Amitava Bhattacharya

On approximate horn formula minimization

The minimization problem for Horn formulas is to find a Horn formula equivalent to a given Horn formula, using a minimum number of clauses. A 2log1-e(n)-inapproximability result is proven, which is the first inapproximability result for this problem. We also consider several other versions of Horn minimization. The more general version which allows for the introduction of new variables is known to be too difficult as its equivalence problem is co-NP-complete. Therefore, we propose a variant called Steiner-minimization, which allows for the introduction of new variables in a restricted manner. Steiner-minimization of Horn formulas is shown to be MAX-SNP-hard. In the positive direction, a o(n), namely, O(n log log n/(log n)1/4)-approximation algorithm is given for the Steiner-minimization of definite Horn formulas. The algorithm is based on a new result in algorithmic extremal graph theory, on partitioning bipartite graphs into complete bipartite graphs, which may be of independent interest. Inapproximability results and approximation algorithms are also given for restricted versions of Horn minimization, where only clauses present in the original formula may be used.

Online Algorithms with Discrete Visibility - Exploring Unknown Polygonal Environments

The context of this work is the exploration of unknown polygonal environments with obstacles. Both the outer boundary and the boundaries of obstacles are piecewise linear. The boundaries can be nonconvex. The exploration problem can be motivated by the following application. Imagine that a robot has to explore the interior of a collapsed building, which has crumbled due to an earthquake, to search for human survivors. It is clearly impossible to have a knowledge of the buildings interior geometry prior to the exploration. Thus, the robot must be able to see, with its onboard vision sensors, all points in the buildings interior while following its exploration path. In this way, no potential survivors will be missed by the exploring robot. The exploratory path must clearly reflect the topology of the free space, and, therefore, such exploratory paths can be used to guide future robot excursions (such as would arise in our example from a rescue operation).

An integrality theorem of root systems

Let @F be an irreducible root system and @D be a base for @F; it is well known that any root in @F is an integral combination of the roots in @D. In comparison to this fact, we establish the following result: Any indecomposable subset T of @F is contained in the Z-span of an indecomposable linearly independent subset of T.

Alternating reachability and integer sum of closed alternating trails: the 3rd annual uri N. peled memorial lecture

We consider a graph with colored edges and study the following two problems. (i) Suppose first that the number of colors is two, say red and blue. A nonnegative real vector on the edges is said to be balanced if the red sum equals the blue sum at every vertex. A balanced subgraph is a subgraph whose characteristic vector is balanced (i.e., red degree equals blue degree at every vertex). By a sum (respectively, fractional sum) of cycles we mean a nonnegative integral (respectively, nonnegative rational) combination of characteristic vectors of cycles. Similarly, we define sum and fractional sum of balanced subgraphs. We show that a balanced sum of cycles is a fractional sum of balanced subgraphs. (ii) Next we consider the problem of finding a necessary and sufficient condition for the existence of a balanced subgraph containing a given edge. This problem is easily reduced to the alternating reachability problem, defined as follows. Given an edge colored graph (here we allow ≥2 colors) a trail (vertices may repeat but not edges) is called alternating when successive edges have different colors. Given a set of vertices called terminals, the alternating reachability problem is to find an alternating trail connecting distinct terminals, if one exists. By reduction to the classical case of searching for an augmenting path with respect to a matching we show that either there exists an alternating trail connecting distinct terminals or there exists an obstacle, called a Tutte set, to the existence of such trails. We also give a Gallai-Edmonds decomposition of the set of nonterminals. This work started when Uri Peled and Murali Srinivasan met in Caesarea Edmond Benjamin de Rothschild Foundation Institute for Interdisciplinary Applications of Computer Science at the University of Haifa, Israel during May---June 2003. This led to many interesting questions and some of them are still open. In this talk we would like to discuss some of them.

Problems on matchings and independent sets of a graph

Let G be a finite simple graph. For X subset of V(G), the difference of X, d(X) := vertical bar X vertical bar vertical bar N(X)vertical bar where N(X) is the neighborhood of X and max {d(X) : X subset of V(G)} is called the critical difference of G. X is called a critical set if d(X) equals the critical difference and ker(G) is the intersection of all critical sets. diadem(G) is the union of all critical independent sets. An independent set S is an inclusion minimal set with d(S) > 0 if no proper subset of S has positive difference. A graph G is called a Konig-Egervdry graph if the sum of its independence number alpha(G) and matching number mu(G) equals vertical bar V(G)vertical bar. In this paper, we prove a conjecture which states that for any graph the number of inclusion minimal independent set S with d(S) > 0 is at least the critical difference of the graph. We also give a new short proof of the inequality vertical bar ker(G)vertical bar + vertical bar diadem(G)vertical bar <= 2 alpha(G). A characterization of unicyclic non-Konig-Egervary graphs is also presented and a conjecture which states that for such a graph G, the critical difference equals alpha(G) mu(G), is proved. We also make an observation about ker(G) using Edmonds-Gallai Structure Theorem as a concluding remark

Some approaches for solving the general (t,k) -design existence problem and other related problems

In this short survey, some approaches that can be used to solve the generalized (t,k)-design problem are considered. Special cases of the generalized (t,k)-design problem include well-known conjectures for t-designs, degree sequences of graphs and hypergraphs, and partial Steiner systems. Also described are some related problems such as the characterization of f-vectors of pure simplicial complexes, which are well known but little understood. Some suggestions how enumerative and polyhedral techniques may help are also described.

An Integrality Theorem of Root Systems

Abstract Let R and Z denote the set of reals and the set of integers respectively and let E be a finite dimensional vector space over R with usual innerproduct (∗,∗). Let S and T be two subsets of E. If S is contained in the Z-span of T- or equivalently, if every vector of S is an integral combination of vectors in T- then we say that S is generated by T. A subset S of E is called decomposable if there is a proper subset T of S such that for all x ∈ T and for all y ∈ S \ T , ( x , y ) = 0; otherwise it is called indecomposable. Let Φ be a root system and Δ be a base for Φ it is well known that any root in Φ is an integral combination of the roots in Δ. A natural question to ask in this connection is the following: If S is a linearly dependent subset of Φ, can there be a linearly independent subset of S which generates S ? We answer this question affirmatively.