# Current work

Here is a brief summary of my current research projects. Please feel free to reach out if you are interested in details.

### 1) On collections of BDDs and “order-associated” diagrams.

**Summary:** Designing a heuristic to find a “good” shared order
of layers for a pair of Binary Decision Diagrams (BDDs) and applying
it to build an exact solution to a hard optimization problem: a variant of facility location.

## [ More details ]

The project focuses on Binary Decision Diagrams and their applications in optimization. This data structure was developed to efficiently manipulate Boolean functions, and sometimes it seems handy to represent a “logical” (binary) constraint as a diagram. So, some optimization problems can be naturally reformulated as linked network flows through a collection of diagrams (and we are looking for a “Consistent path” through several diagrams). Informally speaking, the latter can be solved relatively easily if the diagrams have their layers in the same order. Good order of layers may make a diagram small, but in a bad case the size of the diagram grows exponentially. Finding a best order of layers is NP-hard, even for a single diagram. The project is structured into two large, more or less independent parts.

First, we build a heuristic to “align” the diagrams. The central idea is simple: when we swap two adjacent layers in a diagram, their size change. But instead of working with the original diagrams, which can be computationally expensive, we can just keep track of the upper bounds on the layer sizes. This gives rise to a smaller auxiliary problem that sometimes does allow to find good shared order of layers in reasonable time.

In the second part of the project we actually use this idea to attack a hard combinatorial problem, a variant of the facility location. We demonstrate how to parameterize the problem using a collection of BDDs and compare several ways to obtain an (exact) optimal solution, revealing that Consistent Path representation along with the proposed “alignment” heuristic might allow to obtain some performance benefits (especially when we’d need to re-solve the problem with different numerical data) and sensitivity information.

ποΈ **Working paper:** Bochkarev, Alexey, and J. Cole Smith. βOn Aligning Non-Order-Associated Binary Decision Diagrams.β Under revision in *INFORMS Journal on Computing*. (preprint)

π¬ **Presentation:** at INFORMS Annual Meeting 2020. (slides)

πΎ **Code and data:** work in progress; code docs, repository

π» **Software stack:** Python, R/ggplot, graphviz (dot), Gurobi solver, PBS, GNU parallel, bash, make, sphinx.

### 2) An RL-powered heuristic for Dynamic Shortest-Path Interdiction

**Summary:** Designing algorithms to “play the game” of Dynamic Shortest-Path
Interdiction (which is
NP-hard). We are trying to explore the problem state space using the techniques
involving random simulations and reinforcement learning.

## [ More details ]

We are considering a dynamic game between two agents, “Evader” and “Interdictor”, over a directed weighted graph. The purpose of the Evader is to traverse a graph between “source” and “terminal” nodes at the minimum possible cost, given the other player’s actions. The “Interdictor” is seeking to maximize the Evader’s cost by “attacking” certain number of edges of the graph (which results in the arc cost increasing by a pre-defined amount). The players take turns, where the Evader’s turn implies traversal of an arc, and the Interdictor’s turn is either an attack or a pass. This variant of the game is known to be NP-hard, and an exact algorithm boils down to enumerating all the relevant states in a dynamic programming fashion.

While existing research discusses bounds for
the optimal game cost, the literature on heuristics
(algorithms that would actually play that game, or propose a
*policy* for the players) has been generally lacking. We
look to fill in this gap by leveraging some ideas from the
realm of simulations and game playing research.

ποΈ **Working paper:** in preparation, with Dr. J. Cole Smith.

π¬ **Presentation:** INOC-2022 in Aachen. See also Book of Abstracts, Session 4B (Interdiction), p. 129.

π» **Software stack:** Julia, graphviz (dot), R/ggplot, PBS, make (so far).