Revisiting the Equivalence Problem for Finite Multitape Automata

The decidability of determining equivalence of deterministic multitape automata (or transducers) was a longstanding open problem until it was resolved by Harju and Karhum\"{a}ki in the early 1990s. Their proof of decidability yields a co_NP upper bound, but apparently not much more is known about the complexity of the problem. In this paper we give an alternative proof of decidability, which follows the basic strategy of Harju and Karhumaki but replaces their use of group theory with results on matrix algebras. From our proof we obtain a simple randomised algorithm for deciding language equivalence of deterministic multitape automata and, more generally, multiplicity equivalence of nondeterministic multitape automata. The algorithm involves only matrix exponentiation and runs in polynomial time for each fixed number of tapes. If the two input automata are inequivalent then the algorithm outputs a word on which they differ

Tight bounds and conjectures for the isolation lemma

Given a hypergraph $H$ and a weight function $w: V \rightarrow \{1, \dots, M\}$ on its vertices, we say that $w$ is isolating if there is exactly one edge of minimum weight $w(e) = \sum_{i \in e} w(i)$. The Isolation Lemma is a combinatorial principle introduced in Mulmuley et. al (1987) which gives a lower bound on the number of isolating weight functions. Mulmuley used this as the basis of a parallel algorithm for finding perfect graph matchings. It has a number of other applications to parallel algorithms and to reductions of general search problems to unique search problems (in which there are one or zero solutions). The original bound given by Mulmuley et al. was recently improved by Ta-Shma (2015). In this paper, we show improved lower bounds on the number of isolating weight functions, and we conjecture that the extremal case is when $H$ consists of $n$ singleton edges. When $M \gg n$ our improved bound matches this extremal case asymptotically. We are able to show that this conjecture holds in a number of special cases: when $H$ is a linear hypergraph or is 1-degenerate, or when $M = 2$. We also show that it holds asymptotically when $M \gg n \gg 1$

Space Complexity of Perfect Matching in Bounded Genus Bipartite Graphs

We investigate the space complexity of certain perfect matching problems over bipartite graphs embedded on surfaces of constant genus (orientable or non-orientable). We show that the problems of deciding whether such graphs have (1) a perfect matching or not and (2) a unique perfect matching or not, are in the logspace complexity class \SPL. Since \SPL\ is contained in the logspace counting classes \oplus\L (in fact in \modk\ for all $k\geq 2$), \CeqL, and \PL, our upper bound places the above-mentioned matching problems in these counting classes as well. We also show that the search version, computing a perfect matching, for this class of graphs is in \FL^{\SPL}. Our results extend the same upper bounds for these problems over bipartite planar graphs known earlier. As our main technical result, we design a logspace computable and polynomially bounded weight function which isolates a minimum weight perfect matching in bipartite graphs embedded on surfaces of constant genus. We use results from algebraic topology for proving the correctness of the weight function.Comment: 23 pages, 13 figure

On the Lattice Isomorphism Problem

We study the Lattice Isomorphism Problem (LIP), in which given two lattices L_1 and L_2 the goal is to decide whether there exists an orthogonal linear transformation mapping L_1 to L_2. Our main result is an algorithm for this problem running in time n^{O(n)} times a polynomial in the input size, where n is the rank of the input lattices. A crucial component is a new generalized isolation lemma, which can isolate n linearly independent vectors in a given subset of Z^n and might be useful elsewhere. We also prove that LIP lies in the complexity class SZK.Comment: 23 pages, SODA 201

Revisiting the Equivalence Problem for Finite Multitape Automata

Abstract. The decidability of determining equivalence of deterministic multitape automata was a longstanding open problem until it was resolved by Harju and Karhumäki in the early 1990s. Their proof of decidability yields a co-NP upper bound, but apparently not much more is known about the complexity of the problem. In this paper we give an alternative proof of decidability which follows the basic strategy of Harju and Karhumäki, but replaces their use of group theory with results on matrix algebras. From our proof we obtain a simple randomised algorithm for deciding equivalence of deterministic multitape automata, as well as automata with transition weights in the field of rational numbers. The algorithm involves only matrix exponentiation and runs in polynomial time for each fixed number of tapes. If the two input automata are inequivalent then the algorithm outputs a word on which they differ

Dynamic Programming for Graphs on Surfaces

We provide a framework for the design and analysis of dynamic programming algorithms for surface-embedded graphs on n vertices and branchwidth at most k. Our technique applies to general families of problems where standard dynamic programming runs in 2^{O(k log k)} n steps. Our approach combines tools from topological graph theory and analytic combinatorics. In particular, we introduce a new type of branch decomposition called "surface cut decomposition", generalizing sphere cut decompositions of planar graphs introduced by Seymour and Thomas, which has nice combinatorial properties. Namely, the number of partial solutions that can be arranged on a surface cut decomposition can be upper-bounded by the number of non-crossing partitions on surfaces with boundary. It follows that partial solutions can be represented by a single-exponential (in the branchwidth k) number of configurations. This proves that, when applied on surface cut decompositions, dynamic programming runs in 2^{O(k)} n steps. That way, we considerably extend the class of problems that can be solved in running times with a single-exponential dependence on branchwidth and unify/improve most previous results in this direction.Comment: 28 pages, 3 figure

Space Complexity of the Directed Reachability Problem over Surface-Embedded Graphs

The graph reachability problem, the computational task of deciding whether there is a path between two given nodes in a graph is the canonical problem for studying space bounded computations. Three central open questions regarding the space complexity of the reachabil-ity problem over directed graphs are: (1) improving Savitch’s O(log2 n) space bound, (2) designing a polynomial-time algorithm with O(n1−) space bound, and (3) designing an unambiguous non-deterministic log-space (UL) algorithm. These are well-known open questions in complex-ity theory, and solving any one of them will be a major breakthrough. We will discuss some of the recent progress reported on these questions for certain subclasses of surface-embedded directed graphs

The Matching Problem in General Graphs is in Quasi-NC

We show that the perfect matching problem in general graphs is in Quasi-NC. That is, we give a deterministic parallel algorithm which runs in $O(\log^3 n)$ time on $n^{O(\log^2 n)}$ processors. The result is obtained by a derandomization of the Isolation Lemma for perfect matchings, which was introduced in the classic paper by Mulmuley, Vazirani and Vazirani [1987] to obtain a Randomized NC algorithm. Our proof extends the framework of Fenner, Gurjar and Thierauf [2016], who proved the analogous result in the special case of bipartite graphs. Compared to that setting, several new ingredients are needed due to the significantly more complex structure of perfect matchings in general graphs. In particular, our proof heavily relies on the laminar structure of the faces of the perfect matching polytope.Comment: Accepted to FOCS 2017 (58th Annual IEEE Symposium on Foundations of Computer Science

On the Lattice Isomorphism Problem

Abstract We study the Lattice Isomorphism Problem (LIP), in which given two lattices L 1 and L 2 the goal is to decide whether there exists an orthogonal linear transformation mapping L 1 to L 2 . Our main result is an algorithm for this problem running in time n O(n) times a polynomial in the input size, where n is the rank of the input lattices. A crucial component is a new generalized isolation lemma, which can isolate n linearly independent vectors in a given subset of Z n and might be useful elsewhere. We also prove that LIP lies in the complexity class SZK