arXiv subjects

Kazuhisa Makino

Publications and source records attributed to Kazuhisa Makino.

46 records · Page 3Linked to original sources

arXiv2016

A Convex Programming-based Algorithm for Mean Payoff Stochastic Games with Perfect Information

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-games exists, even when $|V_R|=0$. In fact, a pseudo-polynomial algorithm for BWR-games would already imply their polynomial solvability. In this short note, we show that BWR-games can be solved via convex programming in pseudo-polynomial time if the number of random positions is a constant.

cs.DS↗

arXiv2016

Optimal Composition Ordering Problems for Piecewise Linear Functions

In this paper, we introduce maximum composition ordering problems. The input is $n$ real functions $f_1,\dots,f_n:\mathbb{R}\to\mathbb{R}$ 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 $σ:[n]\to[n]$ which maximizes $f_{σ(n)}\circ f_{σ(n-1)}\circ\dots\circ f_{σ(1)}(c)$, where $[n]=\{1,\dots,n\}$. The maximum partial composition ordering problem is to compute a permutation $σ:[n]\to[n]$ and a nonnegative integer $k~(0\le k\le n)$ which maximize $f_{σ(k)}\circ f_{σ(k-1)}\circ\dots\circ f_{σ(1)}(c)$. We propose $O(n\log n)$ 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 partial composition ordering problem can be solved in polynomial time if $f_i$ is of form $\max\{a_ix+b_i,c_i\}$ for some constants $a_i\,(\ge 0)$, $b_i$ and $c_i$. 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.

cs.DS↗

arXiv2015

A Nested Family of $k$-total Effective Rewards for Positional Games

We consider Gillette's two-person zero-sum stochastic games with perfect information. For each $k \in \ZZ_+$ we introduce an effective reward function, called $k$-total. For $k = 0$ and $1$ this function is known as {\it mean payoff} and {\it total reward}, respectively. We restrict our attention to the deterministic case. For all $k$, we prove the existence of a saddle point which can be realized by uniformly optimal pure stationary strategies. We also demonstrate that $k$-total reward games can be embedded into $(k+1)$-total reward games.

cs.DM↗

arXiv2015

A Potential Reduction Algorithm for Two-person Zero-sum Mean Payoff Stochastic Games

We suggest a new algorithm for two-person zero-sum undiscounted stochastic games focusing on stationary strategies. Given a positive real $ε$, let us call a stochastic game $ε$-ergodic, if its values from any two initial positions differ by at most $ε$. The proposed new algorithm outputs for every $ε>0$ in finite time either a pair of stationary strategies for the two players guaranteeing that the values from any initial positions are within an $ε$-range, or identifies two initial positions $u$ and $v$ and corresponding stationary strategies for the players proving that the game values starting from $u$ and $v$ are at least $ε/24$ apart. In particular, the above result shows that if a stochastic game is $ε$-ergodic, then there are stationary strategies for the players proving $24ε$-ergodicity. This result strengthens and provides a constructive version of an existential result by Vrieze (1980) claiming that if a stochastic game is $0$-ergodic, then there are $ε$-optimal stationary strategies for every $ε> 0$. The suggested algorithm is based on a potential transformation technique that changes the range of local values at all positions without changing the normal form of the game.

cs.GT↗

arXiv2014

Posimodular Function Optimization

Given a posimodular function $f: 2^V \to \mathbb{R}$ on a finite set $V$, we consider the problem of finding a nonempty subset $X$ of $V$ that minimizes $f(X)$. Posimodular functions often arise in combinatorial optimization such as undirected cut functions. In this paper, we show that any algorithm for the problem requires $Ω(2^{\frac{n}{7.54}})$ oracle calls to $f$, where $n=|V|$. It contrasts to the fact that the submodular function minimization, which is another generalization of cut functions, is polynomially solvable. When the range of a given posimodular function is restricted to be $D=\{0,1,...,d\}$ for some nonnegative integer $d$, we show that $Ω(2^{\frac{d}{15.08}})$ oracle calls are necessary, while we propose an $O(n^dT_f+n^{2d+1})$-time algorithm for the problem. Here, $T_f$ denotes the time needed to evaluate the function value $f(X)$ for a given $X \subseteq V$. We also consider the problem of maximizing a given posimodular function. We show that $Ω(2^{n-1})$ oracle calls are necessary for solving the problem, and that the problem has time complexity $Θ(n^{d-1}T_f) $ when $D=\{0,1,..., d\}$ is the range of $f$ for some constant $d$.

cs.DS↗

arXiv2014

On Randomized Fictitious Play for Approximating Saddle Points Over Convex Sets

Given two bounded convex sets $X\subseteq\RR^m$ and $Y\subseteq\RR^n,$ specified by membership oracles, and a continuous convex-concave function $F:X\times Y\to\RR$, we consider the problem of computing an $\eps$-approximate saddle point, that is, a pair $(x^*,y^*)\in X\times Y$ such that $\sup_{y\in Y} F(x^*,y)\le \inf_{x\in X}F(x,y^*)+\eps.$ Grigoriadis and Khachiyan (1995) gave a simple randomized variant of fictitious play for computing an $\eps$-approximate saddle point for matrix games, that is, when $F$ is bilinear and the sets $X$ and $Y$ are simplices. In this paper, we extend their method to the general case. In particular, we show that, for functions of constant "width", an $\eps$-approximate saddle point can be computed using $O^*(\frac{(n+m)}{\eps^2}\ln R)$ random samples from log-concave distributions over the convex sets $X$ and $Y$. It is assumed that $X$ and $Y$ have inscribed balls of radius $1/R$ and circumscribing balls of radius $R$. As a consequence, we obtain a simple randomized polynomial-time algorithm that computes such an approximation faster than known methods for problems with bounded width and when $\eps \in (0,1)$ is a fixed, but arbitrarily small constant. Our main tool for achieving this result is the combination of the randomized fictitious play with the recently developed results on sampling from convex sets.

cs.GT↗

arXiv2014

Parameterized Edge Hamiltonicity

We study the parameterized complexity of the classical Edge Hamiltonian Path problem and give several fixed-parameter tractability results. First, we settle an open question of Demaine et al. by showing that Edge Hamiltonian Path is FPT parameterized by vertex cover, and that it also admits a cubic kernel. We then show fixed-parameter tractability even for a generalization of the problem to arbitrary hypergraphs, parameterized by the size of a (supplied) hitting set. We also consider the problem parameterized by treewidth or clique-width. Surprisingly, we show that the problem is FPT for both of these standard parameters, in contrast to its vertex version, which is W-hard for clique-width. Our technique, which may be of independent interest, relies on a structural characterization of clique-width in terms of treewidth and complete bipartite subgraphs due to Gurski and Wanke.

cs.DS↗

arXiv2011

Derandomizing HSSW Algorithm for 3-SAT

We present a (full) derandomization of HSSW algorithm for 3-SAT, proposed by Hofmeister, Schöning, Schuler, and Watanabe in [STACS'02]. Thereby, we obtain an O(1.3303^n)-time deterministic algorithm for 3-SAT, which is currently fastest.

cs.CC↗

arXiv2009

Deductive Inference for the Interiors and Exteriors of Horn Theories

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

cs.AI↗

arXiv2002

New Results on Monotone Dualization and Generating Hypergraph Transversals

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 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, this is feasible in polynomial time with O(chi(n) * log n) suitably guessed bits, where chi(n) is given by χ(n)^chi(n) = n; note that chi(n) = o(log n). This result sheds new light on the complexity of this important problem.

cs.DS↗