The Graph Isomorphism Problem and approximate categories

It is unknown whether two graphs can be tested for isomorphism in polynomial time. A classical approach to the Graph Isomorphism Problem is the d-dimensional Weisfeiler-Lehman algorithm. The d-dimensional WL-algorithm can distinguish many pairs of graphs, but the pairs of non-isomorphic graphs constructed by Cai, Furer and Immerman it cannot distinguish. If d is fixed, then the WL-algorithm runs in polynomial time. We will formulate the Graph Isomorphism Problem as an Orbit Problem: Given a representation V of an algebraic group G and two elements v_1,v_2 in V, decide whether v_1 and v_2 lie in the same G-orbit. Then we attack the Orbit Problem by constructing certain approximate categories C_d(V), d=1,2,3,... whose objects include the elements of V. We show that v_1 and v_2 are not in the same orbit by showing that they are not isomorphic in the category C_d(V) for some d. For every d this gives us an algorithm for isomorphism testing. We will show that the WL-algorithms reduce to our algorithms, but that our algorithms cannot be reduced to the WL-algorithms. Unlike the Weisfeiler-Lehman algorithm, our algorithm can distinguish the Cai-Furer-Immerman graphs in polynomial time.Comment: 29 page

A linear time algorithm for the orbit problem over cyclic groups

The orbit problem is at the heart of symmetry reduction methods for model checking concurrent systems. It asks whether two given configurations in a concurrent system (represented as finite strings over some finite alphabet) are in the same orbit with respect to a given finite permutation group (represented by their generators) acting on this set of configurations by permuting indices. It is known that the problem is in general as hard as the graph isomorphism problem, whose precise complexity (whether it is solvable in polynomial-time) is a long-standing open problem. In this paper, we consider the restriction of the orbit problem when the permutation group is cyclic (i.e. generated by a single permutation), an important restriction of the problem. It is known that this subproblem is solvable in polynomial-time. Our main result is a linear-time algorithm for this subproblem.Comment: Accepted in Acta Informatica in Nov 201

A Hybrid Search Algorithm for the Whitehead Minimization Problem

The Whitehead Minimization problem is a problem of finding elements of the minimal length in the automorphic orbit of a given element of a free group. The classical algorithm of Whitehead that solves the problem depends exponentially on the group rank. Moreover, it can be easily shown that exponential blowout occurs when a word of minimal length has been reached and, therefore, is inevitable except for some trivial cases. In this paper we introduce a deterministic Hybrid search algorithm and its stochastic variation for solving the Whitehead minimization problem. Both algorithms use search heuristics that allow one to find a length-reducing automorphism in polynomial time on most inputs and significantly improve the reduction procedure. The stochastic version of the algorithm employs a probabilistic system that decides in polynomial time whether or not a word is minimal. The stochastic algorithm is very robust. It has never happened that a non-minimal element has been claimed to be minimal

Implementation of Distributed Time Exchange Based Cooperative Forwarding

In this paper, we design and implement time exchange (TE) based cooperative forwarding where nodes use transmission time slots as incentives for relaying. We focus on distributed joint time slot exchange and relay selection in the sum goodput maximization of the overall network. We formulate the design objective as a mixed integer nonlinear programming (MINLP) problem and provide a polynomial time distributed solution of the MINLP. We implement the designed algorithm in the software defined radio enabled USRP nodes of the ORBIT indoor wireless testbed. The ORBIT grid is used as a global control plane for exchange of control information between the USRP nodes. Experimental results suggest that TE can significantly increase the sum goodput of the network. We also demonstrate the performance of a goodput optimization algorithm that is proportionally fair.Comment: Accepted in 2012 Military Communications Conferenc

Invariant Theory, Tensors and Computational Complexity

The main problem addressed in this dissertation is the problem of giving strong upper bounds on the degree of generators for invariant rings. In the cases of matrix invariants and matrix semi-invariants, we give polynomial upper bounds. An exciting consequence of these bounds is a polynomial time algorithm for rational identity testing. We use an approach inspired by ideas from Popov and Derksen to reduce the problem to finding invariants that define the null cone. The theory of blow-ups of matrix spaces and non-commutative rank is crucial in finding invariants that define the null cone. We also give a polynomial time algorithm for deciding if the orbit closures of two points intersect for matrix invariants and semi-invariants. In addition, we give some applications for proving lower bounds on the border rank of tensors.PHDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/144049/1/visu_1.pd

Computing Multiplicities of Lie Group Representations

For fixed compact connected Lie groups H \subseteq G, we provide a polynomial time algorithm to compute the multiplicity of a given irreducible representation of H in the restriction of an irreducible representation of G. Our algorithm is based on a finite difference formula which makes the multiplicities amenable to Barvinok's algorithm for counting integral points in polytopes. The Kronecker coefficients of the symmetric group, which can be seen to be a special case of such multiplicities, play an important role in the geometric complexity theory approach to the P vs. NP problem. Whereas their computation is known to be #P-hard for Young diagrams with an arbitrary number of rows, our algorithm computes them in polynomial time if the number of rows is bounded. We complement our work by showing that information on the asymptotic growth rates of multiplicities in the coordinate rings of orbit closures does not directly lead to new complexity-theoretic obstructions beyond what can be obtained from the moment polytopes of the orbit closures. Non-asymptotic information on the multiplicities, such as provided by our algorithm, may therefore be essential in order to find obstructions in geometric complexity theory.Comment: 10 page

Automorphic orbits in free groups

Let $F_n$ be the free group of a finite rank $n$. We study orbits $Orb_{\phi}(u)$, where $u$ is an element of the group $F_n$, under the action of an automorphism $\phi$. If an orbit like that is finite, we determine precisely what its cardinality can be if $u$ runs through the whole group $F_n$, and $\phi$ runs through the whole group $Aut(F_n)$. Another problem that we address here is related to Whitehead's algorithm that determines whether or not a given element of a free group of finite rank is an automorphic image of another given element. It is known that the first part of this algorithm (reducing a given free word to a free word of minimum possible length by elementary Whitehead automorphisms) is fast (of quadratic time with respect to the length of the word). On the other hand, the second part of the algorithm (applied to two words of the same minimum length) was always considered very slow. We give here an improved algorithm for the second part, and we believe this algorithm always terminates in polynomial time with respect to the length of the words. We prove that this is indeed the case if the free group has rank 2.Comment: 10 page

Space debris cataloging of GEO objects by using Meta-Heuristic methods

Currently several thousands of objects are being tracked in the Medium Earth Orbit (MEO) and Geosynchronous Earth Orbit (GEO) regions through optical means. The problem faced in this framework is that of Multiple Target Tracking (MTT). The MTT problem becomes an NP-hard combinatorial optimization problem as soon as its dimension S becomes S ≥ 3. In regions with a high density of objects the MTT problem will have to have this dimension in order to avoid ambiguous solutions. With the advent of improved sensors and a eightened interest in the problem of space debris, it is expected that the number of tracked objects will grow by an order of magnitude in the near future. This research aims to identify an algorithm capable of addressing the problem of space debris cataloging in the MEO and GEO regions, in particular for highly dense regions, without possessing a restrictive computational complexity. In an attempt to find an approximate solution of sufficient quality several Population Based Meta Heuristic (PBMH) algorithms are implemented and tested on simulated optical measurements. In addition to this, a novel way of orbit determination is presented which is based on an existing S = 2 tracklet association method. These first results show promise as one of the tested algorithms (the Elitist Genetic Algorithm (EGA)) consistently displays the desired behavior of finding good approximate solutions before reaching the optimum. Furthermore, the results suggest that the algorithm has a polynomial time complexity when finding approximate solutions. The algorithm is also applied to real observations, where it also performs as desired