[Paper Review] Deterministic Parameterized Algorithms for Matching and Packing Problems

Meirav Zehavi|arXiv (Cornell University)|Nov 3, 2013

Optimization and Packing Problems1 references11 citations

TL;DR

This paper presents deterministic parameterized algorithms for weighted $q$-dimensional $p$-matching ((q,p)-WDM) and weighted $q$-set $p$-packing ((q,p)-WSP), achieving improved time complexities: $O^{*}(2.85043^{(q-1)p})$ for (q,p)-WDM, $O^{*}(8.04143^p)$ for unweighted (3,p)-WDM, and $O^{*}((0.56201 \cdot 2.85043^q)^p)$ for (q,p)-WSP. It also introduces a kernel of size $O(e^q q(p-1)^q)$, improving over the prior $O(q! q(p-1)^q)$ bound.

ABSTRACT

We present three deterministic parameterized algorithms for well-studied packing and matching problems, namely, Weighted q-Dimensional p-Matching ((q,p)-WDM) and Weighted q-Set p-Packing ((q,p)-WSP). More specifically, we present an O*(2.85043^{(q-1)p}) time deterministic algorithm for (q,p)-WDM, an O*(8.04143^p) time deterministic algorithm for the unweighted version of (3,p)-WDM, and an O*((0.56201\cdot 2.85043^q)^p) time deterministic algorithm for (q,p)-WSP. Our algorithms significantly improve the previously best known O* running times in solving (q,p)-WDM and (q,p)-WSP, and the previously best known deterministic O* running times in solving the unweighted versions of these problems. Moreover, we present kernels of size O(e^qq(p-1)^q) for (q,p)-WDM and (q,p)-WSP, improving the previously best known kernels of size O(q!q(p-1)^q) for these problems.

Motivation & Objective

• To develop faster deterministic parameterized algorithms for the weighted $q$-dimensional $p$-matching ((q,p)-WDM) and weighted $q$-set $p$-packing ((q,p)-WSP) problems.
• To improve upon the previously best-known $O^{*}$ time bounds for these problems, particularly in the deterministic setting.
• To present tighter kernelization bounds for both (q,p)-WDM and (q,p)-WSP, reducing the size from $O(q! q(p-1)^q)$ to $O(e^q q(p-1)^q)$.
• To achieve significant improvements in running time for the unweighted (3,p)-WDM case, reducing the exponent base from 8.125 to 8.04143.

Proposed method

• Designing a deterministic algorithm for (q,p)-WDM based on a novel recursive decomposition and weight-based pruning, achieving a running time of $O^{*}(2.85043^{(q-1)p})$.
• Optimizing the algorithm for the unweighted (3,p)-WDM case by exploiting structural constraints, yielding a running time of $O^{*}(8.04143^p)$.
• Extending the approach to (q,p)-WSP via a transformation that preserves optimal weight sums, resulting in an $O^{*}((0.56201 \cdot 2.85043^q)^p)$ time algorithm.
• Introducing a kernelization algorithm (WDM-Ker) that reduces the input size to $O(e^q q(p-1)^q)$ by selecting representative tuples based on weight and disjointness constraints.
• Adapting WDM-Ker to (q,p)-WSP via a minor modification (WSP-Ker), maintaining the same kernel size and correctness guarantees.
• Using a recursive lemma to prove that any optimal solution in the original instance corresponds to a solution of at least equal weight in the kernelized instance.

Experimental results

Research questions

• RQ1 Can the $O^{*}$ running time for (q,p)-WDM be improved beyond the previously best-known $O^{*}(4^{(q-1)p+o(qp)})$ in the deterministic setting?
• RQ2 Can tighter kernel sizes be achieved for (q,p)-WDM and (q,p)-WSP, reducing the dependency on $q!$ to $e^q$?
• RQ3 Is it possible to further optimize the running time for the unweighted (3,p)-WDM case beyond the known deterministic bounds?
• RQ4 Can the same algorithmic framework be extended to (q,p)-WSP with a significantly improved $O^{*}$ time complexity?

Key findings

• The paper presents a deterministic algorithm for (q,p)-WDM with a running time of $O^{*}(2.85043^{(q-1)p})$, improving upon the previous best deterministic bound of $O^{*}(4^{(q-1)p+o(qp)})$.
• For the unweighted (3,p)-WDM problem, the algorithm runs in $O^{*}(8.04143^p)$ time, a significant improvement over the prior deterministic bound of $O^{*}(8.125^p)$.
• An $O^{*}((0.56201 \cdot 2.85043^q)^p)$ time deterministic algorithm is developed for (q,p)-WSP, improving over previous deterministic bounds.
• The paper introduces a kernel of size $O(e^q q(p-1)^q)$ for both (q,p)-WDM and (q,p)-WSP, improving the prior $O(q! q(p-1)^q)$ bound.
• The kernelization algorithm WDM-Ker runs in time $O([ ext{min}(|{rak S}|, e^q (p-1)^q)]^{ ilde{w}-1} |{rak S}| q^2 log|igcup U|)$, ensuring correctness and size bounds.
• Theoretical analysis confirms that any optimal solution in the original instance has a corresponding solution of equal or greater weight in the kernelized instance, preserving optimality.