Kleitos Papadopoulos

Kleitos Papadopoulos

Broadcast domination assigns a nonnegative integer power to every vertex of a graph so that every vertex is within the assigned power of some broadcasting vertex, and the objective is to minimize the sum of the powers. Heggernes and Lokshtanov proved that the problem is polynomial-time solvable on arbitrary connected unweighted graphs by showing that some optimal efficient broadcast has a domination graph that is a path or a cycle, and by reducing the general case to an $O(n^6)$-time algorithm. This paper gives an efficient algorithm of the path-case. Instead of building one auxiliary acyclic graph for every possible left endpoint vertex, we build a single directed acyclic graph whose states are oriented broadcast balls together with their two possible residual sides. The resulting path-case algorithm runs in $O(n^3)$ time and $O(n^3)$ space on an $n$-vertex graph. Combining this routine with the same peel-one-ball reduction of Heggernes and Lokshtanov yields an exact $O(n^5)$-time algorithm for optimal broadcast domination on arbitrary connected unweighted graphs. This resolves the quintic-time conjecture for general graphs attributed to Heggernes and Sæther and recorded in subsequent surveys of broadcast domination.

Kleitos Papadopoulos

The generalized egg dropping problem is a classic challenge in sequential decision-making. Standard dynamic programming evaluates the minimax minimum number of tests in $\mathcal{O}(K \cdot N^2)$ time. A known approach formulates the testable thresholds as a partial sum of binomial coefficients and applies binary search to reduce the time complexity to $\mathcal{O}(K \log N)$. In this paper, we demonstrate that binary search over the complete sequential test domain is suboptimal. By restricting a binary search over multiples of $K$, we isolate a dynamic structural envelope that guarantees convergence. We prove that this boundary balances the search depth against the combinatorial evaluation cost, cancelling the $K$ variable to strictly bound the search phase to $\mathcal{O}(\log N)$. Combined with an incremental traversal, our algorithm eliminates the standard bottlenecks. Furthermore, we formulate an explicit $\mathcal{O}(1)$ space policy to dynamically reconstruct the optimal decision tree.