypergraphs and examine several of its consequences, including bou...

Moore's generalization of the game of Nim is played as follows. Given two integer parameters $n, k$ such that $1 \leq k \leq n$, and $n$ piles of tokens. Two players take turns. By one move a player reduces at least one and at most $k$ piles. The player who makes the last move wins. The P-positions of this game were characterized by Moore in 1910 a...

Higher-order Markov Random Fields, which can capture important properties of natural images, have become increasingly important in computer vision. While graph cuts work well for first-order MRF’s, until recently they have rarely been effective for higher-order MRF’s. Ishikawa’s graph cut technique [1], [2] shows great promise for many higher-order...

We consider several graphs classes defined in terms of conditions on cliques and stable sets, including CIS, split, equistable, and other related classes. We pursue a systematic study of the relations between them. As part of this study, we introduce two generalizations of CIS graphs, obtain a new characterization of split graphs, and a characteriz...

In the standard {\sc Nim} with $n$ heaps, a player by one move can reduce (by a positive amount) exactly one heap of his choice. In this paper we consider the game of {\em complementary {\sc Nim}} ({\sc Co-Nim}), in which a player by one move can reduce at least one and at most $n-1$ heaps, of his choice. An explicit formula for the Sprague-Grundy...

We study the inventory and distribution operations encountered in oil and petrochemical industry. We show some special cases for the NP-complete problem, and propose polynomial time solution methods. We propose two approaches for the main problem. One of them makes use of the minimum cost flow formulation of the same problem under some assumptions,...

We present a model and discrete event simulation of USCG Air Stations, accounting for the mission demands and maintenance procedures pertaining to USCG aircraft. The simulation provides aircraft availability distributions and mission performance metrics based on varying input scenarios, including changes in the number of stationed aircraft and main...

Recently, Milani\v{c} and Trotignon introduced the class of equistarable graphs as graphs without isolated vertices admitting positive weights on the edges such that a subset of edges is of total weight $1$ if and only if it forms a maximal star. Based on equistarable graphs, counterexamples to three conjectures on equistable graphs were constructe...

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 effec...

We suggest a new algorithm for two-person zero-sum undiscounted stochastic games focusing on stationary strategies. Given a positive real \(\epsilon \), let us call a stochastic game \(\epsilon \)-ergodic, if its values from any two initial positions differ by at most \(\epsilon \). The proposed new algorithm outputs for every \(\epsilon >0\) in fi...

We consider Gillette's two-person zero-sum stochastic games with perfect information. For each $k \in \mathbb{Z}_+$ we introduce a payoff function, called the $k$-total reward. For $k = 0$ and $1$ these payoffs are known as mean payoff and total reward, respectively. We restrict our attention to the deterministic case, the so called cyclic games. F...

We consider the problem of finding upper and lower bounds for the probability of the union of events when the probabilities of the single events and the probabilities of the intersections of up to m events are given. It is known that the best possible bounds can be obtained by solving linear programming problems with a number of variables that is e...

Motivated by challenges related to domination, connectivity, and information propagation in social and other networks, we initiate the study of the Vector Connectivity problem. This problem takes as input a graph G and an integer k v for every vertex v of G, and the objective is to find a vertex subset S of minimum cardinality such that every verte...

A pseudo-Boolean function is a real-valued function f(x) = f(x1; x2; : : : ; xn) of n binary variables, that is, a mapping from f0; 1gn to R. For a pseudo-Boolean function f(x) on f0; 1gn, we say that g(x; y) is a quadratization of f if g(x; y) is a quadratic polynomial depending on x and on m auxiliary binaryvariables y1; y2; : : : ; ym such that...

We survey current term-wise techniques for quadratizing high-degree pseudo-Boolean functions and introduce a new one, which allows multiple splits of terms. We also introduce the first aggregative approach, which splits a collection of terms based on their common parts.

A model was created for the United States Coast Guard (USCG) to maximize aircraft fleet operational performance subject to budgetary constraints, or, conversely, to minimize aircraft fleet operational costs subject to performance targets. This is a two-stage model: The first stage, prior work, is a simulation model of each USCG Air Station generati...

We study a difficult real life scheduling problem encountered in oil and petrochemical industry, involving inventory and distribution operations, which requires integrated scheduling. The problem itself is NP-complete, however we show some special cases, and propose polynomial time solution methods. These could be used as a starting point for a heu...

For any positive integer parameters a and b, Gurvich recently introduced a generalization mexb of the standard minimum excludant mex = mex1, along with a game NIM(a, b) that extends further Fraenkel’s NIM = NIM(a, 1), which in its turn is a generalization of the classical Wythoff NIM = NIM(1, 1). It was shown that P-positions (the kernel) of NIM(a,...

A CNF is minimal if no shorter CNF representing the same function exists, where by CNF length we mean either the number of clauses or the total number of literals (sum of clause lengths). In this paper we develop a decomposition approach that can be in certain situations applied to a CNF formula when proving its minimality. We give two examples in...

We consider two-person zero-sum stochastic mean payoff games with perfect information, or BWR-games, given by a digraph G = (V, E), with local rewards r: E → ℝ, and three types of vertices: black V B , white V W , and random V R forming a partition of V. It is a long-standing open question whether a polynomial time algorithm for BWR-games exists, o...

It is shown that the discount factor needed to solve an undiscounted mean payoff stochastic game to optimality is exponentially close to 1, even in one-player games with a single random node and polynomially bounded rewards and transition probabilities. For the class of the so-called irreducible games with perfect information and a constant number...

We consider two-person zero-sum mean payoff undiscounted stochastic games and obtain sufficient conditions for the existence of a saddle point in uniformly optimal stationary strategies. Namely, these conditions enable us to bring the game, by applying potential transformations, to a canonical form in which locally optimal strategies are globally o...

A circulant is a Cayley graph over a cyclic group. A well-covered graph is a graph in which all maximal stable sets are of the same size, or in other words, they are all maximum. A CIS graph is a graph in which every maximal stable set and every maximal clique intersect. It is not difficult to show that a circulant G is a CIS graph if and only if G...

Recently, it was shown that Chess-like games may have no uniform (subgame perfect) Nash equilibria in pure positional strategies. Moreover, Nash equilibria may fail to exist already in two-person games in which all infinite plays are equivalent and ranked as the worst outcome by both players. In this paper, we extend this negative result further, p...

Graph cut algorithms [9], commonly used in computer vision, solve a first-order MRF over binary variables. The state of the art for this NP-hard problem is QPBO [1,2], which finds the values for a subset of the variables in the global minimum. While QPBO is very effective overall there are still many difficult problems where it can only label a sma...

\begin{abstract} We study the quality of stable matchings from the individuals' viewpoint. To each matching we associate its \emph{rank-profile} describing the individuals' satisfaction with the matching. We provide a complete and computationally efficient characterization of the rank-profiles that can arise from men-optimal, women-optimal, and arb...

It was recently shown that every totally tight two-person game form is acyclic, dominance-solvable, and hence, Nash-solvable too. In this paper, we exhibit an example showing that the first two implications fail for the three-person (n=3n=3) game forms. Yet, we show that the last one (total tightness implies Nash-solvability) still holds for n=3n=3...

We consider nn-person positional games with perfect information modeled by finite directed graphs that may have directed cycles, assuming that all infinite plays form a single outcome cc, in addition to the standard outcomes a1,…,ama1,…,am formed by the terminal positions. (For example, in the case of Chess or Backgammon n=2n=2 and cc is a draw.) T...

We study the hardness of approximation of clause minimum and literal minimum representations of pure Horn functions in $n$ variables. We show that unless P=NP, it is not possible to approximate in polynomial (depending in $n$) time the minimum number of clauses and the minimum number of literals of pure Horn CNF representations to within factors of...

Higher-order Markov Random Fields, which can capture important properties of natural images, have become increasingly important in computer vision. While graph cuts work well for first-order MRF's, until recently they have rarely been effective for higher-order MRF's. Ishikawa's graph cut technique [8, 9] shows great promise for many higher-order M...

A two-person positional game form g (with perfect information and without moves of chance) is modeled by a finite directed graph (digraph) whose vertices and arcs are interpreted as positions and moves, respectively. All simple directed cycles of this digraph together with its terminal positions form the set A of the outcomes. Each non-terminal pos...

Learning from examples is a frequently arising challenge, with a large number of algorithms proposed in the classification and data mining literature. The evaluation of the quality of such algorithms is usually carried out ex post, on an experimental basis: their perfor- mance is measured either by cross validation on benchmark data sets, or by cli...

Given a graph G=(V,E) and a weight function on the edges w:E→ℝ, we consider the polyhedron P(G,w) of negative-weight flows on G, and get a complete characterization of the vertices and extreme directions of P(G,w). Based on this characterization, and using a construction developed in Khachiyan et al. (Discrete Comput. Geom. 39(1–3):174–190, 2008),...

In this paper, we consider two-player zero-sum stochastic mean payoff games with perfect information modeled by a digraph with black, white, and random vertices. These BWR-games games are polynomially equivalent with the classical Gillette games, which include many well-known subclasses, such as cyclic games, simple stochastic games, stochastic par...

We consider the problem of combining a given set of diagnostic tests into an inspection system to classify items of interest (cases) with maximum accuracy such that the cost of performing the tests does not exceed a given budget constraint. One motivating application is sequencing diagnostic tests for container inspection, where the diagnostic test...

Written by prominent experts in the field, this monograph provides the first comprehensive, unified presentation of the structural, algorithmic and applied aspects of the theory of Boolean functions. The book focuses on algebraic representations of Boolean functions, especially disjunctive and conjunctive normal form representations. This framework...

A friendship graph is a graph in which every two distinct vertices have exactly one common neighbor. All finite friendship graphs are known, each of them consists of triangles having a common vertex. We extend friendship graphs to two-graphs, a two-graph being an ordered pair G = (G 0, G 1) of edge-disjoint graphs G 0 and G 1 on the same vertex-set...

In this paper, we consider two-person zero-sum stochastic mean payoff games with perfect information, or BWR-games, given by a digraph G = (V = V B  ∪ V W  ∪ V R , E), with local rewards r: E ® \mathbb Rr: E \to {\mathbb R}, and three types of vertices: black V B , white V W , and random V R . The game is played by two players, White and Bl...

It is known that a two-person game form gg is Nash-solvable if and only if it is tight. We strengthen the concept of tightness as follows: a game form is called totally tight if each of its 2×2 subfo