On the Hardness of Entropy Minimization and Related Problems

We investigate certain optimization problems for Shannon information measures, namely, minimization of joint and conditional entropies $H(X,Y)$, $H(X|Y)$, $H(Y|X)$, and maximization of mutual information $I(X;Y)$, over convex regions. When restricted to the so-called transportation polytopes (sets of distributions with fixed marginals), very simple proofs of NP-hardness are obtained for these problems because in that case they are all equivalent, and their connection to the well-known \textsc{Subset sum} and \textsc{Partition} problems is revealed. The computational intractability of the more general problems over arbitrary polytopes is then a simple consequence. Further, a simple class of polytopes is shown over which the above problems are not equivalent and their complexity differs sharply, namely, minimization of $H(X,Y)$ and $H(Y|X)$ is trivial, while minimization of $H(X|Y)$ and maximization of $I(X;Y)$ are strongly NP-hard problems. Finally, two new (pseudo)metrics on the space of discrete probability distributions are introduced, based on the so-called variation of information quantity, and NP-hardness of their computation is shown.Comment: IEEE Information Theory Workshop (ITW) 201

A branch, price, and cut approach to solving the maximum weighted independent set problem

The maximum weight-independent set problem (MWISP) is one of the most well-known and well-studied NP-hard problems in the field of combinatorial optimization. In the first part of the dissertation, I explore efficient branch-and-price (B&P) approaches to solve MWISP exactly. B&P is a useful integer-programming tool for solving NP-hard optimization problems. Specifically, I look at vertex- and edge-disjoint decompositions of the underlying graph. MWISPâÂÂs on the resulting subgraphs are less challenging, on average, to solve. I use the B&P framework to solve MWISP on the original graph G using these specially constructed subproblems to generate columns. I demonstrate that vertex-disjoint partitioning scheme gives an effective approach for relatively sparse graphs. I also show that the edge-disjoint approach is less effective than the vertex-disjoint scheme because the associated DWD reformulation of the latter entails a slow rate of convergence. In the second part of the dissertation, I address convergence properties associated with Dantzig-Wolfe Decomposition (DWD). I discuss prevalent methods for improving the rate of convergence of DWD. I also implement specific methods in application to the edge-disjoint B&P scheme and show that these methods improve the rate of convergence. In the third part of the dissertation, I focus on identifying new cut-generation methods within the B&P framework. Such methods have not been explored in the literature. I present two new methodologies for generating generic cutting planes within the B&P framework. These techniques are not limited to MWISP and can be used in general applications of B&P. The first methodology generates cuts by identifying faces (facets) of subproblem polytopes and lifting associated inequalities; the second methodology computes Lift-and-Project (L&P) cuts within B&P. I successfully demonstrate the feasibility of both approaches and present preliminary computational tests of each

A non-partitionable Cohen-Macaulay simplicial complex

A long-standing conjecture of Stanley states that every Cohen-Macaulay simplicial complex is partitionable. We disprove the conjecture by constructing an explicit counterexample. Due to a result of Herzog, Jahan and Yassemi, our construction also disproves the conjecture that the Stanley depth of a monomial ideal is always at least its depth.Comment: Final version. 13 pages, 2 figure

Hidden Vertices in Extensions of Polytopes

Some widely known compact extended formulations have the property that each vertex of the corresponding extension polytope is projected onto a vertex of the target polytope. In this paper, we prove that for heptagons with vertices in general position none of the minimum size extensions has this property. Additionally, for any d >= 2 we construct a family of d-polytopes such that at least 1/9 of all vertices of any of their minimum size extensions is not projected onto vertices.Comment: 9 pages, to appear in: Operations Research Letter