Question: Which started first, the Spring 1945 offensive in Italy or the Battle of Luzon?

Context paragraphs:
Battle of Baguio (1945): The Battle of Baguio occurred between 21 February and 26 April 1945, and was part of the greater Luzon campaign during the Allied liberation of the Philippines at the end of World War II.  During the battle, American and Philippine forces recaptured the city of Baguio on the island of Luzon from a Japanese occupation force.  One of the last tank engagements of the Philippine campaign took place during the battle.  Baguio later became the scene of the final surrender of Japanese forces in the Philippines in September 1945.

Rhodesian Armoured Corps: The Rhodesian Armoured Corps—the "Black Devils"—was the only standing armoured battalion of the Rhodesian Security Forces.  During World War II, it took part in the Allied Spring 1945 offensive and the Battle of Monte Cassino as part of South Africa's 6th Armoured Division.  The unit was among the first to enter a liberated Florence in July 1944.  Prior to 1963, its crews were trained in the United Kingdom or Aden Colony and were known as the "Selous Scouts" under the Federation of Rhodesia and Nyasaland.  After Rhodesia's Unilateral Declaration of Independence, maintaining the armoured vehicle fleet became a responsibility of the Rhodesian Light Infantry until Major Bruce Rooken-Smith reactivated the former "Rhodesian Armoured Car Regiment" in 1972.  During the Rhodesian Bush War, the regiment fought in several major campaigns and battles, particularly Operation Miracle in September 1979.  It was superseded by the new "Zimbabwe Armoured Corps" between 1980 and 1981.

Operation Grapeshot order of battle: Operation Grapeshot order of battle is a listing of the significant formations that were involved in the Spring 1945 offensive in the Apennine Mountains and the Po valley in northern Italy, April 1945 – May 1945.

Spring 1945 offensive in Italy: The Spring 1945 offensive in Italy, codenamed Operation Grapeshot, was the final Allied attack during the Italian Campaign in the final stages of the Second World War.  The attack into the Lombardy Plain by the 15th Allied Army Group started on 6 April 1945, ending on 2 May with the formal surrender of German forces in Italy.

Operation Roast: Operation Roast was a military operation undertaken by British Commandos, at Comacchio lagoon in north-east Italy, during the Spring 1945 offensive in Italy, part of the Italian Campaign, during the final stages of Second World War.

LXXVI Panzer Corps: The LXXVI Panzer Corps ("LXXVI Panzerkorps", 76th Armoured Corps) was a panzer corps of Nazi Germany during World War II.  The headquarters were formed in France under Army Group D on 29 June 1943 as LXXVI Army Corps but renamed a month later.  In August it shipped to Italy to become part of 10th Army.  It spent the rest of the war in Italy fighting in the Italian Campaign mainly under 10th Army but with short periods from February 1944 (Battle of Anzio) and January 1945 (Spring 1945 offensive in Italy) under 14th Army.  The Corps was commanded for most of its active fighting by General Traugott Herr.

Battle of Luzon: The Battle of Luzon, fought 9 January – 15 August 1945, was a land battle of the Pacific Theater of Operations of World War II by the Allied forces of the U.S., its colony the Philippines, and allies against forces of the Empire of Japan.  The battle resulted in a U.S. and Filipino victory.  The Allies had taken control of all strategically and economically important locations of Luzon by March 1945, although pockets of Japanese resistance held out in the mountains until the unconditional surrender of Japan.  While not the highest in U.S. casualties, it is the highest net casualty battle U.S. forces fought in World War II, with 192,000 to 205,000 Japanese combatants dead (mostly from starvation and disease), 10,000 American combatants killed, and between 120,000 to 140,000 Filipino civilians and combatants killed.

Battle of Bologna: The Battle of Bologna was fought in Bologna, Italy from 9–21 April 1945 during the Second World War, as part of the Spring 1945 offensive in Italy.  The Allied forces were victorious, with the Polish II Corps and supporting Allied units capturing the city on 21 April.

Battle of the Argenta Gap: The Battle of the Argenta Gap was an engagement which formed part of the Allied spring 1945 offensive during the Italian Campaign in the final stages of the Second World War.  It took place in northern Italy from 12–19 April 1945 between troops of British V Corps commanded by Lieutenant-General Charles Keightley and German units of LXXVI Panzer Corps commanded by Lieutenant General ("General der Panzertruppe") Gerhard von Schwerin.

191st Army Band: The 191st Army Band, the "Band of the Wild West", is a military band of the U.S. Army Reserve posted at Camp Parks, California.  Activated in 1943 as the 91st Infantry Division Band, the unit deployed to Europe in World War II where it fought in the Spring 1945 offensive in Italy.  It received its current designation as the 191st Army Band in 2008.  The 191st Army Band is a subordinate unit of the U.S. Army's 63rd Regional Support Command.

Supporting paragraphs: Spring 1945 offensive in Italy Battle of Luzon
Answer: The Battle of Luzon